The Analysis of Biological Data [2 ed.] 978-1-319156-71-8

Table of contents :
Cover......Page 1
Halftitle Page......Page 2
Title Page......Page 3
Copyright......Page 4
Dedication......Page 5
Contents in brief......Page 6
Contents......Page 8
Preface......Page 22
A word about the data......Page 24
Acknowledgments......Page 26
About the Authors......Page 28
1 Statistics and samples......Page 29
1.1 What is statistics?......Page 30
EXAMPLE 1.2 Raining cats......Page 32
Populations and samples......Page 33
Properties of good samples......Page 34
Random sampling......Page 35
How to take a random sample......Page 36
The sample of convenience......Page 37
Volunteer bias......Page 38
Data in the real world......Page 39
Categorical and numerical variables......Page 40
Explanatory and response variables......Page 41
1.4 Frequency distributions and probability distributions......Page 43
1.5 Types of studies......Page 45
1.6 Summary......Page 46
PRACTICE PROBLEMS......Page 47
ASSIGNMENT PROBLEMS......Page 51
1 INTERLEAF Biology and the history of statistics......Page 55
2 Displaying data......Page 57
How to draw a bad graph......Page 59
How to draw a good graph......Page 60
Showing categorical data: frequency table and bar graph......Page 63
A bar graph is usually better than a pie chart......Page 65
Showing numerical data: frequency table and histogram......Page 66
Describing the shape of a histogram......Page 69
How to draw a good histogram......Page 70
Other graphs for numerical data......Page 71
Showing association between categorical variables......Page 72
Showing association between numerical variables: scatter plot......Page 75
Showing association between a numerical and a categorical variable......Page 76
Line graph......Page 80
Maps......Page 81
Follow similar principles for display tables......Page 83
2.6 Summary......Page 86
PRACTICE PROBLEMS......Page 87
ASSIGNMENT PROBLEMS......Page 97
3 Describing data......Page 107
EXAMPLE 3.1 Gliding snakes......Page 109
Variance and standard deviation......Page 110
Coefficient of variation......Page 112
Calculating mean and standard deviation from a frequency table......Page 113
Effect of changing measurement scale......Page 114
EXAMPLE 3.2 I’d give my right arm for a female......Page 116
The interquartile range......Page 117
The box plot......Page 118
EXAMPLE 3.3 Disarming fish......Page 120
Mean versus median......Page 121
Standard deviation versus interquartile range......Page 123
Displaying cumulative relative frequencies......Page 124
The proportion is like a sample mean......Page 126
3.6 Summary......Page 128
Table of formulas for descriptive statistics......Page 129
PRACTICE PROBLEMS......Page 130
ASSIGNMENT PROBLEMS......Page 137
EXAMPLE 4.1 The length of human genes......Page 145
Estimating mean gene length with a random sample......Page 146
The sampling distribution of Y¯......Page 147
The standard error of Y¯ from data......Page 150
4.3 Confidence intervals......Page 152
The 2SE rule of thumb......Page 153
4.4 Error bars......Page 155
4.5 Summary......Page 157
Standard error of the mean......Page 158
PRACTICE PROBLEMS......Page 159
ASSIGNMENT PROBLEMS......Page 162
2 INTERLEAF Pseudoreplication......Page 170
5 Probability......Page 172
5.1 The probability of an event......Page 173
5.2 Venn diagrams......Page 175
5.3 Mutually exclusive events......Page 176
Discrete probability distributions......Page 177
Continuous probability distributions......Page 178
The addition rule......Page 180
The probabilities of all possible mutually exclusive outcomes add to one......Page 181
The general addition rule......Page 182
5.6 Independence and the multiplication rule......Page 183
Multiplication rule......Page 184
Independence of more than two events......Page 185
EXAMPLE 5.7 Sex and birth order......Page 187
EXAMPLE 5.8 Is this meat taken?......Page 190
Conditional probability......Page 193
Sampling without replacement......Page 194
Bayes’ theorem......Page 195
5.10 Summary......Page 197
PRACTICE PROBLEMS......Page 198
ASSIGNMENT PROBLEMS......Page 204
6 Hypothesis testing......Page 210
Null hypothesis......Page 212
To reject or not to reject......Page 213
Stating the hypotheses......Page 214
The null distribution......Page 215
Quantifying uncertainty: the P-value......Page 217
Reporting the results......Page 219
Type I and Type II errors......Page 220
The test......Page 222
Interpreting a nonsignificant result......Page 224
6.5 One-sided tests......Page 225
6.6 Hypothesis testing versus confidence intervals......Page 228
6.7 Summary......Page 229
PRACTICE PROBLEMS......Page 231
ASSIGNMENT PROBLEMS......Page 234
3 INTERLEAF Why statistical significance is not the same as biological importance......Page 241
7 Analyzing proportions......Page 243
Formula for the binomial distribution......Page 245
Number of successes in a random sample......Page 246
Sampling distribution of the proportion......Page 248
EXAMPLE 7.2 Sex and the X......Page 250
Approximations for the binomial test......Page 253
Confidence intervals for proportions—the Agresti–Coull method......Page 254
Confidence intervals for proportions—the Wald method......Page 255
7.4 Deriving the binomial distribution......Page 256
7.5 Summary......Page 257
Binomial test......Page 258
PRACTICE PROBLEMS......Page 259
ASSIGNMENT PROBLEMS......Page 264
4 INTERLEAF Correlation does not require causation......Page 268
8 Fitting probability models to frequency data......Page 271
EXAMPLE 8.1 No weekend getaway......Page 273
Observed and expected frequencies......Page 275
The χ2 test statistic......Page 276
The sampling distribution of χ2 under the null hypothesis......Page 277
Calculating the P-value......Page 278
Critical values for the χ2 distribution......Page 279
8.3 Assumptions of the χ2 goodness-of-fit test......Page 282
EXAMPLE 8.4 Gene content of the human X chromosome......Page 283
EXAMPLE 8.5 Designer two-child families?......Page 285
8.6 Random in space or time: the Poisson distribution......Page 288
Testing randomness with the Poisson distribution......Page 289
Comparing the variance to the mean......Page 293
8.7 Summary......Page 294
Poisson distribution......Page 295
PRACTICE PROBLEMS......Page 296
ASSIGNMENT PROBLEMS......Page 300
5 INTERLEAF Making a plan......Page 307
9 Contingency analysis associations between categorical variables......Page 309
9.1 Associating two categorical variables......Page 311
Odds......Page 312
Odds ratio......Page 313
Standard error and confidence interval for odds ratio......Page 314
Odds ratio vs. relative risk......Page 316
EXAMPLE 9.4 The gnarly worm gets the bird......Page 320
Hypotheses......Page 321
The χ2 statistic......Page 322
A shortcut for calculating the expected frequencies......Page 323
Assumptions of the χ2 contingency test......Page 324
Correction for continuity......Page 325
EXAMPLE 9.5 The feeding habits of vampire bats......Page 326
9.6 G-tests......Page 328
9.7 Summary......Page 329
The χ2 contingency test......Page 330
G-test......Page 331
PRACTICE PROBLEMS......Page 332
ASSIGNMENT PROBLEMS......Page 338
Review Problems 1......Page 345
10 The normal distribution......Page 350
10.1 Bell-shaped curves and the normal distribution......Page 352
10.2 The formula for the normal distribution......Page 355
10.3 Properties of the normal distribution......Page 356
Using the standard normal table......Page 358
Using the standard normal to describe any normal distribution......Page 360
10.5 The normal distribution of sample means......Page 363
Calculating probabilities of sample means......Page 364
EXAMPLE 10.6 Young adults and the Spanish flu......Page 366
EXAMPLE 10.7 The only good bug is a dead bug......Page 369
10.8 Summary......Page 372
Normal approximation to the binomial distribution......Page 373
PRACTICE PROBLEMS......Page 374
ASSIGNMENT PROBLEMS......Page 380
6 INTERLEAF Controls in medical studies......Page 386
11 Inference for a normal population......Page 388
Student’s t-distribution......Page 389
Finding critical values of the t-distribution......Page 390
The 95% confidence interval for the mean......Page 393
The 99% confidence interval for the mean......Page 394
EXAMPLE 11.3 Human body temperature......Page 396
The effects of larger sample size: body temperature revisited......Page 399
11.4 Assumptions of the one-sample t-test......Page 401
Confidence limits for the variance......Page 402
Confidence limits for the standard deviation......Page 403
Assumptions......Page 404
11.6 Summary......Page 405
Confidence interval for variance......Page 406
PRACTICE PROBLEMS......Page 407
ASSIGNMENT PROBLEMS......Page 411
12 Comparing two means......Page 417
12.1 Paired sample versus two independent samples......Page 419
Estimating mean difference from paired data......Page 421
Paired t-test......Page 424
Assumptions......Page 426
EXAMPLE 12.3 Spike or be spiked......Page 427
Confidence interval for the difference between two means......Page 428
Two-sample t-test......Page 430
Assumptions......Page 431
A two-sample t-test when standard deviations are unequal......Page 432
EXAMPLE 12.4 So long; thanks to all the fish......Page 433
EXAMPLE 12.5 Mommy’s baby, Daddy’s maybe......Page 436
12.6 Interpreting overlap of confidence intervals......Page 438
Levene’s test for homogeneity of variances......Page 439
12.8 Summary......Page 441
Confidence interval for the difference between two means (two samples)......Page 442
Welch’s approximate t-test......Page 443
Levene’s test......Page 444
PRACTICE PROBLEMS......Page 446
ASSIGNMENT PROBLEMS......Page 456
7 INTERLEAF Which test should I use?......Page 465
13 Handling violations of assumptions......Page 468
Graphical methods......Page 470
Formal test of normality......Page 473
Violations of normality......Page 475
Unequal standard deviations......Page 476
Log transformation......Page 477
Arcsine transformation......Page 480
Confidence intervals with transformations......Page 481
A caveat: Avoid multiple testing with transformations......Page 482
Sign test......Page 483
The Wilcoxon signed-rank test......Page 487
EXAMPLE 13.5 Sexual cannibalism in sagebrush crickets......Page 488
Tied ranks......Page 491
Large samples and the normal approximation......Page 492
13.6 Assumptions of nonparametric tests......Page 493
13.7 Type I and Type II error rates of nonparametric methods......Page 494
13.8 Permutation tests......Page 495
Assumptions of permutation tests......Page 498
13.9 Summary......Page 499
Mann-Whitney U-test......Page 501
PRACTICE PROBLEMS......Page 502
ASSIGNMENT PROBLEMS......Page 514
Review Problems 2......Page 527
14 Designing experiments......Page 535
Confounding variables......Page 537
Experimental artifacts......Page 538
EXAMPLE 14.2 Reducing HIV transmission......Page 539
Design components......Page 540
Simultaneous control group......Page 541
Randomization......Page 542
Blinding......Page 543
Replication......Page 545
Blocking......Page 547
Extreme treatments......Page 550
EXAMPLE 14.5 Lethal combination......Page 552
Match and adjust......Page 554
Plan for precision......Page 556
Plan for power......Page 558
Plan for data loss......Page 559
14.8 Summary......Page 560
Planned sample size for a 95% confidence interval of the difference between two proportions......Page 561
Planned sample size for 2 × 2 contingency test of 80% power at α = 0.05......Page 562
Planned sample size for a two-sample t-test of 80% power at α = 0.05......Page 563
PRACTICE PROBLEMS......Page 564
ASSIGNMENT PROBLEMS......Page 568
8 INTERLEAF Data dredging......Page 572
15 Comparing means of more than two groups......Page 575
EXAMPLE 15.1 The knees who say night......Page 577
ANOVA in a nutshell......Page 578
ANOVA tables......Page 579
Partitioning the sum of squares......Page 580
Calculating the mean squares......Page 581
The variance ratio, F......Page 582
Variation explained: R2......Page 584
ANOVA with two groups......Page 585
Nonparametric alternatives to ANOVA......Page 586
Planned comparison between two means......Page 588
EXAMPLE 15.4 Wood wide web......Page 590
Testing all pairs of means using the Tukey-Kramer method......Page 591
Assumptions......Page 593
15.5 Fixed and random effects......Page 594
EXAMPLE 15.6 Walking-stick limbs......Page 595
ANOVA calculations......Page 596
Variance components......Page 597
Assumptions......Page 598
15.7 Summary......Page 599
Kruskal-Wallis test......Page 601
Tukey-Kramer test of all pairs of means......Page 602
Repeatability and variance components......Page 603
PRACTICE PROBLEMS......Page 604
ASSIGNMENT PROBLEMS......Page 613
9 INTERLEAF Experimental and statistical mistakes......Page 624
16 Correlation between numerical variables......Page 626
The correlation coefficient......Page 628
Standard error......Page 632
Approximate confidence interval......Page 633
EXAMPLE 16.2 What big inbreeding coefficients you have......Page 635
16.3 Assumptions......Page 638
16.4 The correlation coefficient depends on the range......Page 640
EXAMPLE 16.5 The miracles of memory......Page 641
Assumptions of Spearman’s correlation......Page 644
16.6 The effects of measurement error on correlation......Page 645
16.7 Summary......Page 646
Confidence interval (approximate) for a population correlation......Page 647
Spearman’s rank correlation test......Page 648
Correlation corrected for measurement error......Page 649
PRACTICE PROBLEMS......Page 650
ASSIGNMENT PROBLEMS......Page 659
10 INTERLEAF Publication bias......Page 668
17 Regression......Page 671
EXAMPLE 17.1 The lion’s nose......Page 673
The method of least squares......Page 674
Formula for the line......Page 675
Calculating the slope and intercept......Page 676
Predicted values......Page 677
Residuals......Page 678
Confidence interval for the slope......Page 679
Confidence intervals for predictions......Page 680
Extrapolation......Page 681
EXAMPLE 17.3 Prairie Home Campion......Page 683
The t-test of regression slope......Page 684
The ANOVA approach......Page 685
Using R2 to measure the fit of the line to data......Page 686
17.4 Regression toward the mean......Page 687
Outliers......Page 689
Detecting non-normality and unequal variance......Page 691
17.6 Transformations......Page 693
17.7 The effects of measurement error on regression......Page 696
A curve with an asymptote......Page 697
Quadratic curves......Page 698
Formula-free curve fitting......Page 699
17.9 Logistic regression: fitting a binary response variable......Page 701
17.10 Summary......Page 705
Regression intercept......Page 707
Confidence interval for the predicted individual Y at a given X (prediction intervals)......Page 708
R squared (R2)......Page 709
PRACTICE PROBLEMS......Page 711
ASSIGNMENT PROBLEMS......Page 723
11 INTERLEAF Using species as data points......Page 739
Review Problems 3......Page 743
18 Multiple explanatory variables......Page 754
Modeling with linear regression......Page 756
Generalizing linear regression......Page 757
General linear models......Page 759
Analyzing data from a randomized block design......Page 761
Fitting the model to data......Page 762
18.3 Analyzing factorial designs......Page 764
EXAMPLE 18.3 Interaction zone......Page 765
Model formula......Page 766
Testing the factors......Page 767
The importance of distinguishing fixed and random factors......Page 768
EXAMPLE 18.4 Mole-rat layabouts......Page 769
Testing interaction......Page 770
Fitting a model without an interaction term......Page 771
18.5 Assumptions of general linear models......Page 773
18.6 Summary......Page 775
PRACTICE PROBLEMS......Page 777
ASSIGNMENT PROBLEMS......Page 783
19 Computer-intensive methods......Page 789
EXAMPLE 19.1 How did he know? The non-randomness of haphazard choice......Page 791
EXAMPLE 19.2 The language center in chimps’ brains......Page 795
Bootstrap standard error......Page 797
Confidence intervals by bootstrapping......Page 798
Bootstrapping with multiple groups......Page 799
Assumptions and limitations of the bootstrap......Page 800
19.3 Summary......Page 802
PRACTICE PROBLEMS......Page 803
ASSIGNMENT PROBLEMS......Page 810
20 Likelihood......Page 814
20.1 What is likelihood?......Page 816
Phylogeny estimation......Page 817
Gene mapping......Page 818
Probability model......Page 819
The likelihood formula......Page 820
The maximum likelihood estimate......Page 821
Likelihood-based confidence intervals......Page 823
Probability model......Page 825
The likelihood formula......Page 826
Bias......Page 827
Testing a population proportion......Page 829
20.6 Summary......Page 831
Log-likelihood ratio test for a single parameter......Page 832
PRACTICE PROBLEMS......Page 833
ASSIGNMENT PROBLEMS......Page 839
21 Meta-analysis combining information from multiple studies......Page 844
Why repeat a study?......Page 846
EXAMPLE 21.2 Aspirin and myocardial infarction......Page 847
EXAMPLE 21.3 The Transylvania effect......Page 849
Define the question......Page 851
Review the literature......Page 852
Compute effect sizes......Page 853
Calculate confidence intervals and test hypotheses......Page 855
Look for associations......Page 856
21.5 File-drawer problem......Page 858
21.6 How to make your paper accessible to meta-analysis......Page 859
21.7 Summary......Page 860
Mantel-Haenszel test......Page 861
PRACTICE PROBLEMS......Page 862
ASSIGNMENT PROBLEMS......Page 864
Chapter 2......Page 865
Chapter 4......Page 866
Chapter 7......Page 867
Chapter 8......Page 868
Chapter 9......Page 869
Chapter 11......Page 870
Chapter 13......Page 871
Chapter 14......Page 872
Chapter 15......Page 873
Chapter 17......Page 874
Chapter 19......Page 875
Chapter 20......Page 876
Chapter 21......Page 877
Using statistical tables......Page 878
Statistical Table A: The χ2 distribution......Page 879
Statistical Table B: The standard normal (Z) distribution......Page 883
Statistical Table C: Student’s t-distribution......Page 885
Statistical Table D: The F-distribution......Page 888
Statistical Table E: Mann-Whitney U-distribution......Page 895
Statistical Table F: Tukey-Kramer q-distribution......Page 897
Statistical Table G: Critical values for the Spearman’s rank correlation......Page 898
Literature Cited......Page 902
Chapter 1......Page 930
Chapter 2......Page 931
Chapter 3......Page 936
Chapter 4......Page 939
Chapter 5......Page 940
Chapter 6......Page 943
Chapter 7......Page 944
Chapter 8......Page 947
Chapter 9......Page 952
Review 1......Page 959
Chapter 10......Page 963
Chapter 11......Page 965
Chapter 12......Page 968
Chapter 13......Page 973
Review 2......Page 977
Chapter 14......Page 981
Chapter 15......Page 983
Chapter 16......Page 989
Chapter 17......Page 991
Review 3......Page 996
Chapter 18......Page 1001
Chapter 19......Page 1003
Chapter 20......Page 1004
Chapter 21......Page 1006
Chapter 5......Page 1008
Chapter 11......Page 1009
Chapter 16......Page 1010
Chapter 20......Page 1011
Chapter 21......Page 1012
A......Page 1013
B......Page 1014
C......Page 1016
D......Page 1017
E......Page 1018
F......Page 1019
G......Page 1020
H......Page 1021
I......Page 1022
L......Page 1023
M......Page 1024
N......Page 1027
P......Page 1028
R......Page 1031
S......Page 1033
T......Page 1035
U......Page 1036
W......Page 1037
Z......Page 1038
Inside Back Cover......Page 1039
Back Cover......Page 1040

Citation preview

The Analysis of Biological Data

The Analysis of Biological Data Second Edition

Michael C. Whitlock and Dolph Schluter

The Analysis of Biological Data, Second Edition

Publisher: Ben Roberts Proofreader: Kathi Townes Art Studio: Lineworks, Inc.; Lori Heckelman Cover Designer: Emiko Paul Photo Researcher: Jennifer Simmons, Lumina Datamatics, Inc. Permissions Assistant: Michael McCarthy Compositor: Kristina Elliott at TECHarts ©2015 by W. H. Freeman and Company Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the W. H. Freeman and Company Rights and Permissions Department. Grateful acknowledgment for third-party permissions, which have been granted for material in this title not owned by Macmillan Learning, appears in the Photo Credits section, which represents an extension of this copyright page. Cover Photo: ©www.pheromonegallery.com ISBN: 978-1-319156-71-8 Library of Congress Cataloging-in-Publication Data Whitlock, Michael, author. The analysis of biological data / Michael C. Whitlock and Dolph Schluter. -- Second edition. pages cm Includes bibliographical references and index. ISBN 978-1-319156-71-8 1. Biometry--Textbooks. I. Schluter, Dolph, author. II. Title. QH323.5.W48 2015 570.1’5195--dc23 2014010300

10 9 8 7 6 5 4

W. H. Freeman and Company One New York Plaza Suite 4500 10004-1562 New York, NY www.macmillanhighered.com

To Sally and Wilson, Andrea and Maggie

Contents in brief Preface Acknowledgments About the authors

PART 1

INTRODUCTION TO STATISTICS

1. Statistics and samples INTERLEAF 1

Biology and the history of statistics

2. Displaying data 3. Describing data 4. Estimating with uncertainty INTERLEAF 2

Pseudoreplication

5. Probability 6. Hypothesis testing INTERLEAF 3

PART 2

Why statistical significance is not the same as biological importance

PROPORTIONS AND FREQUENCIES

7. Analyzing proportions INTERLEAF 4

Correlation does not require causation

8. Fitting probability models to frequency data INTERLEAF 5

Making a plan

9. Contingency analysis: associations between categorical variables 235 Review Problems 1 PART 3

COMPARING NUMERICAL VALUES

10. The normal distribution INTERLEAF 6

Controls in medical studies

11. Inference for a normal population 12. Comparing two means INTERLEAF 7

Which test should I use?

13. Handling violations of assumptions Review Problems 2 14. Designing experiments INTERLEAF 8

Data dredging

15. Comparing means of more than two groups INTERLEAF 9

PART 4

Experimental and statistical mistakes

REGRESSION AND CORRELATION

16. Correlation between numerical variables INTERLEAF 10

Publication bias

17. Regression INTERLEAF 11

Using species as data points

Review Problems 3 PART 5

MODERN STATISTICAL METHODS

18. Multiple explanatory variables 19. Computer-intensive methods 20. Likelihood 21. Meta-analysis: combining information from multiple studies Statistical tables Literature cited Answers to practice problems Photo credits Index

Contents Preface Acknowledgments About the authors

PART 1

INTRODUCTION TO STATISTICS

1. Statistics and samples 1.1 What is statistics? 1.2 Sampling populations Example 1.2: Raining cats Populations and samples Properties of good samples Random sampling How to take a random sample The sample of convenience Volunteer bias Data in the real world 1.3 Types of data and variables Categorical and numerical variables Explanatory and response variables 1.4 Frequency distributions and probability distributions 1.5 Types of studies 1.6 Summary Practice problems Assignment problems

TERLEAF 1

Biology and the history of statistics

2. Displaying data 2.1 Guidelines for effective graphs How to draw a bad graph How to draw a good graph 2.2 Showing data for one variable Showing categorical data: frequency table and bar graph Example 2.2A: Crouching tiger Making a good bar graph A bar graph is usually better than a pie chart

Showing numerical data: frequency table and histogram Example 2.2B: Abundance of desert bird species Describing the shape of a histogram How to draw a good histogram Other graphs for numerical data 2.3 Showing association between two variables Showing association between categorical variables Example 2.3A: Reproductive effort and avian malaria Showing association between numerical variables: scatter plot Example 2.3B: Sins of the father Showing association between a numerical and a categorical variable Example 2.3C: Blood responses to high elevation 2.4 Showing trends in time and space Line graph Example 2.4A: Bad science can be deadly Maps Example 2.4B: Biodiversity hotspots 2.5 How to make good tables Follow similar principles for display tables 2.6 Summary Practice problems Assignment problems

3. Describing data 3.1 Arithmetic mean and standard deviation Example 3.1: Gliding snakes The sample mean Variance and standard deviation Rounding means, standard deviations, and other quantities Coefficient of variation Calculating mean and standard deviation from a frequency table Effect of changing measurement scale 3.2 Median and interquartile range Example 3.2: I’d give my right arm for a female The median The interquartile range The box plot 3.3 How measures of location and spread compare Example 3.3: Disarming fish Mean versus median Standard deviation versus interquartile range 3.4 Cumulative frequency distribution Percentiles and quantiles Displaying cumulative relative frequencies

3.5 Proportions Calculating a proportion The proportion is like a sample mean 3.6 Summary 3.7 Quick Formula Summary Practice problems Assignment problems

4. Estimating with uncertainty 4.1 The sampling distribution of an estimate Example 4.1: The length of human genes Estimating mean gene length with a random sample The sampling distribution of Y¯ 4.2 Measuring the uncertainty of an estimate Standard error The standard error of Y¯ The standard error of Y¯ from data 4.3 Confidence intervals The 2SE rule of thumb 4.4 Error bars 4.5 Summary 4.6 Quick Formula Summary Practice problems Assignment problems

TERLEAF 2

Pseudoreplication

5. Probability 5.1 The probability of an event 5.2 Venn diagrams 5.3 Mutually exclusive events 5.4 Probability distributions Discrete probability distributions Continuous probability distributions 5.5 Either this or that: adding probabilities The addition rule The probabilities of all possible mutually exclusive outcomes add to one The general addition rule 5.6 Independence and the multiplication rule Multiplication rule Example 5.6A: Smoking and high blood pressure “And” versus “or” Independence of more than two events Example 5.6B: Mendel’s peas

5.7 Probability trees Example 5.7: Sex and birth order 5.8 Dependent events Example 5.8: Is this meat taken? 5.9 Conditional probability and Bayes’ theorem Conditional probability The general multiplication rule Sampling without replacement Bayes’ theorem Example 5.9: Detection of Down syndrome 5.10 Summary Practice problems Assignment problems

6. Hypothesis testing 6.1 Making and using hypotheses Null hypothesis Alternative hypothesis To reject or not to reject 6.2 Hypothesis testing: an example Example 6.2: The right hand of toad Stating the hypotheses The test statistic The null distribution Quantifying uncertainty: the P-value Draw the appropriate conclusion Reporting the results 6.3 Errors in hypothesis testing Type I and Type II errors 6.4 When the null hypothesis is not rejected Example 6.4: The genetics of mirror-image flowers The test Interpreting a non-significant result 6.5 One-sided tests 6.6 Hypothesis testing versus confidence intervals 6.7 Summary Practice problems Assignment problems

TERLEAF 3

PART 2

Why statistical significance is not the same as biological importance

PROPORTIONS AND FREQUENCIES

7. Analyzing proportions

7.1 The binomial distribution Formula for the binomial distribution Number of successes in a random sample Sampling distribution of the proportion 7.2 Testing a proportion: the binomial test Example 7.2: Sex and the X Approximations for the binomial test 7.3 Estimating proportions Example 7.3: Radiologists’ missing sons Estimating the standard error of a proportion Confidence intervals for proportions—the Agresti–Coull method Confidence intervals for proportions—the Wald method 7.4 Deriving the binomial distribution 7.5 Summary 7.6 Quick Formula Summary Practice problems Assignment problems

TERLEAF 4

Correlation does not require causation

8. Fitting probability models to frequency data 8.1 Example of a probability model: the proportional model Example 8.1: No weekend getaway 8.2 χ2 goodness-of-fit test Null and alternative hypotheses Observed and expected frequencies The χ2 test statistic The sampling distribution of χ2 under the null hypothesis Calculating the P-value Critical values for the χ2 distribution 8.3 Assumptions of the χ2 goodness-of-fit test 8.4 Goodness-of-fit tests when there are only two categories Example 8.4: Gene content of the human X chromosome 8.5 Fitting the binomial distribution Example 8.5: Designer two-child families? 8.6 Random in space or time: the Poisson distribution Formula for the Poisson distribution Testing randomness with the Poisson distribution Example 8.6: Mass extinctions Comparing the variance to the mean 8.7 Summary 8.8 Quick Formula Summary Practice problems

Assignment problems

TERLEAF 5

Making a plan

9. Contingency analysis: associations between categorical variables 9.1 Associating two categorical variables 9.2 Estimating association in 2 × 2 tables: odds ratio Odds Example 9.2: Take two aspirin and call me in the morning? Odds ratio Standard error and confidence interval for odds ratio 9.3 Estimating association in 2 × 2 table: relative risk Odds ratio vs. relative ristk Example 9.3: Your litter box and your brain 9.4 The χ2 contingency test Example 9.4: The gnarly worm gets the bird Hypotheses Expected frequencies assuming independence The χ2 statistic Degrees of freedom P-value and conclusion A shortcut for calculating the expected frequencies The χ2 contingency test is a special case of the χ2 goodness-of-fit test Assumptions of the χ2 contingency test Correction for continuity 9.5 Fisher’s exact test Example 9.5: The feeding habits of vampire bats 9.6 G-tests 9.7 Summary 9.8 Quick Formula Summary Practice problems Assignment problems

Review Problems 1 PART 3:

COMPARING NUMERICAL VALUES

10. The normal distribution 10.1 Bell-shaped curves and the normal distribution 10.2 The formula for the normal distribution 10.3 Properties of the normal distribution 10.4 The standard normal distribution and statistical tables Using the standard normal table Using the standard normal to describe any normal distribution Example 10.4: One small step for man?

10.5 The normal distribution of sample means Calculating probabilities of sample means 10.6 Central limit theorem Example 10.6: Young adults and the Spanish flu 10.7 Normal approximation to the binomial distribution Example 10.7: The only good bug is a dead bug 10.8 Summary 10.9 Quick Formula Summary Practice problems Assignment problems

TERLEAF 6

Controls in medical studies

11. Inference for a normal population 11.1 The t-distribution for sample means Student’s t-distribution Finding critical values of the t-distribution 11.2 The confidence interval for the mean of a normal distribution Example 11.2: Eye to eye The 95% confidence interval for the mean The 99% confidence interval for the mean 11.3 The one-sample t-test Example 11.3: Human body temperature The effects of larger sample size—body temperature revisited 11.4 Assumptions of the one-sample t-test 11.5 Estimating the standard deviation and variance of a normal population Confidence limits for the variance Confidence limits for the standard deviation Assumptions 11.6 Summary 11.7 Quick Formula Summary Practice problems Assignment problems

12. Comparing two means 12.1 Paired sample versus two independent samples 12.2 Paired comparison of means Estimating mean difference from paired data Example 12.2: So macho it makes you sick? Paired t-test Assumptions 12.3 Two-sample comparison of means Example 12.3: Spike or be spiked Confidence interval for the difference between two means Two-sample t-test

Assumptions A two-sample t-test when standard deviations are unequal 12.4 Using the correct sampling units Example 12.4: So long; thanks to all the fish 12.5 The fallacy of indirect comparison Example 12.5: Mommy’s baby, Daddy’s maybe 12.6 Interpreting overlap of confidence intervals 12.7 Comparing variances The F-test of equal variances Levene’s test for homogeneity of variances 12.8 Summary 12.9 Quick Formula Summary Practice problems Assignment problems

TERLEAF 7

Which test should I use?

13. Handling violations of assumptions 13.1 Detecting deviations from normality Graphical methods Example 13.1: The benefits of marine reserves Formal test of normality 13.2 When to ignore violations of assumptions Violations of normality Unequal standard deviations 13.3 Data transformations Log transformation Arcsine transformation The square-root transformation Other transformations Confidence intervals with transformations A caveat: Avoid multiple testing with transformations 13.4 Nonparametric alternatives to one-sample and paired t-tests Sign test Example 13.4: Sexual conflict and the origin of new species The Wilcoxon signed-rank test 13.5 Comparing two groups: the Mann–Whitney U-test Example 13.5: Sexual cannibalism in sagebrush crickets Tied ranks Large samples and the normal approximation 13.6 Assumptions of nonparametric tests 13.7 Type I and Type II error rates of nonparametric methods 13.8 Permutation tests Assumptions of permutation tests

13.9 Summary 13.10 Quick Formula Summary Practice problems Assignment problems

Review Problems 2 14. Designing experiments 14.1 Why do experiments? Confounding variables Experimental artifacts 14.2 Lessons from clinical trials Example 14.2: Reducing HIV transmission Design components 14.3 How to reduce bias Simultaneous control group Randomization Blinding 14.4 How to reduce the influence of sampling error Replication Balance Blocking Example 14.4A: Holey waters Extreme treatments Example 14.4B: Plastic hormones 14.5 Experiments with more than one factor Example 14.5: Lethal combination 14.6 What if you can’t do experiments? Match and adjust 14.7 Choosing a sample size Plan for precision Plan for power Plan for data loss 14.8 Summary 14.9 Quick Formula Summary Practice problems Assignment problems

TERLEAF 8

Data dredging

15. Comparing means of more than two groups 15.1 The analysis of variance Example 15.1: The knees who say night Hypotheses ANOVA in a nutshell ANOVA tables

Partitioning the sum of squares Calculating the mean squares The variance ratio, F Variation explained: R2 ANOVA with two groups 15.2 Assumptions and alternatives The robustness of ANOVA Data transformations Nonparametric alternatives to ANOVA 15.3 Planned comparisons Planned comparison between two means 15.4 Unplanned comparisons Example 15.4: Wood wide web Testing all pairs of means using the Tukey–Kramer method Assumptions 15.5 Fixed and random effects 15.6 ANOVA with randomly chosen groups Example 15.6: Walking stick limbs ANOVA calculations Variance components Repeatability Assumptions 15.7 Summary 15.8 Quick Formula Summary Practice problems Assignment problems

TERLEAF 9

PART 4

Experimental and statistical mistakes

REGRESSION AND CORRELATION

16. Correlation between numerical variables 16.1 Estimating a linear correlation coefficient The correlation coefficient Example 16.1: Flipping the bird Standard error Approximate confidence interval 16.2 Testing the null hypothesis of zero correlation Example 16.2: What big inbreeding coefficients you have 16.3 Assumptions 16.4 The correlation coefficient depends on the range 16.5 Spearman’s rank correlation Example 16.5: The miracles of memory Procedure for large n

Assumptions of Spearman’s correlation 16.6 The effects of measurement error on correlation 16.7 Summary 16.8 Quick Formula Summary Practice problems Assignment problems

TERLEAF 10

Publication bias

17. Regression 17.1 Linear regression Example 17.1: The lion’s nose The method of least squares Formula for the line Calculating the slope and intercept Populations and samples Predicted values Residuals Standard error of slope Confidence interval for the slope 17.2 Confidence in predictions Confidence intervals for predictions Extrapolation 17.3 Testing hypotheses about a slope Example 17.3: Prairie home campion The t-test of regression slope The ANOVA approach Using R2 to measure the fit of the line to data 17.4 Regression toward the mean 17.5 Assumptions of regression Outliers Detecting nonlinearity Detecting non-normality and unequal variance 17.6 Transformations 17.7 The effects of measurement error on regression 17.8 Nonlinear regression A curve with an asymptote Quadratic curves Formula-free curve fitting Example 17.8: The incredible shrinking seal 17.9 Logistic regression: fitting a binary response variable 17.10 Summary 17.11 Quick Formula Summary Practice problems

Assignment problems

TERLEAF 11

Using species as data points Review Problems 3

PART 5

MODERN STATISTICAL METHODS

18. Multiple explanatory variables 18.1 ANOVA and linear regression are linear models Modeling with linear regression Generalizing linear regression General linear models 18.2 Analyzing experiments with blocking Analyzing data from a randomized block design Example 18.2: Zooplankton depredation Model formula Fitting the model to data 18.3 Analyzing factorial designs Example 18.3: Interaction zone Model formula Testing the factors The importance of distinguishing fixed and random factors 18.4 Adjusting for the effects of a covariate Example 18.4: Mole-rat layabouts Testing interaction Fitting a model without an interaction term 18.5 Assumptions of general linear models 18.6. Summary Practice problems Assignment problems

19. Computer-intensive methods 19.1 Hypothesis testing using simulation Example 19.1: How did he know? The nonrandomness of haphazard choice 19.2 Bootstrap standard errors and confidence intervals Example 19.2: The language center in chimps’ brains Bootstrap standard error Confidence intervals by bootstrapping Bootstrapping with multiple groups Assumptions and limitations of the bootstrap 19.3 Summary Practice problems Assignment problems

20. Likelihood

20.1 What is likelihood? 20.2 Two uses of likelihood in biology Phylogeny estimation Gene mapping 20.3 Maximum likelihood estimation Example 20.3: Unruly passengers Probability model The likelihood formula The maximum likelihood estimate Likelihood-based confidence intervals 20.4 Versatility of maximum likelihood estimation Example 20.4: Conservation scoop Probability model The likelihood formula The maximum likelihood estimate Bias 20.5 Log-likelihood ratio test Likelihood ratio test statistic Testing a population proportion 20.6 Summary 20.7 Quick Formula Summary Practice problems Assignment problems

21. Meta-analysis: combining information from multiple studies 21.1 What is meta-analysis? Why repeat a study? 21.2 The power of meta-analysis Example 21.2: Aspirin and myocardial infarction 21.3 Meta-analysis can give a balanced view Example 21.3: The Transylvania effect 21.4 The steps of a meta-analysis Define the question Example 21.4: Testosterone and aggression Review the literature Compute effect sizes Determine the average effect size Calculate confidence intervals and test hypotheses Look for effects of study quality Look for associations 21.5 File-drawer problem 21.6 How to make your paper accessible to meta-analysis 21.7 Summary 21.8 Quick Formula Summary

Practice problems Assignment problems

Statistical tables Using statistical tables Statistical Table A: The χ2 distribution Statistical Table B: The standard normal (Z) distribution Statistical Table C: The Student t-distribution Statistical Table D: The F-distribution Statistical Table E: Mann–Whitney U-distribution Statistical Table F: Tukey–Kramer q-distribution Statistical Table G: Critical values for the Spearman’s rank correlation

Literature cited Answers to practice problems Photo credits Index

Preface Modern biologists need the powerful tools of data analysis. As a result, an increasing number of universities offer, or even require, a basic data analysis course for all their biology and premedical students. We have been teaching such a course at the University of British Columbia for the last two decades. Over this period, we have sought a textbook that covered the material we needed in an introductory course at just the right level. We found that most texts were too technical and encyclopedic, or else they didn’t go far enough, missing methods that were crucial to the practice of modern biology and medicine. We wanted a book that had a strong emphasis on intuitive understanding to convey meaning, rather than an overreliance on formulas. We wanted to teach by example, and the examples needed to be interesting. Most importantly, we needed a biology book, addressing topics important to biologists and health care providers handling real data. We couldn’t find the book that we needed, so we decided to write this one to fill the gap. We include several unusual features that we have discovered to be helpful for effectively reaching our audience: Interesting biology examples. Our teaching has shown us that biology students learn data analysis best in the context of interesting examples drawn from the medical and biological literature. Statistics is a means to an end, a tool to learn about the human and natural world. By emphasizing what we can learn about the science, the power and value of statistics becomes plain. Plus, it’s just more fun for everyone concerned. Every chapter has several biological or medical examples of key concepts, and each example is prefaced by a substantial description of the biological setting. The examples are illustrated with photos of the real organisms, so that students can look at what they’re learning about. The emphasis on real and interesting examples carries into the problem sets; for each chapter, there are dozens of questions based on real data about biological and medical issues. In this Second Edition, we have added approximately 200 new problems. These include a new type of problem, called Calculation Practice, that takes the student step-by-step through the important procedures described in the chapter. The corresponding answers are provided at the back of the book so that the students can check their success at each step. Two or three such problems are at the start of most of the chapter problem sets. We’ve also added three new sections of review problems (after Chapters 9, 13, and 17) to allow cumulative review of the important concepts up to that point in the book. Intuitive explanations of key concepts. Statistical reasoning requires a lot of new ways of thinking. Students can get lost in the barrage of new jargon and multitudinous tests. We have found that starting from an intuitive foundation, away from all the details, is extremely valuable. We take an intuitive approach to basic questions: What’s a good sample? What’s a confidence interval? Why do an experiment? The first several chapters establish this basic knowledge, and the rest of the book builds on it.

Practical data analysis. As its title suggests, this book focuses on data rather than the mathematical foundations of statistics. We teach how to make good graphical displays, and we emphasize that a good graph is the beginning point of any good data analysis. We give equal time to estimation and hypothesis testing, and we avoid treating the P-value as an end in itself. The book does not demand a knowledge of mathematics beyond simple algebra. We focus on practicality over nuance, on biological usefulness over theoretical hand wringing. We not only teach the “right” way of doing something but also highlight some of the pitfalls that might be encountered. We demonstrate how to carry out the calculations for most methods so that the steps are not mysterious to students. At the same time, we know that a computer will be available for most calculations. Hence, we focus on the concepts of biological data analysis and how statistics can help extract scientific insight from data. With the power of modern computers at hand, the challenge in analyzing data becomes knowing what method to use and why.1 We imagine and hope that every course using this book will have a component encouraging students to use computer statistical packages. We are also aware that the diversity of such packages is immense, and so we have not tied the book to any particular program. We have heard most often from instructors who use the R package with this book, and we now provide R scripts for all the examples in this book at the book’s web site (http://whitlockschluter.zoology.ubc.ca). Practical experimental design. A biologist cannot do good statistics—or good science— without a practical understanding of experimental design. Unlike most books, our book covers basic topics in experimental design, such as controls, randomization, pseudoreplication, and blocking, and we do it in a practical, intuitive way. Up to date on the basics. Believe it or not, the best confidence interval for the proportion is not the one you probably learned as an undergraduate. Nonparametric statistics do not effectively test for differences in means (or medians, for that matter) without some fairly strong assumptions that we normally hear little about. For these and many other topics, we have updated the coverage of basic, everyday topics in statistics. Coverage of modern topics. Modern biology uses a larger toolkit than the one available a generation ago. In this book, we go beyond most introductory books by establishing the conceptual principles of important topics, such as likelihood, nonlinear regression, permutation tests, meta-analysis, and the bootstrap. Useful summaries. Near the end of each chapter is a short, clear summary of the key concepts, and most chapters end with Quick Formula Summaries that put most equations in one easy-tofind place. Interleaves. Between chapters are short essays that we call interleaves. These interleaves cover a variety of conceptual topics in a nontechnical way—ideas that are nevertheless crucial for the interpretation of statistical results in scientific research. Several of them focus on common mistakes in the analysis and interpretation of biological data and how to avoid them. Although the interleaves are pressed between the chapters, they complement the material in the core chapters in important ways. We believe that the interleaves discuss some of the most important topics of the book, such as the meaning of statistical significance (Interleaf 3), the difference between correlation and causation (Interleaf 4), why control treatments are necessary (Interleaf

6), and the distortions caused by publication bias (Interleaf 10). Interleaf 7 summarizes what statistical test should be used and when, and it is a good place to start when reviewing for exams. After five years of writing—and a couple more years of work on this new edition—the result is now in your hands. We think The Analysis of Biological Data provides a good background in data analysis for biologists and medical practitioners, covering a broad range of topics in a practical and intuitive way. It works for our classes; we hope that it works for yours, too.

Organization of the book The Analysis of Biological Data is divided into five parts, each with a handful of chapters as indicated in the table of contents. We recommend starting with the first part, because it introduces many basic concepts that are used throughout the book, such as sampling, drawing a good graph, describing data, estimating, hypothesis testing, and concepts in probability. These early chapters are meant to be read in their entirety. After the first block, most chapters progress from the most general topics at the start to more specialized topics by the end. Each chapter is structured so that a basic understanding of the topic may be obtained from the earliest sections. For example, in the chapter on analysis of variance (Chapter 15), the basics are taught in the first two sections; reading Sections 15.1 and 15.2 gives roughly the same material that most introductory statistics texts provide about this method. Sections 15.3–15.6 explain additional twists and other interesting applications. The last block of chapters (Chapters 18–21) is mainly for the adventurous and the curious. These chapters introduce topics, such as likelihood, bootstrapping, and meta-analysis, that are commonly encountered in the biological and medical literature but not often mentioned in an introductory course. These chapters introduce the basic principles of each topic, discuss how the methods work, and point to where you might look to find out more. A basic course could be taught by using only Chapters 1–17 and, within this subset of chapters, by stopping after Sections 5.8, 7.3, 8.4, 9.4, 12.6, 13.6, 15.2, 16.4, and 17.6 in their respective chapters. We suggest that all courses highlight specific topics covered only in the interleaves. Each chapter ends with a series of problems that are designed to test students’ understanding of the concepts and the practical application of statistics. The problems are divided into Practice Problems and Assignment Problems. Three cumulative review problem sets have been added to the Second Edition, placed approximately at locations convenient for midterm and final exam review. Short answers to all Practice Problems and Review Problems are provided in the back of the book; answers to the Assignment Problems are available to instructors only from the publisher. For a copy, contact Karen Carson at [email protected] or 1-800-446-8923. Other teaching resources for the book are available online at http://whitlockschluter.zoology.ubc.ca.

A word about the data

The data used in this book are real, with a few well-marked exceptions. For the most part, these data were obtained directly from published papers. In some cases, we contacted the authors of articles who generously provided the raw data for our use. Often, when raw data were not provided in the original paper, we resorted to obtaining data points by digitizing graphical depictions, such as scatter plots and histograms. Inevitably, the numbers we extracted differ slightly from the original numbers because of measurement error. In rare cases, we generated data by computer that matched the statistical summaries in the paper. In all cases, the results we present are consistent with the conclusions of the original papers. Most of the data sets are available online at http://whitlockschluter.zoology.ubc.ca. 1. “A computer lets you make more mistakes faster than any invention in human history—with the possible exceptions of handguns and tequila.” (Mitch Ratcliffe, in Technology Review, 1992).

Acknowledgments This book would not have happened had the two of us had been left to do it by ourselves. Many people contributed in substantial ways, and we are forever grateful. The clarity and accuracy of its contents were improved by the careful attention of a lot of generous readers, including Arianne Albert, Brad Anholt, Cécile Ané, Eric Baack, Arthur Berg, Chad Brassil, James Bryant, Martin Buntinas, Mark Butler, Carrie Case, C. Ray Chandler, Mark Clemens, Bradley Cosentino, Perry de Valpine, Flo Débarre, Abby Drake, Christiana Drake, Jonathan Dushoff, Steven George, Aleeza Gerstein, George Gilchrist, Brett Goodwin, Steven Green, Tim Hanson, Mike Hickerson, Lisa Hines, Darren Irwin, Nusrat Jahan, Philip Johns, Roger Johnson, Istvan Karsai, Robert Keen, John Kelly, Rex Kenner, Ben Kerr, Laura Kubatko, Joseph G. Kunkel, Bret Larget, Theo Light, Todd Livdahl, Heather Masonjones, Brian C. McCarthy, Kevin Middleton, Eli Minkoff, Robert Montgomerie, Spencer Muse, Courtney Murren, Claudia Neuhauser, Liam O’Brien, Maria Orive, Patrick C. Phillips, Jay Pitocchelli, Danielle Presgraves, James Robinson, Simon Robson, Michael Rosenberg, Noah Rosenberg, Nathan Rank, Bruce Rannala, Mark Rizzardi, Colin Rundel, Michael Russell, Ronald W. Russell, Andrew Schaffner, Andrea Schluter, James Scott, Joel Shore, John Soghigian, William Thomas, Michael Travisano, Thomas Valone, Bruce Walsh, Claus Wilke, Michael Wunder, Grace A. Wyngaard, and Sam Yeaman. Many of these good people read multiple chapters, and we thank all for their invaluable aid and considerate forebearance. Sally Otto, Allan StewartOaten, and Maria Orive earned our undying gratitude by reading and commenting on the entire book. Of course, any errors that remain are our own fault; we didn’t always take everyone’s advice, even perhaps when we should have. If we have forgotten anyone, you have our thanks even if our memories are poor. We owe a debt to the students of BIOL 300 at the University of British Columbia, who class-tested this book over the last several years. The book also benefited by class testing at several colleges and universities in courses by Helen Alexander, Brad Anholt, Eric Baack, Carol Baskauf, Peter Dunn, Matthias Foellmer, Marie-Josée Fortin, George Gilchrist, Michael Grant, Joe Hardin, Karen Harper, Scott Harrison, Mike Hickerson, Stephen Hudman, Nusrat Jahan, Laura Kapitula, Randy Larsen, Terri Lacourse, Susan Lehman, Todd Livdahl, Kelly McCoy, Jean Richardson, Simon Robson, Chris Sanford, Tom Short, Andrew Tierman, Steve Vamosi, Liette Vasseur, Jason Wilson, and Grace A. Wyngaard. George Gilchrist and his students gave us a very detailed and extremely helpful set of comments at a crucial stage of the book. The following students and faculty from UBC and other institutions uncovered errors in earlier versions of the book: Jessica Beaubier, Chad Brassil, Edward Cheung, Lorena Cheung, Stephanie Cheung, Denise Choi, Peter Dunn, Maryam Garabedian, Samrad Ghavimi, Inderjit Grewal, Rutger Hermsen, Sarah Hamanishi, Anne Harris, Yunhyung Ku, Gurpreet Khaira, Joanne Kim, Jung Min Kim, Arleigh Lambert, Alexander Leung, Mira Li, Flora Liu, Dianna Louie, Johnston Mak, Giovanni Marchetti, Luke Miller, Diana Moanga, Sarah Neumann, Jarad Niemi, Tyler Ng, Ruth Ogbamichael, Jasmine Ono, Chris Parrish, Jason Pither, Marion Pearson, Jessica Pham, Trevor Schofield, Melissa Smith, Meredith Soon, Erin Stacey, Daniel Stoebel, Myriam Tremblay, Michelle Uzelac, John Wakeley, Hillary Ward, Chris Wong, Irene Yu, Fei Yuan, Anush Zakaryan, Qiong Zhang, Paul Zhou, and Jon-Paul Zacharias. We send

special thanks to Nick Cox, who graciously read a previous printing of this book with extraordinary care. A number of researchers freely sent us their original data, including Matt Arnegard, Angela Attwood, Audrey Barker-Plotkin, Cynthia Beall, Butch Brodie, Pamela Colosimo, Kimmo Eriksson, Kevin Fowler, Suzanne Gray, Chris Harley, Luke Harmon, Andrew Hendry, Peter Keightley, Fredrik Liljeros, Jean Thierry-Mieg, Jeffrey S. Mogil, Patrik Nosil, Margarita Ramos, Rick Relyea, Locke Rowe, Natarajan Singaravelan, Jake Socha, Jan Soumanan, Brian Starzomski, Richard Svanback, David Tilman, Andrew Trites, Neils van de Ven, Christoph von Beeren, Yitong Wang, Jason Weir, Jack Werren, and Martin Wikelski. The book has benefited enormously from the efforts of a large team of able people: copyediting by John Murdzek (1st ed.), Gunder Hefta (1st ed.), and Kathi Townes (2nd ed.); photo research by Laura Roberts (1st ed.), Terri Wright, Austin MacRae, and Jen Simmons (2nd ed.); art by Tom Webster (1st ed.) and Lori Heckelman (2nd ed.); and design and composition by Mark Ong (1st ed.), Kathi Townes (2nd ed.), and Kristina Elliott (2nd ed.). Eric Baack has our special appreciation for slaving over the problem sets to create the answer keys, as does Holly Kindsvater, who carefully checked all the answer keys for the 2nd edition. Steven Green pointed out several improvements to the answer key and reviewed the answers to the review problem sets. Aleeza Gerstein corrected numerous errors with her careful proofreading. Finally, Ben Roberts deserves our greatest thanks, for all of his support and vision in making this book happen, and especially for Clause 24. The book was started while MCW was supported by the Peter Wall Institute for Advanced Studies at UBC as a Distinguished-Scholar-in-Residence, and the majority of the final stages of the book were written while he was a Sabbatical Scholar at the National Evolutionary Synthesis Center in North Carolina (NSF #EF-0423641). DS began working on the book while a visiting professor in Developmental Biology at Stanford University. The second edition was aided by a sabbatical leave at the University of Texas (MCW) and a Canada Council Senior Killam Fellowship (DS), which included a wonderful stay at La Selva Biological Station. The scholarly support and environment provided by each of these institutions was exceptional—and greatly appreciated. Finally, we would like to give great thanks to all of the people that have taught us the most over the years. MCW would like to thank Dave McCauley, Mike Wade, Nick Barton, Ben Pierce, Kevin Fowler, Patrick Phillips, Sally Otto, and Betty Whitlock.

About the authors Michael Whitlock is an evolutionary biologist and population geneticist known for his work on evolution in spatially structured populations. He is a Professor of Zoology at the University of British Columbia, where he has taught statistics to biology students since 1995. He is a fellow of the American Association for the Advancement of Science and of the American Academy of Arts and Science. Dolph Schluter is Professor and Canada Research Chair in the Zoology Department and Biodiversity Research Center at the University of British Columbia. He is known for his research on the ecology and evolution of Galápagos finches and threespine stickleback. He is a fellow of the Royal Societies of Canada and London and a foreign member of the American Academy of Arts and Sciences.

1

Statistics and samples

Leafcutter ant

What is statistics? Biologists study the properties of living things. Measuring these properties is a challenge, though, because no two individuals from the same biological population are ever exactly alike. We can’t measure everyone in the population, either, so we are constrained by time and funding to limit our measurements to a sample of individuals drawn from the population. Sampling brings uncertainty to the project because, by chance, properties of the sample are not the same as the true values in the population. Thus, measurements made from a sample are affected by who happened to get sampled and who did not. Statistics is the study of methods to describe and measure aspects of nature from samples. Crucially, statistics gives us tools to quantify the uncertainty of these measures—that is, statistics makes it possible to determine the likely magnitude of their departure from the truth. Statistics is about estimation, the process of inferring an unknown quantity of a target population using sample data. Properly applied, the tools for estimation allow us to approximate almost everything about populations using only samples. Examples range from the average flying speed of bumblebees, to the risks of exposure to cell phones, to the variation in beak size of finches on a remote Galápagos island. We can estimate the proportion of people with a particular disease who die per year and the fraction who recover when treated. Estimation is the process of inferring an unknown quantity of a population using sample data. Most importantly, we can assess differences between groups and relationships between variables. For example, we can estimate the effects of different drugs on the possibility of recovery, we can measure the association between the lengths of horns on male antelopes and their success at attracting mates, and we can determine by how much the survival of women and children during shipwrecks differs from that of men. All of these quantities describing populations—namely, averages, proportions, measures of variation, and measures of relationship—are called parameters. Statistical methods tell us how best to estimate these parameters using our measurements of a sample. The parameter is the truth, and the estimate (also known as the statistic) is an approximation of the truth, subject to error. If we were able to measure every possible member of the population, we could know the parameter without error, but this is rarely possible. Instead, we use estimates based on incomplete data to approximate this true value. With the right statistical tools, we can determine just how good our approximations are. A parameter is a quantity describing a population, whereas an estimate or statistic is a related quantity calculated from a sample. Statistics is also about hypothesis testing. A statistical hypothesis is a specific claim regarding a population parameter. Hypothesis testing uses data to evaluate evidence for or against statistical hypotheses. Examples are “The mean effect of this new drug is not different from that of its predecessor,” and “Inhibition of the Tbx4 gene changes the rate of limb development in chick embryos.” Biological data usually get more interesting and informative if they can resolve competing claims about a population parameter.

Statistical methods have become essential in almost every area of biology—as indispensable as the PCR machine, calipers, binoculars, and the microscope. This book presents the ideas and methods needed to use statistics effectively, so that we can improve our understanding of nature. Chapter 1 begins with an overview of samples—how they should be gathered and the conclusions that can be drawn from them. We also discuss the types of variables that can be measured from samples, introducing terms that will be used throughout the book.

Sampling populations Our ability to obtain reliable measures of population characteristics—and to assess the uncertainty of these measures—depends critically on how we sample populations. It is often at this early step in an investigation that the fate of a study is sealed, for better or worse, as Example 1.2 demonstrates. EXAMPLE 1.2 Raining cats In an article published in the Journal of the American Veterinary Medical Association, Whitney and Mehlhaff (1987) presented results on the injury rates of cats that had plummeted from buildings in New York City, according to the number of floors they had fallen. Fear not: no experimental scientist tossed cats from different altitudes to obtain the data for this study. Rather, the cats had fallen (or jumped) of their own accord. The researchers were merely recording the fates of the sample of cats that ended up at a veterinary hospital for repair. The damage caused by such falls was dubbed Feline High-Rise Syndrome, or FHRS.1

Not surprisingly, cats that fell five floors fared worse than those dropping only two, and those falling seven or eight floors tended to suffer even more (see Figure 1.2-1). But the astonishing result was that things got better after that. On average, the number of injuries was reduced in cats that fell more than nine floors. This was true in every injury category. Their injury rates approached that of cats that had fallen only two floors! One cat fell 32 floors and walked away with only a chipped tooth.

FIGURE 1.2-1 A graph plotting the average number of injuries sustained per cat

according to the number of stories fallen. Numbers in parentheses indicate number of cats. Modified from Diamond (1988). This effect cannot be attributed to the ability of cats to right themselves so as to land on their feet—a cat needs less than one story to do that. The authors of the article put forth a more surprising explanation. They proposed that after a cat attains terminal velocity, which happens after it has dropped six or seven floors, the falling cat relaxes, and this change to its muscles cushions the impact when the cat finally meets the pavement.

Remarkable as these results seem, aspects of the sampling procedure raise questions. A clue to the problem is provided by the number of cats that fell a particular number of floors, indicated along the horizontal axis of Figure 1.2-1. No cats fell just one floor, and the number of cats falling increases with each floor from the second floor to the fifth. Yet, surely, every building in New York that has a fifth floor has a fourth floor, too, with open windows no less inviting. What can explain this curious trend? To answer this, keep in mind that the data are a sample of cats. The study was not carried out on the whole population of cats that fell from New York buildings. Our strong suspicion is that the sample is biased. Not all fallen cats were taken to the vet, and the chance of a cat making it to the vet might have been affected by the number of stories it had fallen. Perhaps most cats that tumble out of a first- or second-floor window suffer only indignity, which is untreatable. Any cat appearing to suffer no physical damage from a fall of even a few stories may likewise skip a trip to the vet. At the other extreme, a cat fatally plunging 20 stories might also avoid a trip to the vet, heading to the nearest pet cemetery instead. This example illustrates the kinds of questions of interpretation that arise if samples are biased. If the sample of cats delivered to the vet clinic is, as we suspect, a distorted subset of all the cats that fell, then the measures of injury rate and injury severity will also be distorted. We cannot say whether this bias is enough to cause the surprising downturn in injuries at a high number of stories fallen. At the very least, though, we can say that, if the chances of a cat making it to the vet depends on the number of stories fallen, the relationship between injury rate and number of floors fallen will be distorted. Good samples are a foundation of good science. In the rest of this section, we give an overview of the concept of sampling, what we are trying to accomplish when we take a sample, and the inferences that are possible when researchers get it right.

Populations and samples The first step in collecting any biological data is to decide on the target population. A population is the entire collection of individual units that a researcher is interested in. Ordinarily, a population is composed of a large number of individuals—so many that it is not possible to measure them all. Examples of populations include ■ ■ ■ ■ ■

all cats that have fallen from buildings in New York City, all the genes in the human genome, all individuals of voting age in Australia, all paradise flying snakes in Borneo, and all children in Vancouver, Canada, suffering from asthma.

A sample is a much smaller set of individuals selected from the population.2 The researcher uses this sample to draw conclusions that, hopefully, apply to the whole population. Examples include ■ ■ ■ ■ ■

the fallen cats brought to one veterinary clinic in New York City, a selection of 20 human genes, all voters in an Australian pub, eight paradise flying snakes caught by researchers in Borneo, and a selection of 50 children in Vancouver, Canada, suffering from asthma.

A population is all the individual units of interest, whereas a sample is a subset of units taken from the population. In most of the above examples, the basic unit of sampling is literally a single individual. However, in one example, the sampling unit was a single gene. Sometimes the basic unit of sampling is a group of individuals, in which case a sample consists of a set of such groups. Examples of units that are groups of individuals include a single family, a colony of microbes, a plot of ground in a field, an aquarium of fish, and a cage of mice. Scientists use several terms to indicate the sampling unit, such as unit, individual, subject, or replicate.

Properties of good samples Estimates based on samples are doomed to depart somewhat from the true population characteristics simply by chance. This chance difference from the truth is called sampling error. The spread of estimates resulting from sampling error indicates the precision of an estimate. The lower the sampling error, the higher the precision. Larger samples are less affected by chance and so, all else being equal, larger samples will have lower sampling error and higher precision than smaller samples. Sampling error is the difference between an estimate and the population parameter being estimated caused by chance. Ideally, our estimate is accurate (or unbiased), meaning that the average of estimates that we might obtain is centered on the true population value. If a sample is not properly taken, measurements made on it might systematically underestimate (or overestimate) the population parameter. This is a second kind of error called bias. Bias is a systematic discrepancy between the estimates we would obtain, if we could sample a population again and again, and the true population characteristic. The major goal of sampling is to minimize sampling error and bias in estimates. Figure 1.22 illustrates these goals by analogy with shooting at a target. Each point represents an estimate of the population bull’s-eye (i.e., of the true characteristic). Multiple points represent different estimates that we might obtain if we could sample the population repeatedly. Ideally, all the estimates we might obtain are tightly grouped, indicating low sampling error, and they are centered on the bull’s-eye, indicating low bias. Estimates are precise if the values we might

obtain are tightly grouped and highly repeatable, with different samples giving similar answers. Estimates are accurate if they are centered on the bull’s-eye. Estimates are imprecise, on the other hand, if they are spread out, and they are biased (inaccurate) if they are displaced systematically to one side of the bull’s-eye. The shots (estimates) on the upper right-hand target in Figure 1.2-2 are widely spread out but centered on the bull’s-eye, so we say that the estimates are accurate but imprecise. The shots on the lower left-hand target are tightly grouped but not near the bull’s-eye, so we say that they are precise but inaccurate. Both precision and accuracy are important, because a lack of either means that an estimate is likely to differ greatly from the truth.

FIGURE 1.2-2 Analogy between estimation and target shooting. An accurate estimate is centered around the bull’s-eye, whereas a precise estimate has low spread.

With sampling, we also want to be able to quantify the precision of an estimate. There are several quantities available to measure precision, which we discuss in Chapter 4. The sample of cats in Example 1.2 falls short in achieving some of these goals. If uninjured and dead cats do not make it to the pet hospital, then estimates of injury rate are biased. Injury rates for cats falling only two or three floors are likely to be over-estimated, whereas injury rates for cats falling many stories might be underestimated.

Random sampling The common requirement of the methods presented in this book is that the data come from a random sample. A random sample is a sample from a population that fulfills two criteria. In a random sample, each member of a population has an equal and independent chance of being selected. First, every unit in the population must have an equal chance of being included in the sample. This is not as easy as it sounds. A botanist estimating plant growth might be more likely to find the taller individual plants or to collect those closer to the road. Some members of

animal or human populations may be difficult to collect because they are shy of traps, never answer the phone, ignore questionnaires, or live at greater depths or distances than other members. These hard-to-sample individuals might differ in their characteristics from those of the rest of the population, so underrepresenting them in samples would lead to bias. Second, the selection of units must be independent. In other words, the selection of any one member of the population must in no way influence the selection of any other member.3 This, too, is not easy to ensure. Imagine, for example, that a sample of adults is chosen for a survey of consumer preferences. Because of the effort required to contact and visit each household to conduct an interview, the lazy researcher is tempted to record the preferences of multiple adults in each household and add their responses to those of other adults in the sample. This approach violates the criterion of independence, because the selection of one individual has increased the probability that another individual from the same household will also be selected. This will distort the sampling error in the data if individuals from the same household have preferences more similar to one another than would individuals randomly chosen from the population at large. With non-independent sampling, our sample size is effectively smaller than we think. This, in turn, will cause us to miscalculate the precision of the estimates. In general, the surest way to minimize bias and allow sampling error to be quantified is to obtain a random sample.4 Random sampling minimizes bias and makes it possible to measure the amount of sampling error.

How to take a random sample Obtaining a random sample is easy in principle but can be challenging in practice. A random sample can be obtained by using the following procedure: 1. Create a list of every unit in the population of interest, and give each unit a number between one and the total population size. 2. Decide on the number of units to be sampled (call this number n). 3. Using a random-number generator,5 generate n random integers between one and the total number of units in the population. 4. Sample the units whose numbers match those produced by the random-number generator. An example of this process is shown in Figure 1.2-3. In both panels of the figure, we’ve drawn the locations of all 5699 trees present in 2001 in a carefully mapped tract of Harvard Forest in Massachussets, USA (Barker-Plotkin et al. 2006). Every tree in this population has a unique number between 1 and 5699 to identify it. We used a computerized random-number generator to pick n = 20 random integers between 1 and 5699, where 20 is the desired sample size. The 20 random integers, after sorting, are as follows: 156, 167, 232, 246, 826, 1106, 1476, 1968, 2084, 2222, 2223, 2284, 2790, 2898, 3103, 3739, 4315, 4978, 5258, 5500

FIGURE 1.2-3 The locations of all 5699 trees present in the Prospect Hill Tract of Harvard Forest in 2001 (green circles). The red dots in the left panel are a random sample of 20 trees. The squares in the right panel are a random sample of 20 quadrats (each 20 feet on a side).

These 20 randomly chosen trees are identified by red dots in the left panel of Figure 1.2-3. How realistic are the requirements of a random sample? Creating a numbered list of every individual member of a population might be feasible for patients recorded in a hospital database, for children registered in an elementary-school system, or for some other populations for which a registry has been built. The feat is impractical for most plant populations, however, and unimaginable for most populations of animals or microbes. What can be done in such cases? One answer is that the basic unit of sampling doesn’t have to be a single individual—it can be a group, instead. For example, it is easier to use a map to divide a forest tract into many equal-sized blocks or plots and then to create a numbered list of these plots than it is to produce a numbered list of every tree in the forest. To illustrate this second approach, we divided the Harvard Forest tract into 836 plots of 400 square feet each. With the aid of a random-number generator, we then identified a random sample of 20 plots, which are identified by the squares in the right panel of Figure 1.2-3. The trees contained within a random sample of plots do not constitute a random sample of trees, for the same reason that all of the adults inhabiting a random sample of households do not constitute a random sample of adults. Trees in the same plot are not sampled independently; this can cause problems if trees growing next to one another in the same plot are more similar (or more different) in their traits than trees chosen randomly from the forest. The data in this case must be handled carefully. A simple technique is to take the average of the measurements of all of the individuals within a unit as the single independent observation for that unit. Random numbers should always be generated with the aid of a computer. Haphazard numbers made up by the researcher are not likely to be random (see Example 19.1). Most spreadsheet programs and statistical software packages on the computer include randomnumber generators.

The sample of convenience One undesirable alternative to the random sample is the sample of convenience, a sample

based on individuals that are easily available to the researcher. The researchers must assume (i.e., dream) that a sample of convenience is unbiased and independent like a random sample, but there is no way to guarantee it. A sample of convenience is a collection of individuals that are easily available to the researcher. The main problem with the sample of convenience is bias, as the following examples illustrate: ■ If the only cats measured are those brought to a veterinary clinic, then the injury rate of cats that have fallen from high-rise buildings is likely to be underestimated. Uninjured and fatally injured cats are less likely to make it to the vet and into the sample. ■ The spectacular collapse of the North Atlantic cod fishery in the last century was caused in part by overestimating cod densities in the sea, which led to excessive allowable catches by fishing boats (Walters and Maguire 1996). Density estimates were too high because they relied heavily on the rates at which the fishing boats were observed to capture cod. However, the fishing boats tended to concentrate in the few remaining areas where cod were still numerous, and they did not randomly sample the entire fishing area (Rose and Kulka 1999). A sample of convenience might also violate the assumption of independence if individuals in the sample are more similar to one another in their characteristics than individuals chosen randomly from the whole population. This is likely if, for example, the sample includes a disproportionate number of individuals who are friends or who are related to one another.

Volunteer bias Human studies in particular must deal with the possibility of volunteer bias, which is a bias resulting from a systematic difference between the pool of volunteers (the volunteer sample) and the population to which they belong. The problem arises when the behavior of the subjects affects their chance of being sampled. In a large experiment to test the benefits of a polio vaccine, for example, participating schoolchildren were randomly chosen to receive either the vaccine or a saline solution (serving as the control). The vaccine proved effective, but the rate at which children in the saline group contracted polio was found to be higher than in the general population. Perhaps parents of children who had not been exposed to polio prior to the study, and therefore had no immunity, were more likely to volunteer their children for the study than parents of kids who had been exposed (Brownlee 1955, Bland 2000). Compared with the rest of the population, volunteers might be ■ more health conscious and more proactive; ■ low-income (if volunteers are paid); ■ more ill, particularly if the therapy involves risk, because individuals who are dying anyway might try anything; ■ more likely to have time on their hands (e.g., retirees and the unemployed are more likely to answer telephone surveys);

■ more angry, because people who are upset are sometimes more likely to speak up; or ■ less prudish, because people with liberal opinions about sex are more likely to speak to surveyors about sex. Such differences can cause substantial bias in the results of studies. Bias can be minimized, however, by careful handling of the volunteer sample, but the resulting sample is still inferior to a random sample.

Data in the real world In this book we use real data, hard-won from observational or experimental studies in the lab and field and published in the literature. Do the samples on which the studies are based conform to the ideals outlined above? Alas, the answer is often no. Random samples are desired but often not achieved by researchers working in the trenches. Sometimes, the only data available are a sample of convenience or a volunteer sample, as the falling cats in Example 1.2 demonstrate. Scientists deal with this problem by taking every possible step to obtain random samples. If random sampling is impossible, then it is important to acknowledge that the problem exists and to point out where biases might arise in their studies.6 Ultimately, further studies should be carried out that attempt to control for any sampling problems evident in earlier work.

Types of data and variables With a sample in hand, we can begin to measure variables. A variable is any characteristic or measurement that differs from individual to individual. Examples include running speed, reproductive rate, and genotype. Estimates (e.g., average running speed of a random sample of 10 lizards) are also variables, because they differ by chance from sample to sample. Data are the measurements of one or more variables made on a sample of individuals. Variables are characteristics that differ among individuals.

Categorical and numerical variables Variables can be categorical or numerical. Categorical variables describe membership in a category or group. They describe qualitative characteristics of individuals that do not correspond to a degree of difference on a numerical scale. Categorical variables are also called attribute or qualitative variables. Examples of categorical variables include Categorical data are qualitative characteristics of individuals that do not have magnitude on a numerical scale. ■ ■ ■ ■ ■ ■ ■

survival (alive or dead), sex chromosome genotype (e.g., XX, XY, XO, XXY, or XYY), method of disease transmission (e.g., water, air, animal vector, or direct contact), predominant language spoken (e.g., English, Mandarin, Spanish, Indonesian, etc.), life stage (e.g., egg, larva, juvenile, subadult, or adult), snakebite severity score (e.g., minimal severity, moderate severity, or very severe), and size class (e.g., small, medium, or large).

A categorical variable is nominal if the different categories have no inherent order. Nominal means “name.” Sex chromosome genotype, method of disease transmission, and predominant language spoken are nominal variables. In contrast, the values of an ordinal categorical variable can be ordered. Unlike numerical data, the magnitude of the difference between consecutive values is not known. Ordinal means “having an order.” Life stage, snakebite severity score, and size class are ordinal categorical variables. A variable is numerical when measurements of individuals are quantitative and have magnitude. These variables are numbers. Measurements that are counts, dimensions, angles, rates, and percentages are numerical. Examples of numerical variables include ■ ■ ■ ■

core body temperature (e.g., degrees Celsius, °C), territory size (e.g., hectares), cigarette consumption rate (e.g., average number per day), age at death (e.g., years),

■ number of mates, and ■ number of amino acids in a protein. Numerical data are quantitative measurements that have magnitude on a numerical scale. Numerical data are either continuous or discrete. Continuous numerical data can take on any real-number value within some range. Between any two values of a continuous variable, an infinite number of other values are possible. In practice, continuous data are rounded to a predetermined number of digits, set for convenience or by the limitations of the instrument used to take the measurements. Core body temperature, territory size, and cigarette consumption rate are continuous variables. In contrast, discrete numerical data come in indivisible units. Number of amino acids in a protein and numerical rating of a statistics professor in a student evaluation are discrete numerical measurements. Number of cigarettes consumed on a specific day is a discrete variable, but the rate of cigarette consumption is a continuous variable when calculated as an average number per day over a large number of days. In practice, discrete numerical variables are often analyzed as though they were continuous, if there is a large number of possible values. Just because a variable is indexed by a number does not mean it is a numerical variable. Numbers might also be used to name categories (e.g., family 1, family 2, etc.). Numerical data can be reduced to categorical data by grouping, though the result contains less information (e.g., “above average” and “below average”).

Explanatory and response variables One major use of statistics is to relate one variable to another, by examining associations between variables and differences between groups. Measuring an association is equivalent to measuring a difference, statistically speaking. Showing that “the proportion of survivors differs between treatment categories” is the same as showing that the variables “survival” and “treatment” are associated. Often when association between two variables is investigated, a goal is to assess how well one of the variables, deemed the explanatory variable, predicts or affects the other variable, called the response variable. When conducting an experiment, the treatment variable (the one manipulated by the researcher) is the explanatory variable, and the measured effect of the treatment is the response variable. For example, the administered dose of a toxin in a toxicology experiment would be the explanatory variable, and organism survival would be the response variable. When neither variable is manipulated by the researcher, their association might nevertheless be described by the “effect” of one of the variables (the explanatory) on the other (the response), even though the association itself is not direct evidence for causation. For example, when exploring the possibility that high blood pressure affects the risk of stroke in a sample of people, blood pressure is the explanatory variable and incidence of strokes is the response variable. When natural groups of organisms, such as populations or species, are compared in some measurement, such as body mass, the group variable (population or species) is typically the explanatory variable and the measurement is the response variable. In more complicated studies involving more than two variables, there may be more than one explanatory or response variable. Sometimes you will hear variables referred to as “independent” and “dependent.” These are

the same as explanatory and response variables, respectively. Strictly speaking, if one variable depends on the other, then neither is independent, so we prefer to use explanatory and response throughout this book.

Frequency distributions and probability distributions Different individuals in a sample will have different measurements. We can see this variability with a frequency distribution. The frequency of a specific measurement in a sample is the number of observations having a particular value of the measurement.7 The frequency distribution shows how often each value of the variable occurs in the sample. The frequency distribution describes the number of times each value of a variable occurs in a sample. Figure 1.4-1 shows the frequency distribution for the measured beak depths of a sample of 100 finches from a Galápagos island population.8

FIGURE 1.4-1 The frequency distribution of beak depths in a sample of 100 finches from a Galápagos island population (Boag and Grant 1984). The vertical axis indicates the frequency, the number of observations in each 0.5-mm interval of beak depth.

The large-beaked ground finch on the Galápagos Islands.

We use the frequency distribution of a sample to inform us about the distribution of the variable (beak depth) in the population from which it came. Looking at a frequency distribution gives us an intuitive understanding of the variable. For example, we can see which values of beak depth are common and which are rare, we can get an idea of the average beak depth, and we can start to understand how variable beak depth is among the finches living on the island. The distribution of a variable in the whole population is called its probability distribution. The real probability distribution of a population in nature is almost never known. Researchers typically use theoretical probability distributions to approximate the real probability distribution. For a continuous variable like beak depth, the distribution in the population is often approximated by a theoretical probability distribution known as the normal distribution. The normal distribution drawn in Figure 1.4-2, for example, approximates the probability distribution of beak depths in the finch population from which the sample of 100 birds was drawn.

FIGURE 1.4-2 A normal distribution. This probability distribution is often used to approximate the distribution of a variable in the population from which a sample has been drawn.

The normal distribution is the familiar “bell curve.” It is the most important probability distribution in all of statistics. You’ll learn a lot more about it in Chapter 10. Most of the methods presented in this book for analyzing data depend on the normal distribution in some way.

Types of studies Data in biology are obtained from either an experimental study or an observational study. In an experimental study, the researcher assigns different treatment groups or values of an explanatory variable randomly to the individual units of study. A classic example is the clinical trial, where different treatments are assigned randomly to patients in order to compare responses. In an observational study, on the other hand, the researcher has no control over which units fall into which groups. A study is experimental if the researcher assigns treatments randomly to individuals, whereas a study is observational if the assignment of treatments is not made by the researcher. Studies of the health consequences of cigarette smoking in humans are all observational studies, because it is ethically impossible to assign smoking and nonsmoking treatments to human beings to assess the effects of smoking. The individuals in the sample have made the decision themselves about whether to smoke. The only experimental studies of the health consequences of smoking have been carried out on nonhuman subjects, such as mice, where researchers can assign smoking and nonsmoking treatments randomly to individuals. The distinction between experimental studies and observational studies is that experimental studies can determine cause-and-effect relationships between variables, whereas observational studies can only point to associations. An association between smoking and lung cancer might be due to the effects of smoking per se, or perhaps to an underlying predisposition to lung cancer in those individuals prone to smoking. It is difficult to distinguish these alternatives with observational studies alone. For this reason, experimental studies of the health hazards of smoking in nonhuman animals have helped make the case that cigarette smoking is dangerous to human health. But experimental studies are not always possible, even on animals. Smoking in humans, for example, is also associated with a higher suicide rate (Hemmingsson and Kriebel 2003). Is this association caused by the effects of smoking, or is it caused by the effects of some other variable? Just because a study was carried out in the laboratory does not mean that the study is an experimental study in the sense described here. A complex laboratory apparatus and careful conditions may be necessary to obtain measurements of interest, but such a study is still observational if the assignment of treatments is out of the control of the researcher.

Summary ■ Statistics is the study of methods for measuring aspects of populations from samples and for quantifying the uncertainty of the measurements. ■ Much of statistics is about estimation, which infers an unknown quantity of a population using sample data. ■ Statistics also allows hypothesis testing, a method to determine how well hypotheses about a population parameter fit the sample data. ■ Sampling error is the chance difference between an estimate describing a sample and the corresponding parameter of the whole population. Bias is a systematic discrepancy between an estimate and the population quantity. ■ The goals of sampling are to increase the accuracy and precision of estimates and to ensure that it is possible to quantify precision. ■ In a random sample, every individual in a population has the same chance of being selected, and the selection of individuals is independent. ■ A sample of convenience is a collection of individuals easily available to a researcher, but it is not usually a random sample. ■ Volunteer bias is a systematic discrepancy in a quantity between the pool of volunteers and the population. ■ Variables are measurements that differ among individuals. ■ Variables are either categorical or numerical. A categorical variable describes which category an individual belongs to, whereas a numerical variable is expressed as a number. ■ The frequency distribution describes the number of times each value of a variable occurs in a sample. A probability distribution describes the number of times each value occurs in a population. Probability distributions in populations can often be approximated by a normal distribution. ■ In studies of association between two variables, one variable is typically used to predict the value of another variable and is designated as the explanatory variable. The other variable is designated as the response variable. ■ In experimental studies, the researcher is able to assign subjects randomly to different treatments or groups. In observational studies, the assignment of individuals to treatments is not controlled by the researcher.

PRACTICE PROBLEMS Answers to the practice problems are provided at the end of the book, starting on page 747. 1. Which of the following numerical variables are continuous? Which are discrete? a. Number of injuries sustained in a fall b. Fraction of birds in a large sample infected with avian flu virus c. Number of crimes committed by a randomly sampled individual d. Logarithm of body mass 2. The peppered moth (Biston betularia) occurs in two types: peppered (speckled black and white) and melanic (black). A researcher wished to measure the proportion of melanic individuals in the peppered moth population in England, to examine how this proportion changed from year to year in the past. To accomplish this, she photographed all the peppered moth specimens available in museums and large private collections and grouped them by the year in which they had been collected. Based on this sample, she calculated the proportion of melanic individuals in every year. The people who collected the specimens, she knew, would prefer to collect whichever type was rarest in any given year, since those would be the most valuable. a. Can the specimens from any given year be considered a random sample from the moth population? b. If not a random sample, what type of sample is it? c. What type of error might be introduced by the sampling method when estimating the proportion of melanic moths?

3. What feature of an estimate—precision or accuracy—is most strongly affected when individuals differing in the variable of interest do not have an equal chance of being selected? 4. In a study of stress levels in U.S. army recruits stationed in Iraq, researchers obtained a complete list of the names of recruits in Iraq at the time of the study. They listed the recruits alphabetically and then numbered them consecutively. One hundred random numbers between one and the total number of recruits were then generated using a random-number

generator on a computer. The 100 recruits whose numbers corresponded to those generated by the computer were interviewed for the study. a. What is the population of interest in this study? b. The 100 recruits interviewed were randomly sampled as described. Is the sample affected by sampling error? Explain. c. What are the main advantages of random sampling in this example? d. What effect would a larger sample size have had on sampling error? 5. An important quantity in conservation biology is the number of plant and animal species inhabiting a given area. To survey the community of small mammals inhabiting Kruger National Park in South Africa, a large series of live traps were placed randomly throughout the park for one week in the main dry season of 2004. Traps were set each evening and checked the following morning. Individuals caught were identified, tagged (so that new captures could be distinguished from recaptures), and released. At the end of the survey, the total number of small mammal species in the park was estimated by the total number of species captured in the survey. a. What is the parameter being estimated in the survey? b. Is the sample of individuals captured in the traps likely to be a random sample? Why or why not? In your answer, address the two criteria that define a sample as random. c. Is the number of species in the sample likely to be an unbiased estimate of the total number of small mammal species in the park? Why or why not? 6. In a recent study, researchers took electrophysiological measurements from the brains of two rhesus macaques (monkeys). Forty neurons were tested in each monkey, yielding a total of 80 measurements. a. Do the 80 neurons constitute a random sample? Why or why not? b. If the 80 measurements were analyzed as though they constituted a random sample, what consequences would this have for the estimate of the measurement in the monkey population? 7. Identify which of the following variables are discrete and which are continuous: a. Number of warts on a toad b. Survival time after poisoning c. Temperature of porridge d. Number of bread crumbs in 10 meters of trail e. Length of wolves’ canines 8. A study was carried out in women to determine whether the psychological consequences of having an abortion differ from those experienced by women who have lost their fetuses by other causes at the same stage of pregnancy. a. Which is the explanatory variable in this study, and which is the response variable? b. Was this an observational study or an experimental study? Explain. 9. For each of the following studies, say which is the explanatory variable and which is the response variable. Also, say whether the study is observational or experimental. a. Forestry researchers wanted to compare the growth rates of trees growing at high altitude to that of trees growing at low altitude. They measured growth rates using the space between tree rings in a set of trees harvested from a natural forest.

b. Researchers randomly assign diabetes patients to two groups. In the first group, the patients receive a new drug tasploglutide, whereas patients in the other group receive standard treatment without the new drug. The researchers compared the rate of insulin release in the two groups. c. Psychologists tested whether the frequency of illegal drug use differs between people suffering from schizophrenia and those not having the disease. They measured drug use in a group of schizophrenia patients and compared it with that in a similar sized group of randomly chosen people. d. Spinner Hansen et al. (2008) studied a species of spider whose females often eat males that are trying to mate with them. The researchers removed a leg from each male spider in one group (to make them weaker and more vulnerable to being eaten) and left the males in another group undamaged. They studied whether survival of males in the two groups differed during courtship. e. Bowen et al. (2012) studied the effects of advanced communication therapy for patients whose communication skills had been affected by previous strokes. They randomly assigned two therapies to stroke patients. One group received advanced communication therapy and the other received only social visits without formal therapy. Both groups otherwise received normal, best-practice care. After six months, the communication ability (as measured by a standardized quantitative test score) was measured on all patients. 10. Each of the examples (a–e) in problem 9 involves estimating or testing an association between two variables. For each of the examples, list the two variables and state whether each is categorical or numerical. 11. A random sample of 500 households was identified in a major North American city using the municipal voter registration list. Five hundred questionnaires went out, directed at one adult in each household, and surveyed respondents about attitudes regarding the municipal recycling program. Eighty of the 500 surveys were filled out and returned to the researchers. a. Can the 80 households that returned questionnaires be regarded as a random sample of households? Explain. b. What type of bias might affect the survey outcome? 12. State whether the following represent cases of estimation or hypothesis testing. a. A random sample of quadrats in Olympic National Forest is taken to determine the average density of Ensatina salamanders. b. A study is carried out to determine whether there is extrasensory perception (ESP), by counting the number of cards guessed correctly by a subject isolated from a second person who is drawing cards randomly from a normal card deck. The number of correct guesses is compared with the number we would expect by chance if there were no such thing as ESP. c. A trapping study measures the rate of fruit fall in forest clear-cuts. d. An experiment is conducted to calculate the optimal number of calories per day to feed captive sugar gliders (Petaurus breviceps) to maintain normal body mass and good health. e. A clinical trial is carried out to determine whether taking large doses of vitamin C benefits health of advanced cancer patients. f. An observational study is carried out to determine whether hospital emergency room admissions increase during nights with a full moon compared with other nights. 13. A researcher dissected the retinas of 20 randomly sampled fish belonging to each of two

subspecies of guppy in Venezuela. Using a sophisticated laboratory apparatus, he measured the two groups of fish to find the wavelengths of visible light to which the cones of the retina were most sensitive. The goal was to explore whether fish from the two subspecies differed in the wavelength of light that they were most sensitive to. a. What are the two variables being associated in this study? b. Which is the explanatory variable and which is the response variable? c. Is this an experimental study or an observational study? Why?

ASSIGNMENT PROBLEMS 14. Identify whether the following variables are numerical or categorical. If numerical, state whether the variable is discrete or continuous. If categorical, state whether the categories have a natural order (ordinal) or not (nominal). a. Number of sexual partners in a year b. Petal area of rose flowers c. Heartbeats per minute of a Tour de France cyclist, averaged over the duration of the race d. Birth weight e. Stage of fruit ripeness (e.g., underripe, ripe, or overripe) f. Angle of flower orientation relative to position of the sun g. Tree species h. Year of birth i. Gender 15. In the vermilion flycatcher, the males are brightly colored and sing frequently and prominently. Females are more dull-colored and make less sound. In a field study of this bird, a researcher attempted to estimate the fraction of individuals of each sex in the population. She based her estimate on the number of individuals of each sex detected while walking through suitable habitat. Is her sample of birds detected likely to be a random sample? Why or why not?

16. Not all telephone polls carried out to estimate voter or consumer preferences make calls to cell phones. One reason is that in the USA, automated calls (“robocalls”) to cell phones are not permitted and interviews conducted by humans are more costly. a. How might the strategy of leaving out cell phones affect the goal of obtaining a random sample of voters or consumers? b. Which criterion of random sampling is most likely to be violated by the problems you identified in part (a): equal chance of being selected, or the independence of the selection of individuals? c. Which attribute of estimated consumer preference is most affected by the problem you identified in (a): accuracy or precision? 17. The average age of piñon pine trees in the coast ranges of California was investigated by

placing 500 ten-hectare plots randomly on a distribution map of the species in California using a computer. Researchers then found the location of each random plot in the field, and then they measured the age of every piñon pine tree within each of the 10-hectare plots. The average age within the plot was used as the unit measurement. These unit measurements were then used to estimate the average age of California piñon pines. a. What is the population of interest in this study? b. Why did the researchers take an average of the ages of trees within each plot as their unit measurement, rather than combine into a single sample the ages of all the trees from all the plots? 18. Refer to problem 17. a. Is the estimate of age based on 500 plots influenced by sampling error? Why? b. How would the sampling error of the estimate of mean age change if the investigators had used a sample of only 100 plots? 19. In each of the following examples, indicate which variable is the explanatory variable and which is the response variable. a. The anticoagulant warfarin is often used as a pesticide against house mice, Mus musculus. Some populations of the house mouse have acquired a mutation in the vkorc1 gene from hybridizing with the Algerian mouse, M. spretus (Song et al. 2011). In the Algerian mice, this gene confers resistance to warfarin. In a hypothetical follow-up study, researchers collected a sample of house mice to determine whether presence of the vkorc1 mutation is associated with warfarin resistance in house mice as well. They fed warfarin to all the mice in a sample and compared survival between the individuals possessing the mutation and those not possessing the mutation. b. Cooley et al. (2009) randomly assigned either of two treatments, naturopathic care (diet counseling, breathing techniques, vitamins, and a herbal medicine) or standardized psychotherapy (psychotherapy with breathing techniques and a placebo added), to 81 individuals having moderate to severe anxiety. Anxiety scores decreased an average of 57% in the naturopathic group and 31% in the psychotherapy group. c. Individuals highly sensitive to rewards tend to experience more food cravings and are more likely to be overweight or develop eating disorders than other people. Beaver et al. (2006) recruited 14 healthy volunteers and scored their reward sensitivity using a questionnaire (they were asked to answer yes or no to questions like: “I’m always willing to try something new if I think it will be fun”). The subjects were then presented with images of appetizing foods (e.g., chocolate cake, pizza) while activity of their fronto– striatal– amygdala–midbrain was measured using functional MRI. Reward sensitivity of subjects was found to correlate with brain activity in response to the images. d. Endostatin is a naturally occurring protein in humans and mice that inhibits the growth of blood vessels. O’Reilly et al. (1997) investigated its effects on growth of cancer tumors, whose growth and spread requires blood vessel proliferation. Mice having lung carcinoma tumors were randomly divided into groups that were treated with doses of 0, 2.5, 10, or 20 mg/kg of endostatin injected once daily. They found that higher doses of endostatin led to inhibition of tumor growth. 20. For each of the studies presented in problem 19, indicate whether the study is an experimental or observational study. 21. In a study of heart rate in ocean-diving birds, researchers harnessed 10 randomly sampled,

wild-caught cormorants to a laboratory contraption that monitored vital signs. Each cormorant was subjected to six artificial “dives” over the following week (one dive per day). A dive consisted of rapidly immersing the head of the bird in water by tipping the harness. In this way, a sample of 60 measurements of heart rate in diving birds was obtained. Do these 60 measurements represent a random sample? Why or why not? 22. Researchers sent out a survey to U.S. war veterans that asked a series of questions, including whether individuals surveyed were smokers or nonsmokers (Seltzer et al. 1974). They found that nonsmokers were 27% more likely than smokers to respond to a survey within 30 days (based on the much larger number of smokers and nonsmokers who eventually responded). Hypothetically, if the study had ended after 30 days, what effect would this have on the estimate of the proportion of veterans who smoke? (Use terminology you learned in this chapter to describe the effect.) 23. During World War II, the British Royal Air Force estimated the density of bullet holes on different sections of planes returning to base from aerial sorties. Their goal was to use this information to determine which plane sections most needed additional protective shields. (It was not possible to reinforce the whole plane, because it would weigh too much.) They found that the density of holes was highest on the wings and lowest on the engines and near the cockpit, where the pilot sits (their initial conclusion, that therefore the wings should be reinforced, was later shown to be mistaken). What is the main problem with the sample: bias or large sampling error? What part of the plane should have been reinforced?

24. In a study of diet preferences in leafcutter ants, a researcher presented 20 randomly chosen ant colonies with leaves from the two most common tree species in the surrounding forest. The leaves were placed in piles of 100, one pile for each tree species, close to colony entrances. Leaves were cut so that each was small enough to be carried by a single ant. After 24 hours, the researcher returned and counted the number of leaves remaining of the original 100 of each species. Some of the results are shown in the following table. Tree species

Number of leaves removed

Spondius mombin Sapium thelocarpum

1561 851

Total

2412

Using these results, the researcher estimated the proportion of Spondius leaves taken as 0.65 and concluded that the ants have a preference for leaves of this species. a. Identify the two variables whose association is displayed in the table. Which is the explanatory variable and which is the response variable? Are they numeric or categorical?

b. Why do the 2412 leaves used in the calculation of the proportion not represent a random sample? c. Would treating the 2412 leaves as a random sample most likely affect the accuracy of the estimate of diet preference or the precision of the estimate? d. If not the leaves, what units were randomly sampled in the study? 25. Garaci et al. (2012) examined a sample of people with and without multiple sclerosis (MS) to test the controversial idea that the disease is caused by blood flow restriction resulting from a vein condition known as chronic cerebrospinal venous insufficiency (CCSVI). Of 39 randomly sampled patients with MS, 25 were found to have CCSVI and 14 were not. Of 26 healthy control subjects, 14 were found to have CCSVI and 12 were not. The researchers found an association between CCSVI and MS. a. What is the explanatory variable and what is the response variable? b. Is this an experimental study or an observational study? c. Where might hypothesis testing have been used in the study?

1 INTERLEAF

Biology and the history of statistics

Sir Ronald Fisher

The formal study of probability began in the mid-17th century when Blaise Pascal and Pierre de Fermat started to describe the mathematical rules for determining the best gambling strategies. Gambling and insurance continued to motivate the development of probability for the next couple of centuries. The application of probability to data did not happen until much later. The importance of variation in the natural world, and by extension to samples from that world, was made obvious by Charles Darwin’s theory of natural selection. Darwin highlighted the importance of variation in biological systems as the raw material of evolution. Early followers of Darwin saw the need for quantitative descriptions of variation and the need to incorporate the effects of sampling error in biology. This led to the development of modern statistics. In many ways, therefore, modern statistics was an offshoot of evolutionary biology. One of the first pioneers in statistical data analysis was Francis Galton, who began to apply probability thinking to the study of all sorts of data. Galton was a real polymath, thinking about nearly everything and collecting and analyzing data at every chance. He said, “Whenever you can, count.”1 He invented the study of fingerprints, he tested whether prayer increased the life span of preachers compared with others of the middle class, and he quantified the heritable nature of many important traits. He once recorded his idea of the attractiveness of women seen from the window of a train headed from London to Glasgow, finding that “attractiveness” (at least according to Galton) declined as a function of distance from London. Galton is best known, though, for his twin interests in data analysis and evolution (he was the first cousin of

Darwin). He invented the idea of regression, which we will learn more about in Chapter 17. Galton was also responsible for establishing a lab that brought more researchers into the study of both statistics and biology, including Karl Pearson and Ronald Fisher. Karl Pearson, like Galton, was interested in many spheres of knowledge. Pearson was motivated by biological data—in particular, by heredity and evolution. Pearson was responsible for our most often-used measure of the correlation between numerical variables. In fact, the correlation coefficient that we will learn about in Chapter 16 is often referred to as Pearson’s correlation coefficient. He also made many contributions to the study of regression analysis and invented the χ2 contingency test (Chapter 9). He also invented the term standard deviation (Chapter 3). Last, but definitely not least, Ronald Fisher was one of the great geniuses of the 20th century. Fisher is well known in evolutionary biology as one of the three founders of the field of theoretical population genetics. Among his many contributions are the demonstration that Mendelian inheritance is compatible with the continuous variation we see in many traits, the accepted theory for why most animals conceive equal numbers of male and female offspring, and a great deal of the mathematical machinery we use to describe the process of evolution. But his contributions did not end there. He is probably also the most important figure in the history of statistics. He developed the analysis of variance (Chapter 15), likelihood (Chapter 20), the Pvalue (Chapter 6), randomized experiments (Chapter 14), multiple regression, and many other tools used in data analysis. His mathematical knowledge was made practical by a lifelong association with biologists, particularly agricultural scientists at the Rothamsted Experimental Station in England. Fisher solved problems associated with the analysis of real data as he encountered them, and he generalized them for application to many other related questions. Moreover, Fisher developed experimental designs that would give more information than might otherwise be possible. What this short hagiography is intended to demonstrate is that the early history of statistics is tightly bound up with biology. Biological questions motivated the development of most of the basic statistical tools, and biologists helped to develop those tools. Biology and statistics naturally go hand in hand.

2

Displaying data

The human eye is a natural pattern detector, adept at spotting trends and exceptions in visual displays. For this reason, biologists spend hours creating and examining visual summaries of their data—graphs and, to a lesser extent, tables. Effective graphs enable visual comparisons of measurements between groups, and they expose relationships between different variables. They are also the principal means of communicating results to a wider audience. Florence Nightingale (1858) was one of the first persons to put graphs to good use. In her famous wedge diagrams, redrawn in the figure above, she visualized the causes of death of British troops during the Crimean War. The number of cases is indicated by the area of a wedge, and the cause of death by color. The diagrams showed convincingly that disease was the main cause of soldier deaths during the wars, not wounds or other causes. With these vivid graphs, she successfully campaigned for military and public health measures that saved many lives. Effective graphs are a prerequisite for good data analysis, revealing general patterns in the

data that bare numbers cannot. Therefore, the first step in any data analysis or statistical procedure is to graph the data and look at it. Humans are a visual species, with brains evolved to process visual information. Take advantage of millions of years of evolution, and look at visual representations of your data before doing anything else. We’ll follow this prescription throughout the book. In this chapter, we explain how to produce effective graphical displays of data and how to avoid common pitfalls. We’ll then review which types of graphs best show the data. The choice will depend on the type of data, numerical or categorical, and whether the goal is to show measurements of one variable or the association between two variables. There is often more than one way to show the same pattern in data, and we will compare and evaluate successful and unsuccessful approaches. We will also mention a few tips for constructing tables, which should also be laid out to show patterns in data.

Guidelines for effective graphs Graphs are vital tools for analyzing data. They are also used to communicate patterns in data to a wider audience in the form of reports, slide shows, and web content. The two purposes, analysis and presentation, are largely coincident because the most revealing displays will be the best both for identifying patterns in the data and for communicating these patterns to others. Both purposes require displays that are clear, honest, and efficient. To motivate principles of effective graphs, let’s highlight some common ways in which researchers might get it wrong.

How to draw a bad graph Figure 2.1-1 shows the results of an experiment in which maize plants were grown in pots under three nitrogen regimes and two soil water contents. Height of bars represents the average maize yield (dry weight per plant) at the end of the experiment under the six combinations of water and nitrogen. The data are real (Quaye et al. 2009), but we made the bad graph intentionally to highlight four common defects. Examine the graph before reading further and try to recognize some of them. Many graphics packages on the computer make it easy to produce flawed graphs like this one, which is probably why we still encounter them so often.

FIGURE 2.1-1 An example of a defective graph showing mean plant height of maize grown in pots under different nitrogen and water treatments.

Mistake #1: Where are the data? Each bar in Figure 2.1-1 represents average yield of four plant pots assigned to that nitrogen and water treatment. The data points—yields of all the experimental units (pots)—are nowhere to be seen. This means we are unable to see the variation in yield between pots and compare it with the magnitude of differences between treatments. It means that any unusual observations that might distort the calculation of average yield remain hidden. It would be a challenge to add the data points to this particular graph because the bars are in the way. We’ll say more later about when bars are appropriate and when they are not. Mistake #2: Patterns in the data are difficult to see. The three dimensions and angled perspective make it difficult to judge bar height by eye, which means that average plant growth is difficult to compare between treatments. In his classic book on information graphics, Tufte

(1983) referred to 3-D and other visual embellishments as “chartjunk.” Chartjunk adds clutter that dilutes information and interferes with the ability of the eye and brain to “see” patterns in data. Mistake #3: Magnitudes are distorted. The vertical axis on the graph, plant yield, ranges from 2 to 9 g/plant, rather than 0 to 9, which means that bar height is out of proportion to actual magnitudes. Mistake #4: Graphical elements are unclear. Text and other figure elements are too small to read easily.

How to draw a good graph A few straightforward principles will help to make sure that your graphs do not end up with the kind of problems illustrated in Figure 2.1-1. We follow these four rules ourselves in the remainder of the book. ■ ■ ■ ■

Show the data. Make patterns in the data easy to see. Represent magnitudes honestly. Draw graphical elements clearly.

Show the data, first and foremost (Tufte 1983). A good graph allows you to visualize the measurements and helps the eye detect patterns in the data. Showing the data makes it possible to evaluate the shape of the distribution of data points and to compare measurements between groups. It helps you to spot potential problems, such as extreme observations, which will be useful as you decide the next step of your data analysis. Figure 2.1-2 gives an example of what it means to show data. The study examined the role of the neurotransmitter serotonin1 in bringing about a transition in social behavior, from solitary to gregarious, in a desert locust (Anstey et al. 2009). This behavior change is a critical point in the production of huge locust swarms that blacken skies and ravage crops in many parts of the world. Each data point is the serotonin level of one of 30 locusts experimentally caged at high density for 0, 1, or 2 hours, with 0 representing the control. The panel on the left of Figure 2.1-2 shows the data (this type of graph is called a strip chart or dot plot). The panel on the right of Figure 2.1-2 hides the data, using bars to show only treatment averages.

FIGURE 2.1-2 A graph that shows the data (left) and a graph that hides the data (right). Data points are serotonin levels in the central nervous system of desert

locusts, Schistocerca gregaria, that were experimentally crowded for 0 (the control group), 1, and 2 hours. The data points in the left panel were perturbed a small amount to the left or right to minimize overlap and make each point easier to see. The horizontal bars in the left panel indicate the mean (average) serotonin level in each group. The graph on the right shows only the mean serotonin level in each treatment (indicated by bar height). Note that the vertical axis does not have the same scale in the two graphs.

In the left panel of Figure 2.1-2, we can see lots of scatter in the data in each treatment group and plenty of overlap between groups. We see that most points fall below the treatment average, and that each group has a few extreme observations. Nevertheless, we can see a clear shift in serotonin levels of locusts between treatments. All this information is missing from the right panel of Figure 2.1-2, which uses more ink yet shows only the averages of each treatment group. Make patterns easy to see. Try displaying your data in different ways, possibly with different types of graphs, to find the best way to communicate the findings. Is the main pattern in the data recognizable right away? If not, try again with a different method. Stay away from 3-D effects and elaborate chartjunk that obscures the patterns in the data. In the rest of the chapter we’ll compare alternative ways of graphing the same data sets and discuss their effectiveness. Avoid putting too much information into one graph. Remember the purpose of a graph: to communicate essential patterns to eyes and brains. The purpose is not to cram as much data as possible into each graph. Think about getting the main point across with one or two key graphs in the main body of your presentation. Put the remainder into an appendix or online supplement if it is important to show them to a subset of your audience. Represent magnitudes honestly. This sounds easy, but misleading graphics are common in the scientific literature. One of the most important decisions concerns the smallest value on the vertical axis of a graph (the “baseline”). A bar graph must always have a baseline at zero, because the eye instinctively reads bar height and area as proportional to magnitude. The upper bar graph in Figure 2.1-3 shows an example, depicting government spending on education each year since 1998 in British Columbia. The area of each bar is not proportional to the magnitude of the value displayed. As a result, the graph exaggerates the differences. The figure falsely suggests that spending increased twenty-fold over time, but the real increase is less than 20%. It is more honest to plot the bars with a baseline of zero, as in the lower graph in Figure 2.1-3 (the revised graph also removed the 3-D effects and the numbers above bars to make the pattern easier to see).

FIGURE 2.1-3 Upper graph: A bar graph, taken from data from a British Columbia government brochure, indicating spending per student in different years. Lower graph: A revised presentation of the same data, in which the magnitude of the spending is proportional to the height and area of bars. This revision also removed the 3-D effects and the numbers above bars to make the pattern easier to see. The upper graph is modified from data from British Columbia Ministry of Education (2004).

Other types of graphs, such as strip charts, don’t always need a zero baseline if the main goal is to show differences between treatments rather than proportional magnitudes. Draw graphical elements clearly. Clearly label the axes and choose unadorned, simple typefaces and colors. Text should be legible even after the graph is shrunk to fit the final document. Always provide the units of measurement in the axis label. Use clearly distinguishable graphical symbols if you plot with more than one kind. Don’t always accept the default output of statistical or spreadsheet programs. Up to a tenth of your male audience is red-green color-blind, so choose colors that differ in intensity and apply redundant coding to distinguish groups (for example, use distinctive shapes or patterns as well as different colors).2 A good graph is like a good paragraph. It conveys information clearly, concisely, and without distortion. A good graph requires careful editing. Just as in writing, the first draft is rarely as good as the final product.

Showing data for one variable To examine data for single variable, we show its frequency distribution. Recall from Chapter 1 that the frequency of occurrence of a specific measurement in a sample is the number of observations having that particular measurement. The frequency distribution of a variable is the number of occurrences of all values in the data. Relative frequency is the proportion of observations having a given measurement, calculated as the frequency divided by the total number of observations. The relative frequency distribution is the proportion of occurrences of each value in the data set. The relative frequency distribution describes the fraction of occurrences of each value of a variable.

Showing categorical data: frequency table and bar graph Let’s start with displays for a categorical variable. A frequency table is a text display of the number of occurrences of each category in the data set. A bar graph uses the height of rectangular bars to visualize the frequency (or relative frequency) of occurrence of each category. A bar graph uses the height of rectangular bars to display the frequency distribution (or relative frequency distribution) of a categorical variable. Example 2.2A illustrates both kinds of displays. EXAMPLE 2.2A Crouching tiger Conflict between humans and tigers threatens tiger populations, kills people, and reduces public support for conservation. Gurung et al. (2008) investigated causes of human deaths by tigers near the protected area of Chitwan National Park, Nepal. Eighty-eight people were killed by 36 individual tigers between 1979 and 2006, mainly within 1 km of the park edge. Table 2.2-1 lists the main activities of people at the time they were killed. Such information may be helpful to identify activities that increase vulnerability to attack. TABLE 2.2-1 Frequency table showing the activities of 88 people at the time they were attacked and killed by tigers near Chitwan National Park, Nepal, from 1979 to 2006. Activity Frequency (number of people) Collecting grass or fodder for livestock Collecting non-timber forest products Fishing Herding livestock Disturbing tiger at its kill

44 11 8 7 5

Collecting fuel wood or timber Sleeping in a house Walking in forest Using an outside toilet Total

5 5 3 2 88

Table 2.1-1 is a frequency table showing the number of deaths associated with each activity. Here, alternative values of the variable “activity” are listed in a single column, and frequencies of occurrence are listed next to them in a second column. The categories have no intrinsic order, but comparing the frequencies of each activity is made easier by arranging the categories in order of their importance, from the most frequent at the top to the least frequent at the bottom. The table shows that more people were killed while collecting grass and fodder for their livestock than when doing any other activity. The number of deaths under this activity was four times that of the next category of activity (collecting non-timber forest products) and is related to the amount of time people spend carrying out these activities. The differences in frequency stand out even more vividly in the bar graph shown in Figure 2.2-1. In a bar graph, frequency is depicted by the height of rectangular bars. Unlike a frequency table, a bar graph does not usually present the actual numbers. Instead, the graph gives a clear picture of how steeply the numbers drop between categories. Some activities are much more common than others, and we don’t need the actual numbers to see this.

FIGURE 2.2-1 Bar graph showing the activities of people at the time they were attacked and killed by tigers near Chitwan National Park, Nepal, between 1979 and 2006. Total number of deaths: n = 88. The frequencies are taken from Table 2.2-1, which also gives more detailed labels of activities.

Making a good bar graph The top edge of each bar conveys all the information about frequency, but the eye also compares the areas of the bars, which must therefore be of equal width. It is crucial that the baseline of the y-axis is at zero—otherwise, the area and height of bars are out of proportion with actual magnitudes and so are misleading. When the categorical variable is nominal, as in Figure 2.2-1 and Table 2.2-1, the best way to arrange categories is by frequency of occurrence. The most frequent category goes first, the next most frequent category goes second, and so on. This aids in the visual presentation of the information. For an ordinal categorical variable, such as snakebite severity score, the values should be in the natural order (e.g., minimally severe, moderately severe, and very severe). Bars should stand apart, not be fused together. It is a good habit to provide the total number of observations (n) in the figure legend.

A bar graph is usually better than a pie chart The pie chart is another type of graph often used to display frequencies of a categorical variable. This method uses colored wedges around the circumference of a circle to represent frequency or relative frequency. Figure 2.2-2 shows the tiger data again, this time in a pie chart. This graphical method is reminiscent of Florence Nightingale’s wedge diagram shown at the beginning of this chapter.

FIGURE 2.2-2 Pie chart of the activities of people at the time they were attacked and killed by tigers near Chitwan National Park, Nepal. The frequencies are taken from Table 2.2-1. Total number of deaths: n = 88.

The pie chart has received a lot of criticism from experts in information graphics. One reason is that while it is straightforward to visualize the frequency of deaths in the first and most frequent category (Collecting grass/fodder), it is more difficult to compare frequencies in the remaining categories by eye. This problem worsens as the number of categories increases. Another reason is that it is very difficult to compare frequencies between two or more pie charts side by side, especially when there are many categories. To compensate, pie charts are often drawn with the frequencies added as text around the circle perimeter. The result is not better than a table. The shape of a frequency distribution is more readily perceived in a bar graph than a pie chart, and it is easier to compare frequencies between two or more bar graphs than between pie charts. Use the bar graph instead of the pie chart for showing frequencies in categorical data.

Showing numerical data: frequency table and histogram A frequency distribution for a numerical variable can be displayed either in a frequency table or in a histogram. A histogram uses area of rectangular bars to display frequency. The data values are split into consecutive intervals, or “bins,” usually of equal width, and the frequency of observations falling into each bin is displayed. A histogram uses the area of rectangular bars to display the frequency distribution (or relative frequency distribution) of a numerical variable. We discuss how histograms are made in greater detail using the data in Example 2.2B. EXAMPLE 2.2B Abundance of desert bird species How many species are common in nature and how many are rare? One way to address this question is to construct a frequency distribution of species abundance. The data in Table 2.2-2 are from a survey of the breeding birds of Organ Pipe Cactus National Monument in southern Arizona, USA. The measurements were extracted from the North American Breeding Bird Survey, a continent-wide data set of estimated bird numbers (Sauer et al. 2003).

TABLE 2.2-2 Data on the abundance of each species of bird encountered during four surveys in Organ Pipe Cactus National Monument. Species Abundance Greater roadrunner Black-chinned hummingbird Western kingbird Great-tailed grackle Bronzed cowbird Great horned owl Costa’s hummingbird Canyon wren Canyon towhee Harris’s hawk Loggerhead shrike Hooded oriole Northern mockingbird American kestrel Rock dove Bell’s vireo Common raven Northern cardinal House sparrow Ladder-backed woodpecker Red-tailed hawk Phainopepla Turkey vulture Violet-green swallow Lesser nighthawk Scott’s oriole Purple martin Black-throated sparrow Brown-headed cowbird

1 1 1 1 1 2 2 2 2 3 3 4 5 7 7 10 12 13 14 15 16 18 23 23 25 28 33 33 59

Black vulture

64

Lucy’s warbler Gilded flicker

67 77

Brown-crested flycatcher Mourning dove Gambel’s quail Black-tailed gnatcatcher Ash-throated flycatcher

128 135 148 152 173

Curve-billed thrasher Cactus wren Verdin House finch

173 230 282 297

Gila woodpecker White-winged dove

300 625

We treated each bird species in the survey as the unit of interest and the abundance of a species in the survey as its measurement. The range of abundance values was divided into 13 intervals of equal width (0–50, 50–100, and so on), and the number of species falling into each abundance interval was counted and presented in a frequency table to help see patterns (Table 2.2-3). TABLE 2.2-3 Frequency distribution of bird species abundance at Organ Pipe Cactus National Monument. Abundance Frequency (Number of species) 0–50 50–100 100–150 150–200 200–250 250–300 300–350 350–400 400–450 450–500 500–550 550–600 600–650

28 4 3 3 1 2 1 0 0 0 0 0 1

Total

43

Source: Data are from Table 2.2-2.

Although the table shows the numbers, the shape of the frequency distribution is more obvious in a histogram of these same data (Figure 2.2-3). Here, frequency (number of species) in each abundance interval is perceived as bar area.

FIGURE 2.2-3 Histogram illustrating the frequency distribution of bird species abundance at Organ Pipe Cactus National Monument. Total number of bird species: n = 43.

The frequency table and histogram of the bird abundance data reveal that the majority of bird species have low abundance. Frequency falls steeply with increasing abundance.3 The white-winged dove (pictured in Example 2.2B) is exceptionally abundant at Organ Pipe Cactus National Monument, accounting for a large fraction of all individual birds encountered in the survey.

Describing the shape of a histogram The histogram reveals the shape of a frequency distribution. Some of the most common shapes are displayed in Figure 2.2-4. Any interval of the frequency distribution that is noticeably more frequent than surrounding intervals is called a peak. The mode is the interval corresponding to the highest peak. For example, a bell-shaped frequency distribution has a single peak (the mode) in the center of the range of observations. A frequency distribution having two distinct peaks is said to be bimodal. The mode is the interval corresponding to the highest peak in the frequency distribution.

FIGURE 2.2-4 Some possible shapes of frequency distributions.

A frequency distribution is symmetric if the pattern of frequencies on the left half of the histogram is the mirror image of the pattern on the right half. The uniform distribution and the

bell-shaped distribution in Figure 2.2-4 are symmetric. If a frequency distribution is not symmetric, we say that it is skewed. The distribution in Figure 2.2-4 labeled “Asymmetric” has left or negative skew: it has a long tail extending to the left. The distribution in Figure 2.2-4 labeled “Bimodal” is also asymmetric but is positively skewed: its long tail is to the right.4 The abundance data for desert bird species also have positive skew (Figure 2.2-3), which means they have a long tail extending to the right. Skew refers to asymmetry in the shape of a frequency distribution for a numerical variable. Extreme data points lying well away from the rest of the data are called outliers. The histogram of desert bird abundance (Figure 2.2-3) includes one extreme observation (the whitewinged dove) that falls well outside the range of abundance of other bird species. The whitewinged dove, therefore, is an outlier. Outliers are common in biological data. They can result from mistakes in recording the data or, as in the case of the white-winged dove, they may represent real features of nature. Outliers should always be investigated. They should be removed from the data only if they are found to be errors. An outlier is an observation well outside the range of values of other observations in the data set.

How to draw a good histogram When drawing a histogram, the choice of interval width must be made carefully because it can affect the conclusions. For example, Figure 2.2-5 shows three different histograms that depict the body mass of 228 female sockeye salmon (Oncorhynchus nerka) from Pick Creek, Alaska, in 1996 (Hendry et al. 1999). The leftmost histogram of Figure 2.2-5 was drawn using a narrow interval width. The result is a somewhat bumpy frequency distribution that suggests the existence of two or even more peaks. The rightmost histogram uses a wide interval. The result is a smoother frequency distribution that masks the second of the two dominant peaks. The middle histogram uses an intermediate interval that shows two distinct body-size groups. The fluctuations from interval to interval within size groups are less noticeable.

FIGURE 2.2-5 Body mass of 228 female sockeye salmon sampled from Pick Creek in Alaska (Hendry et al. 1999). The same data are shown in each case, but the interval widths are different: 0.1 kg (left), 0.3 kg (middle), and 0.5 kg (right).

To choose the ideal interval width we must decide whether the two distinct body-size groups are likely to be “real,” in which case the histogram should show both, or whether a

bimodal shape is an artifact produced by too few observations.5 When you draw a histogram, each bar must rise from a baseline of zero, so that the area of each bar is proportional to frequency. Unlike bar graphs, adjacent histogram bars are contiguous, with no spaces between them. This reinforces the perception of a numerical scale, with bars grading from one into the next. In this book we follow convention by placing an observation whose value is exactly at the boundary of two successive intervals into the higher interval. For example, the Gila woodpecker, with a total of 300 individuals observed (Table 2.2-2), is recorded in the interval 300–350, not in the interval 250–300. There are no strict rules about the number of intervals that should be used in frequency tables and histograms. Some computer programs use Sturges’s rule of thumb, in which the number of intervals is 1 + ln(n)/ln(2), where n is the number of observations and ln is the natural logarithm. The resulting number is then rounded up to the higher integer (Venables and Ripley 2002). Many regard this rule as overly conservative, and in this book we tend to use a few more intervals than Sturges. The number of intervals should be chosen to best show patterns and exceptions in the data, and this requires good judgment rather than strict rules. Computers allow you to try several alternatives to help you determine the best option. When breaking the data into intervals for the histogram, use readable numbers for breakpoints—for example, break at 0.5 rather than 0.483. Finally, it is a good idea to provide the total number of individuals in the accompanying legend.

Other graphs for numerical data The histogram is recommended for showing the frequency distribution of a single numerical variable. The box plot and the strip chart are alternatives, but most often these are used to show differences when there are data from two or more groups. We describe these graphs in the next section. Another type of graph, the cumulative frequency distribution, is explained in Chapter 3.

Showing association between two variables Here we illustrate how to show data for two variables simultaneously, rather than one at a time. The goal is to create an image that visualizes association or correlation between two variables and differences between groups. The most suitable type of graph depends on whether both variables are categorical, both are numerical, or one is of each data type.

Showing association between categorical variables If two categorical variables are associated, the relative frequencies for one variable will differ among categories of the other variable. To reveal such association, show the frequencies using a contingency table, a mosaic plot, or a grouped bar graph. Here’s an example. EXAMPLE 2.3A Reproductive effort and avian malaria Is reproduction hazardous to health? If not, then it is difficult to explain why adults in many organisms seem to hold back on the number of offspring they raise. Oppliger et al. (1996) investigated the impact of reproductive effort on the susceptibility to malaria6 in wild great tits (Parus major) breeding in nest boxes. They divided 65 nesting females into two treatment groups. In one group of 30 females, each bird had two eggs stolen from her nest, causing the female to lay an additional egg. The extra effort required might increase stress on these females. The remaining 35 females were left alone, establishing the control group. A blood sample was taken from each female 14 days after her eggs hatched to test for infection by avian malaria.

The association between experimental treatment and the incidence of malaria is displayed in Table 2.3-1. This table is known as a contingency table, a frequency table for two (or more) categorical variables. It is called a contingency table because it shows how the frequencies of the categories in a response variable (the incidence of malaria, in this case) are contingent upon the value of an explanatory variable (the experimental treatment group). A contingency table gives the frequency of occurrence of all combinations of two (or more) categorical variables. TABLE 2.3-1 Contingency table showing the incidence of malaria in female great tits in relation to experimental treatment. Experimental treatment group

Control group

Egg-removal group

Row total

7

15

22

No malaria

28

15

43

Column total

35

30

65

Malaria

Each experimental unit (bird) is counted exactly once in the four “cells” of Table 2.3-1, and so the total count (65) is the number of birds in the study. A cell is one combination of categories of the row and column variables in the table. The explanatory variable (experimental treatment) is displayed in the columns, whereas the response variable, the variable being predicted (incidence of malaria), is displayed in the rows. The frequency of subjects in each treatment group is given in the column totals, and the frequency of subjects with and without malaria is given in the row totals. According to Table 2.3-1, malaria was detected in 15 of the 30 birds subjected to egg removal, but in only seven of the 35 control birds. This difference between treatments suggests that the stress of egg removal, or the effort involved in producing one extra egg, increases female susceptibility to avian malaria. Table 2.3-1 is an example of a 2 × 2 (“two-by-two”) contingency table, because it displays the frequency of occurrence of all combinations of two variables, each having exactly two categories. Larger contingency tables are possible if the variables have more than two categories. Two types of graph work best for displaying the relationship between a pair of categorical variables. The grouped bar graph uses heights of rectangles to graph the frequency of occurrence of all combinations of two (or more) categorical variables. Figure 2.3-1 shows the grouped bar graph for the avian malaria experiments. Grouped bar graphs are like bar graphs for single variables, except that different categories of the response variable (e.g., malaria and no malaria) are indicated by different colors or shades. Bars are grouped by the categories of the explanatory variable treatment (control and egg removal), so make sure that the spaces between bars from different groups are wider than the spaces between bars separating categories of the response variable. We can see from the grouped bar graph in Figure 2.3-1 that incidence of malaria is associated with treatment, because the relative heights of the bars for malaria and no malaria differ between treatments. Most birds in the control group had no malaria (the gold bar is much taller than the red bar), whereas in the experimental group, the frequency of subjects with and without malaria was equal. A grouped bar graph uses the height of rectangular bars to display the frequency distributions (or relative frequency distributions) of two or more categorical variables.

FIGURE 2.3-1 Grouped bar graph for reproductive effort and avian malaria in great tits. The data are from Table 2.3-1, where n = 65 birds.

A mosaic plot is similar to a grouped bar plot except that bars within treatment groups are stacked on top of one another (Figure 2.3-2). Within a stack, bar area and height indicate the relative frequencies (i.e., the proportion) of the responses. This makes it easy to see the association between treatment and response variables: if an association is present in the data, then the vertical position at which the colors meet will differ between stacks. If no association is present, then the meeting point between the colors will be at the same vertical position between stacks. In Figure 2.3-2, for example, few individuals in the control group were infected with malaria, so the red bar (malaria) meets the gold bar (no malaria) at a higher vertical position than in the egg removal stack, where the incidence of malaria was greater. The mosaic plot uses the area of rectangles to display the relative frequency of occurrence of all combinations of two categorical variables.

FIGURE 2.3-2 Mosaic plot for reproductive effort and avian malaria in great tits. Red indicates birds with malaria, whereas gold indicates birds free of malaria. The data are from Table 2.3-1, where n = 65 birds.

Another feature of the mosaic plot is that the width of each vertical stack is proportional to the number of observations in that group. In Figure 2.3-2, the wider stack for the control group reflects the greater total number of individuals in this treatment (35) compared with the number in the egg-removal treatment (30). As a result, the total area of each box is proportional to the relative frequency of that combination of variables in the whole data set.

A mosaic plot provides only relative frequencies, not the absolute frequency of occurrence in each combination of variables. This might be considered a drawback, but keep in mind that the most important goal of graphs is to depict the pattern in the data rather than exact figures. Here, the pattern is the association between treatment and response variables: the difference in the relative frequencies of diseased birds in the two treatments. Of the three methods for presenting the same data—the contingency table, the mosaic plot, and the grouped bar graph—which is best? The answer depends on the circumstances, and it is a good idea to try all three to evaluate their effectiveness in any data set. It is usually easier to see differences in relative frequency between groups when the data are visualized in a grouped bar plot or mosaic plot than in a contingency table. On the other hand, a contingency table might work best if one of the response categories is vastly more frequent than the other, making it difficult to see the bars corresponding to rare categories in a graph, or if the explanatory and response variables have many categories, thus increasing the complexity of the graph. We find that association, or lack of association, is easier to see in a mosaic plot than in a grouped bar graph, but this will not always be the case. Deciding which type of display is most effective for a given circumstance is best done by trying several methods and choosing among them on the basis of information, clarity, and simplicity.

Showing association between numerical variables: scatter plot Use a scatter plot to show the association between two numerical variables. Position along the horizontal axis (the x-axis) indicates the measurement of the explanatory variable. The position along the vertical axis (the y-axis) indicates the measurement of the response variable. The pattern in the resulting cloud of points indicates whether an association between the two variables is positive (in which case the points tend to run from the lower left to the upper right of the graph), negative (the points run from the upper left to the lower right), or absent (no discernible pattern). Example 2.3B shows an example. EXAMPLE 2.3B Sins of the father The bright colors and elaborate courtship displays of the males of many species posed a problem for Charles Darwin: how can such elaborate traits evolve? His answer was that they evolved because females are attracted to them when choosing a mate. But why would females choose those kinds of males? One possible answer: females that choose fancy males have attractive sons as well. A recent laboratory study examined how attractive traits in guppies are inherited from father to son (Brooks 2000). The attractiveness of sons (a score representing the rate of visits by females to corralled males, relative to a standard) was compared with their fathers’ ornamentation (a composite index of several aspects of male color and brightness). The father’s ornamentation is the explanatory variable in the resulting scatter plot of these data (Figure 2.3-3).

FIGURE 2.3-3 Scatter plot showing the relationship between the ornamentation of male guppies and the average attractiveness of their sons. Total number of families: n = 36.

Each dot in the scatter plot is a father-son pair. The father’s ornamentation is the explanatory variable and the son’s attractiveness is the response variable. The plot shows a positive association between these variables (note how the points tend to run from the lower left to the upper right of the graph). Thus, the sexiest sons come from the most gloriously ornamented fathers, whereas unadorned fathers produce less attractive sons on average. A scatter plot is a graphical display of two numerical variables in which each observation is represented as a point on a graph with two axes.

Showing association between a numerical and a categorical variable There are several good methods to show an association between a numerical variable and a categorical variable. Three that we recommend are the strip chart (which we first saw in Figure 2.1-2), the box plot, and the multiple histograms method. Here we compare these methods with an example. We recommend against the common practice of using a bar graph because the bars make it difficult to show the data (bar graphs are ideal for frequency data). Showing an association between a numerical and a categorical variable is the same as showing a difference in the numerical variable between groups. EXAMPLE 2.3C Blood responses to high elevation The amount of oxygen obtained in each breath at high altitude can be as low as one-third that obtained at sea level. Studies have begun to examine whether indigenous people living at high elevations have physiological attributes that compensate for the reduced availability of oxygen. A reasonable expectation is that they should have more hemoglobin, the molecule that binds and transports oxygen in the blood. To test this, researchers sampled blood from males in three high-altitude human populations: the high Andes, high-elevation Ethiopia, and Tibet, along with a sea-level population from the USA (Beall et al. 2002). Results are shown in Figures 2.3-4 and 2.3-5.

FIGURE 2.3-4 Strip chart (left) and box plot (right) showing hemoglobin concentration in males living at high altitude in three different parts of the world: the Andes (71), Ethiopia (128), and Tibet (59). A fourth population of 1704 males living at sea level (USA) is included as a control.

FIGURE 2.3-5 Multiple histograms showing the hemoglobin concentration in males of

the four populations. The number of measurements in each group is given in Figure 2.3-4.

The left panel of Figure 2.3-4 shows the hemoglobin data with a strip chart (sometimes also called a dot plot). In a strip chart, each observation is represented as a dot on a graph showing its numerical measurement on one axis (here, the vertical or y-axis) and the category (group) to which it belongs on the other (here, horizontal or x-axis). A strip chart is like a scatter plot except the explanatory variable is categorical rather than numerical. It is usually necessary to spread, or “jitter,” the points along the horizontal axis to reduce overlap of points, so that they can be more easily seen. The strip chart method worked well in Figure 2.1-2, where there were few data points in each group. However, several of the male populations in Example 2.3C have so many observations that the points overlap too much in the strip chart, making it difficult to see the individual dots and their distribution (left panel of Figure 2.3-4). The strip chart is a graphical display of a numerical variable and a categorical variable in which each observation is represented as a dot. An alternative method to “show the data” is the box plot, which uses lines and rectangles to display a compact summary of the frequency distribution of the data (right panel of Figure 2.34). A box plot doesn’t show most of the data points, but instead uses lines and boxes to indicate where the bulk of the observations lie. The scale of the vertical axis is the same in both panels of Figure 2.3-4 so that you can see the correspondence between boxes and data points. A box plot is a graph that uses lines and a rectangular box to display the median, quartiles, range, and extreme measurements of the data. The line inside each box (right panel of Figure 2.3-4) is the median, the middle measurement of the group of observations. Half the observations lie above the median and half lie below. The lower and upper edge of each box are first and third quartiles. One-fourth of the observations lie below the first quartile (three-fourths lie above). Conversely, three-fourths of the observations lie below the third quartile (one-quarter lie above). Two lines, called whiskers, extend outward from a box at each end. The whiskers stop at the smallest and largest “non-extreme” values in the data. Extreme values are plotted as isolated dots past the ends of the whiskers. We explain these quantities in more detail, and how to calculate them, in Chapter 3. The box plot in Figure 2.3-4 shows key features of the four frequency distributions using just a few graphical elements. We can clearly see in this graph how only men from the high Andes had elevated hemoglobin concentrations, whereas men from high-elevation Ethiopia and Tibet were not noticeably different in hemoglobin concentration from the sea-level group.7 We can see that the shapes of distributions are relatively similar in the four groups of males—the span of the boxes is similar in the four groups, although greatest in the Andean males and least in the USA males. The lengths of the whiskers are also fairly similar. USA males have the most extreme observations, but the shape of the distribution, as indicated by the box and whiskers, is similar to that in the other groups. The third method uses multiple histograms, one for each category, to show the data, as shown in Figure 2.3-5. It is important that the histograms be stacked above one another as shown so that the position and spread of the data are most easily compared. Side-by-side histograms lose most of the advantages of the multiple histogram method for visualizing

association, because differences in the position of bars between groups are difficult to see. Use the same scale along the horizontal axis to allow comparison. Of the three methods for showing association between a numeric and a categorical variable (difference between groups), which is the best? The strip chart shows all the data points, which is ideal when there are only a few observations in each category. The box plot picks out a few of the most important features of the frequency distribution and is more suitable when the number of observations is large. The multiple histogram plot shows more features of the frequency distribution but takes up more space than the other two options. It works best when there are only a few categories. As usual, the best strategy is to try all three methods on your data and judge for that situation which method shows the association most clearly.

Showing trends in time and space Often a variable of interest represents a summary measurement taken at consecutive points in time or space. In this case, line graphs and maps are excellent visual tools.

Line graph A line graph is a powerful tool for displaying trends in time or other ordered series. Typically, one y-measurement is displayed for each point in time or space, which is displayed along the xaxis. Adjacent points along the x-axis are connected by a line segment. Example 2.4A illustrates the line graph. A line graph uses dots connected by line segments to display trends in a measurement over time or other ordered series. EXAMPLE 2.4A Bad science can be deadly Since the introduction of a measles vaccine, the number of cases in the U.K. dropped dramatically. A disease that once killed hundreds of people per year in the U.K. became a negligible risk as most of the population was immunized. However, recent declines in the fraction of people vaccinated, in part from unfounded fears concerning the safety of the vaccine,8 has caused a resurgence in the number of cases of measles (Jansen et al. 2003). The number of cases quarterly between 1995 and 2011 is shown in a line plot in Figure 2.4-1 (data from Health Protection Agency 2012).

FIGURE 2.4-1 Confirmed cases of measles in England and Wales from 1995 to 2011. The four numbers in each year refer to new cases in each quarter.

The trends in the number of measles cases over time are made more visible by the lines connecting the points in Figure 2.4-1. The steepness of the line segments reflects the speed of change in the number of cases from one quarter-year to the next. Notice, for example, how steeply the number of cases rises when an outbreak begins, and then how cases decline just as quickly afterward, as immunity spreads. When the baseline for the vertical axis is zero, as in the present example, the area under the curve between two time points is proportional to the total number of new cases in that period.

Maps A map is the spatial equivalent of the line graph, using a color gradient to display a numerical response variable at multiple locations on a surface. The explanatory variable is location in space. One measurement is displayed for each point or interval of the surface, as shown in Example 2.4B. EXAMPLE 2.4B Biodiversity hotspots South America is renowned for its extraordinary numbers of species. We tend to think of the vast expanse of lowland Amazon rainforest as the seat of this abundance. The data shown in Figure 2.4-2 are the numbers of plant species recorded at many points on a fine grid covering the northern part of South America. Points are colored such that “hotter” colors represent more plant species at each point. The image shows that peak diversities actually occur at the northwest coast, the nearby inland where the Andes mountains meet the lowland rainforest, and along the southeast coast of Brazil.

FIGURE 2.4-2 Map displaying numbers of plant species in northern South America. Colors reflect the numbers of species estimated at many points in a fine grid, with each point consisting of an area 100 km x 100 km. The color scale is on the right, with hotter colors reflecting more species. The horizontal gray line is the equator. Modified from Bass et al. (2010).

The map in Figure 2.4-2 contains an enormous amount of data, yet the pattern is easy to see. The regions of peak diversity, and those of relatively low diversity, are clearly evident. Maps can be used for measurements at points on any two-dimensional surface, including a spatial grid (such as in the plant species richness example) or at political or geological boundaries on a map. They can also be used to indicate measurements at locations on the surface of two- or three-dimensional objects, such as the brain or the body. For example, a visual representation of an MRI scan is also a map.

How to make good tables Tables have two functions: to communicate patterns and to record quantitative data summaries for further use. When the main function of a table is to display the patterns in the data—a “display table”—numerical detail is less important than the effective communication of results. This is the kind of table that would appear in the main body of a report or publication. Compact frequency tables are examples of display tables; for example, look at Table 2.2-3, which shows the number of bird species in a sequence of abundance categories. In this section we summarize strategies for making good display tables. The purpose of the second kind of table is to store raw data and numerical summaries for reference purposes. Such “data tables” are often large and are not ideal for recognizing patterns in data. They are inappropriate for communicating general findings to a wider audience. They are nevertheless often invaluable. Table 2.2-2, which lists the abundances of every bird species in a survey, is an example. Data tables aren’t usually included in the main body of a report. When published, they usually appear as appendices or online supplements, so specialized readers interested in more details can find them.

Follow similar principles for display tables Producing clear, honest, and efficient display tables should follow many of the same principles discussed already for graphs. In particular, ■ Make patterns in the data easy to see. ■ Represent magnitudes honestly. ■ Draw table elements clearly. Make patterns easy to see. Make the table compact and present as few significant digits as are necessary to communicate the pattern. Avoid putting too much data into one table. Arrange the rows and columns of numbers to facilitate pattern detection. For example, a series of numbers listed above one another in a single column are easier to compare with one another than the same numbers listed side by side in different columns. Our earlier recommendations for frequency tables apply here (Section 2.2). For example, list unordered categorical (nominal) data in order of importance (frequency) rather than alphabetically or otherwise. If the categorical variables have a natural order (such as life stages: zygote, fetus, newborn, adolescent, adult), they should be listed in that order. Represent magnitudes honestly. For example, when combining numbers into bins in frequency tables, use intervals of equal width so that the numbers can be more accurately compared. Draw table elements clearly. Clearly label row and column headers, and always provide units of measurement. Choose unadorned, simple fonts. Let’s look at an example of a display table and then consider how it might be improved. The data in Table 2.5-1 were put together by Alvarez et al. (2009) to investigate the idea that a strong preference for consanguineous marriages (inbreeding) within the line of Spanish Habsburg kings, which ruled Spain from 1516 to 1700, contributed to its downfall. The

quantity F is a measure of inbreeding in the offspring. F is zero if king and queen were unrelated, and F is 0.25 if they were brother and sister whose own parents were unrelated. Values may be lower or higher if there was inbreeding further generations back. TABLE 2.5-1 Inbreeding coefficient (F) of Spanish Habsburg kings and queens and survival of their progeny. Miscarriages Neonatal Later Survivors Survival Survival King/Queen F Pregnancies & stillbirths deaths deaths to age 10 (total) (postnatal) Ferdinand of Aragon Elizabeth of Castile Philip I Joanna I Charles I Isabella of Portugal Philip II Elizabeth of Valois Anna of Austria Philip III Margaret of Austria Philip IV Elizabeth of Bourbon Mariana of Austria

0.039

7

2

0

0

5

0.714

1.000

0.037

6

0

0

0

6

1.000

1.000

0.123

7

1

1

2

3

0.429

0.600

0.008

4

1

1

0

2

0.500

1.000

0.218

6

1

0

4

1

0.167

0.200

0.115

8

0

0

3

5

0.625

0.625

0.050

7

0

3

2

2

0.286

0.500

0.254

6

0

1

3

2

0.333

0.400

Source: Data are from Alvarez et al. (2009).

There is a tendency for less related kings and queens to produce a higher proportion of surviving offspring, but it is not so easy to see this in Table 2.5-1. Before reading any further, examine the table and make a note of any deficiencies. How might these deficiencies be overcome by modifying the table? Let’s apply the principles of effective display to improve this table. Consider that the main goal of producing the table should be to show a pattern, in this case a possible association between F and offspring survival. Here is a list of features of Table 2.5-1 that we felt made it difficult to see this pattern. ■ King/queen pairs are not ordered in such a way as to make it easy for the eye to see any association. ■ The main variables of interest, F and survival, are separated by intervening columns. ■ Blank lines are inserted for every new king listed, fragmenting any pattern. ■ The number of decimal places is overly large, making it difficult to read the numbers.

To overcome these problems, we have extracted the most crucial columns and reorganized them in Table 2.5-2. In this revised table, king and queen pairs are ordered by F value of the offspring, and survival has been placed in the adjacent column. Blank lines have been eliminated along with unnecessary columns, and decimals have been rounded to two places. TABLE 2.5-2 Inbreeding coefficient (F) of Spanish kings and queens and survival of their progeny. These data are extracted and reorganized from Table 2.5-1. Survival Survival Number of King/Queen F (postnatal) (total) pregnancies Philip II/Elizabeth of Valois Philip I/Joanna I Ferdinand/Elizabeth of Castile Philip IV/Elizabeth of Bourbon Philip III/Margaret of Austria Charles I/Isabella of Portugal Philip II/Anna of Austria Philip IV/Mariana of Austria

0.01 0.04

1.00 1.00

0.50 1.00

4 6

0.04

1.00

0.71

7

0.05

0.50

0.29

7

0.12 0.12 0.22 0.25

0.63 0.60 0.20 0.40

0.63 0.43 0.17 0.33

8 7 6 6

The revised Table 2.5-2 suggests that survival of more inbred progeny tends to be lower, at least when measured as postnatal survival. The trend appears weaker for total survival, which includes prenatal and neonatal survival. Just as for a graph, a good table must convey information clearly, concisely, and without distortion. A good table requires careful editing. See Ehrenberg (1977) for further insights into how to draw tables.

Summary ■ Graphical displays must be clear, honest, and efficient. ■ Strive to show the data, to make patterns in the data easy to see, to represent magnitudes honestly, and to draw graphical elements clearly. ■ Follow the same rules when constructing tables to reveal patterns in the data. ■ A frequency table is used to display a frequency distribution for categorical or numerical data. ■ Bar graphs and histograms are recommended graphical methods for displaying frequency distributions of categorical and numerical variables: Type of data Categorical data Numerical data

Graphical method Bar graph Histogram

■ Contingency tables describe the association between two (or more) categorical variables by displaying frequencies of all combinations of categories. ■ Recommended graphical methods for displaying associations between variables and differences between groups include the following: Types of data

Graphical method

Two numerical variables

Scatter plot Line plot (space or time) Map (space)

Two categorical variables

Grouped bar graph Mosaic plot

One numerical variable and Strip chart Box plot Multiple histograms one categorical variable Cumulative frequency distributions (Chapter 3)

PRACTICE PROBLEMS 1. Estimate by eye the relative frequency of the shaded areas in each of the following histograms.

2. Using a graphical method from this chapter, draw three frequency distributions: one that is symmetric, one that is skewed, and one that is bimodal. a. Identify the mode in each of your frequency distributions. b. Does your skewed distribution have negative or positive skew? c. Is your bimodal distribution skewed or symmetric? 3. In the southern elephant seal, males defend harems that may contain hundreds of reproductively active females. Modig (1996) recorded the numbers of females in harems in a population on South Georgia Island. The histograms of the data (below, drawn from data in Modig 1996) are unusual because the rarer, larger harems have been divided into wider intervals. In the upper histogram, bar height indicates the relative frequency of harems in the interval. In the lower histogram, bar height is adjusted such that bar area indicates relative frequency. Which histogram is correct? Why?

4. Draw scatter plots for invented data that illustrate the following patterns: a. Two numerical variables that are positively associated b. Two numerical variables that are negatively associated c. Two numerical variables whose relationship is nonlinear 5. A study by Miller et al. (2004) compared the survival of two kinds of Lake Superior rainbow trout fry (babies). Four thousand fry were from a government hatchery on the lake, whereas 4000 more fry came from wild trout. All 8000 fry were released into a stream flowing into the lake, where they remained for one year. After one year, the researchers found 78 survivors. Of these, 27 were hatchery fish and 51 were wild. Display these results in the most appropriate table. Identify the type of table you used. 6. The following data are the occurrences in 2012 of the different taxa in the list of endangered and threatened species under the U.S. Endangered Species Act (U.S. Fish and Wildlife Service 2012). The taxa are listed in no particular order in the table. Taxon Birds Clams Reptiles Fish Crustaceans Mammals Snails Flowering plants Amphibians Insects Arachnids

Number of species 93 83 36 152 22 85 40 782 26 66 12

a. Rewrite the table, but list the taxa in a more revealing order. Explain your reasons behind the ordering you choose.

b. What kind of table did you construct in part (a)? c. Choosing the most appropriate graphical method, display the number of species in each taxon. What kind of graph did you choose? Why? d. Should the baseline for the number of species in your graph in part (c) be 0 or 12, the smallest number in the data set? Why? e. Create a version of this table that shows the relative frequency of endangered species by taxon. 7. Can environmental factors influence the incidence of schizophrenia? A recent project measured the incidence of the disease among children born in a region of eastern China: 192 of 13,748 babies born in the midst of a severe famine in the region in 1960 later developed schizophrenia. This compared with 483 schizophrenics out of 59,088 births in 1956, before the famine, and 695 out of 83,536 births in 1965, after the famine (St. Clair et al. 2005). a. What two variables are compared in this example? b. Are the variables numerical or categorical? If numerical, are they continuous or discrete; if categorical, are they nominal or ordinal? c. Effectively display the findings in a table. What kind of table did you use? d. In each of the three years, calculate the relative frequency (proportion) of children born who later developed schizophrenia. Plot these proportions in a line graph. What pattern is revealed? 8. Human diseases differ in their virulence, which is defined as their ability to cause harm. Scientists are interested in determining what features of different diseases make some more dangerous to their hosts than others. The graph below depicts the frequency distribution of virulence measurements, on a log base 10 scale, of a sample of human diseases (modified from Ewald 1993). Diseases that spread from one victim to another by direct contact between people are shown in the upper graph. Those transmitted from person to person by insect vectors are shown in the lower graph.

a. Identify the type of graph displayed. b. What are the two groups being compared in this graph?

c. What variable is being compared between the two groups? Is it numerical or categorical? d. Explain the units on the vertical (y) axis. e. What is the main result depicted by this graph? 9. Examine the figure below, which indicates the date of first occurrence of rabies in raccoons in the townships of Connecticut, measured by the number of months following March 1, 1991 (modified from Smith et al. 2002).

a. Identify the type of graph shown. b. What is the response variable? c. What is the explanatory variable? d. What was the direction of spread of the disease (from where, to where, approximately)? 10. The following graph is taken from a study of married women who had been raised by adoptive parents (Bereczkei et al. 2004). It shows the facial resemblance between the women and their husbands (first bar), between their husbands and the women’s adoptive fathers (second bar), and between their husbands and the women’s adoptive mothers (third bar). Facial resemblance of a given woman to her husband was scored by 242 “judges,” each of whom was given a photograph of the woman, photos of three other women, and a photo of her husband. The judges were asked to decide from the photos which of the four women was the wife of the husband based on facial similarity. Her resemblance was scored as the percentage of judges who chose correctly. If there was no resemblance between a given woman and her husband, then by chance only one in four judges (25%) should have chosen correctly. Resemblance of the woman’s husband to the wife’s adoptive father and to the wife’s adoptive mother was measured in the same way.

a. Describe the essential findings displayed in the figure. b. Which two principles of good graph design are violated in this figure? 11. Each of the following graphs illustrates an association between two variables. For each graph identify (1) the type of graph, (2) the explanatory and response variables, and (3) the type of data (whether numerical or categorical) for each variable. a. Observed fruiting of individual plants in a population of Campanula americana according to the number of fruits produced previously (Richardson and Stephenson 1991):

b. The maximum density of wood produced at the end of the growing season in white spruce trees in Alaska in different years (modified from Barber et al. 2000):

c. Relative expression levels of Neuropeptide Y (NPY), a gene whose activity correlates with anxiety and is induced by stress, in the brains of people differing in their genotypes at the locus (Zhou et al. 2008).

12. The following data are from the Cambridge Study in Delinquent Development (see Problem 22). They examine the relationship between the occurrence of convictions by the end of the study and the family income of each boy when growing up. Three categories described income level: inadequate, adequate, and comfortable. Income level No convictions Convicted

Inadequate

Adequate

Comfortable

47 43

128 57

90 30

a. What type of table is this? b. Display these same data in a mosaic plot. c. What type of variable is “income level”? How should this affect the arrangement of groups in your mosaic plot in part (b)? d. By viewing the table above and the graph in part (b), describe any apparent association between family income and later convictions. e. In answering part (d), which method (the table or the graph) better revealed the association between conviction status and income level? Explain. 13. Each of the following graphs illustrates an association between two variables. For each graph, identify (1) the type of graph, (2) the explanatory and response variables, and (3) the type of data (whether numerical or categorical) for each variable. a. Taste sensitivity to phenylthiocarbamide (PTC) in a sample of human subjects grouped according to their genotype at the PTC gene—namely, AA, Aa, or aa (Kim et al. 2003):

b. Migratory activity (hours of nighttime restlessness) of young captive blackcaps (Sylvia atricapilla) compared with the migratory activity of their parents (Berthold and Pulido 1994):

c. Sizes of the second appendage (middle leg) of embryos of water striders, in 10 control embryos and 10 embryos dosed with RNAi for the developmental gene Ultrabithorax (Khila et al. 2009).

d. The frequency of injection-heroin users that share or do not share needles according to their known HIV infection status (Wood et al. 2001):

14. Spot the flaw. In an experimental study of gender and wages, Moss-Racusin et al. (2012) presented professors from research-intensive universities each with a job application for a laboratory manager position. The application was randomly assigned a male or female name, and the professors were asked to state the starting salary they would offer the candidate if hired. The average starting salary reported is compared in the following figure between applications with male names and female names. (The vertical lines at the top edge of each bar are “standard error bars”—we’ll learn about them in Chapter 4).

a. Identify at least two of the four principles of good graph design that are violated. b. What alternative graph type is ideal for these data? c. Identify the main pattern in the data (interestingly, this pattern was similar when male professors and female professors were examined separately). 15. How do the insects that pollinate flowers distinguish individual flowers with nectar from empty flowers? One possibility is that they can detect the slightly higher humidity of the air —produced by evaporation—in flowers that contain nectar. von Arx et al. (2012) recently tested this idea by manipulating the humidity of air emitted from artificial flowers that were otherwise identical. The following graph summarizes the number of visits to the two types of flowers by hawk moths (Hyles lineata).

a. What type of graph is this? b. What does the horizontal line in the center of each rectangle represent? c. What do the top and bottom edges of each rectangle represent? d. What are the vertical lines extending above and below each rectangle? e. Is an association apparent between the variables plotted? Explain.

16. For each of the graphs shown below, based on hypothetical data, identify the type of graph and say whether or not the two variables exhibit an association. Explain your answer in each case. FIGURE FOR PROBLEM 16

17. “Animal personality” has been defined as the presence of consistent differences between individuals in behaviors that persist over time. Do sea anemones have it? To investigate, Briffa and Greenaway (2011) measured the consistency of the startle response of individuals of wild beadlet anemones, Actinia equina, in tide pools in the U.K. When disturbed, such as with a mild jet of water (the method used in this study), the anemones retract their feeding tentacles to cover the oral disc, opening them again some time later. The accompanying table records the duration of the startle response (time to reopen, in seconds) of 12 individual anemones. Each anemone was measured twice, 14 days apart. TABLE FOR PROBLEM 17

Anemone:

1

2

3

4

5

6

7

8

9

10 11 12

Occasion one 1065 248 436 350 378 410 232 201 267 687 688 980 Occasion one

939 268 460 261 368 467 303 188 401 690 711 571

a. Choose the best method, and make a graph to show the association between the first and second measurements of startle response. b. Is a strong association present? In other words, does the beadlet anemone have animal personality? 18. Refer to the previous question. a. Draw a frequency distribution of startle durations measured on the first occasion. b. Describe the shape of the frequency distribution: is it skewed or symmetric? If skewed, say whether skew is positive or negative.

ASSIGNMENT PROBLEMS 19. Male fireflies of the species Photinus ignitus attract females with pulses of light. Flashes of longer duration seem to attract the most females. During mating, the male transfers a spermatophore to the female. Besides containing sperm, the spermatophore is rich in protein that is distributed by the female to her fertilized eggs. The data below are measurements of spermatophore mass (in mg) of 35 males (Cratsley and Lewis 2003). 0.047, 0.037, 0.041, 0.045, 0.039, 0.064, 0.064, 0.065, 0.079, 0.070, 0.066, 0.059, 0.075, 0.079, 0.090, 0.069, 0.066, 0.078, 0.066, 0.066, 0.055, 0.046, 0.056, 0.067, 0.075, 0.048, 0.077, 0.081, 0.066, 0.172, 0.080, 0.078, 0.048, 0.096, 0.097 a. Create a graph depicting the frequency distribution of the 35 mass measurements. b. What type of graph did you choose in part (a)? Why? c. Describe the shape of the frequency distribution. What are its main features? d. What term would be used to describe the largest measurement in the frequency distribution? 20. The accompanying graph depicts a frequency distribution of beak widths of 1017 blackbellied seedcrackers, Pyrenestes ostrinus, a finch from West Africa (Smith 1993).

a. What is the mode of the frequency distribution? b. Estimate by eye the fraction of birds whose measurements are in the interval representing the mode. c. There is a hint of a second peak in the frequency distribution between 15 and 16 mm. What strategy would you recommend be used to explore more fully the possibility of a second peak? d. What name is given to a frequency distribution having two distinct peaks?

21. When its numbers increase following favorable environmental conditions, the desert locust, Schistocerca gregaria, undergoes a dramatic transformation from a solitary, cryptic form into a gregarious form that swarms by the billions. The transition is triggered by mechanical stimulation—locusts bumping into one another. The accompanying figure shows the results of a laboratory study investigating the degree of gregariousness resulting from mechanical stimulation of different parts of the body (modified from Simpson et al. 2001).

a. Identify the type of graph displayed. b. Identify the explanatory and response variables. 22. The Cambridge Study in Delinquent Development was undertaken in north London (U.K.) to investigate the links between criminal behavior in young men and the socioeconomic factors of their upbringing (Farrington 1994). A cohort of 395 boys was followed for about 20 years, starting at the age of 8 or 9. All of the boys attended six schools located near the research office. The following table shows the total number of criminal convictions by the boys between the start and end of the study. Number of convictions 0 1 2 3 4 5 6 7

Frequency 265 49 21 19 10 10 2 2

8 9 10 11 12 13 14

4 2 1 4 3 1 2 Total:395

a. What type of table is this? b. How many variables are presented in this table? c. How many boys had exactly two convictions by the end of the study? d. What fraction of boys had no convictions? e. Display the frequency distribution in a graph. Which type of graph is most appropriate? Why? f. Describe the shape of the frequency distribution. Is it skewed or is it symmetric? Is it unimodal or bimodal? Where is the mode in number of criminal convictions? Are there outliers in the number of convictions? g. Does the sample of boys used in this study represent a random sample of British boys? Why or why not? 23. Swordfish have a unique “heater organ” that maintains elevated eye and brain temperatures when hunting in deep, cold water. The following graph illustrates the results of a study by Fritsches et al. (2005) that measured how the ability of swordfish retinas to detect rapid motion, measured by the flicker fusion frequency, changes with eye temperature.

a. What types of variables are displayed? b. What type of graph is this? c. Describe the association between the two variables. Is the relationship between flicker fusion frequency and temperature positive or negative? Is the relationship linear or nonlinear? d. The 20 points in the graph were obtained from measurements of six swordfish. Can we treat the 20 measurements as a random sample? Why or why not?

24. The following graph displays the net number of species listed under the U.S. Endangered Species Act between 1980 and 2002 (U.S. Fish and Wildlife Service 2001):

a. What type of graph is this? b. What does the steepness of each line segment indicate? c. Explain what the graph tells us about the relationship between the number of species listed and time. 25. Spot the flaw. Examine the following figure, which displays the frequency distribution of similarity values (the percentage of amino acids that are the same) between equivalent (homologous) proteins in humans and pufferfish of the genus Fugu (modified from Aparicio et al. 2002).

a. What type of graph is this? b. Identify the main flaw in the construction of this figure. c. What are the main results displayed in the figure? d. Describe the shape of the frequency distribution shown. e. What is the mode in the frequency distribution? 26. The following data give the photosynthetic capacity of nine individual females of the neotropical tree Ocotea tenera, according to the number of fruits produced in the previous reproductive season (Wheelwright and Logan 2004). The goal of the study was to investigate how reproductive effort in females of these trees impacts subsequent growth and photosynthesis. Number of fruits produced

previously

Photosynthetic capacity (µmol O2/m2/s)

10 14 5 24 50 37 89

13.0 11.9 11.5 10.6 11.1 9.4 9.3

162 149

9.1 7.3

a. Graph the association between these two variables using the most appropriate method. Identify the type of graph you used. b. Which variable is the explanatory variable in your graph? Why? c. Describe the association between the two variables in words, as revealed by your graph. 27. Examine the accompanying figure, which displays the percentage of adults over 18 with a “body mass index” greater than 25 in different years (modified from The Economist 2005, with permission). Body mass index is a measure of weight relative to height.

a. What is the main result displayed in this figure? b. Which of the four principles for drawing good graphs are violated here? How are they violated? c. Redraw the figure using the most appropriate method discussed in this chapter. What type of graph did you use? 28. When a courting male of the small Indonesian fish Telmatherina sarasinorum spawns with a

female, other males sometimes sneak in and release sperm, too. The result is that not all of the female’s eggs are fertilized by the courting male. Gray et al. (2007) noticed that courting males occasionally cannibalize fertilized eggs immediately after spawning. Egg eating took place by 61 of 450 courting males who fathered the entire batch; the remaining 389 males did not cannibalize eggs. In contrast, 18 of 35 courting males ate eggs when a single sneaking male also participated in the spawning event. Finally, 16 of 20 males ate eggs when two or more sneaking males were present. a. Display these results in a table that best shows the association between cannibalism and the number of sneaking males. Identify the type of table you used. b. Illustrate the same results using a graphical technique instead. Identify the type of graph you used. 29. The graph at the top of page 62, in red, shows the number of new cases of influenza in New York, Pennsylvania, and New Jersey, according to data from the Centers for Disease Control (Ginsberg et al. 2009). The black line shows predictions based on the number of Google searches of words like “flu” or “influenza.” a. What type of graph is this? b. Describe some of the scientific conclusions you might draw from looking at this graph. c. Can you suggest an improvement to the axes labels? FIGURE FOR PROBLEM 29

Source: Jeremy Ginsberg et al., “Detecting Influenza Epidemics Using Search Engine Query Data,” Nature 457 (2009): 1012–1014. 30. The following graph was drawn using a very popular spreadsheet program in an attempt to show the frequencies of observations in four hypothetical groups. Before reading further, estimate by eye the frequencies in each of the four groups. a. Identify two features of this graph that cause it to violate the principle, “Make patterns in the data easy to see.” b. Identify at least two other features of the graph that make it difficult to interpret. c. The actual frequencies are 10, 20, 30, and 40. Draw a graph that overcomes the problems identified above.

31. In Poland, students are required to achieve a score of 21 or higher on the high-school Polish language “maturity exam” to be eligible for university. The following graph shows the frequency distribution of scores (Freakonomics 2011). a. Examine the graph and identify the most conspicuous pattern in these data. b. Generate a hypothesis to explain the pattern.

32. More than 10% of people carry the parasite Toxoplasma gondii. The following table gives data from Prague on 15- to 29-year-old drivers who had been involved in an accident. The table gives the number of drivers who were infected with Toxoplasma gondii and who were uninfected. These numbers are compared with a control sample of 249 drivers of the same age living in the same area who had not been in an accident.

Drivers with accidents Controls

Infected

Uninfected

21 38

38 211

a. What type of table is this? b. What are the two variables being compared? Which is the explanatory variable and which is the response? c. Depict the data in a graph. Use the results to answer the question: are the two variables associated in this data set? 33. The cutoff birth date for school entry in British Columbia, Canada, is December 31. As a result, children born in December tend to be the youngest in their grade, whereas those born in January tend to be the oldest. Morrow et al. (2012) examined how this relative age difference influenced diagnosis and treatment of attention deficit/hyperactivity disorder

(ADHD). A total of 39,136 boys aged 6 to 12 years and registered in school in 1997–1998 had January birth dates. Of these, 2219 were diagnosed with ADHD in that year. A total of 38,977 boys had December birth dates, of which 2870 were diagnosed with ADHD in that year. Display the association between birth month and ADHD diagnosis using a table or graphical method from this chapter. Is there an association? 34. Examine the following figure, which displays hypothetical measurements of a sample of individuals from several groups.

a. What type of graph is this? b. In which of the groups is the frequency distribution of measurements approximately symmetric? c. Which of the frequency distributions show positive skew? d. Which of the frequency distributions show negative skew? e. Which group has the largest value for the upper quartile? f. Which group has the smallest value for the median? g. Which group has the most extreme observation? 35. The following data are from Mattison et al. (2012), who carried out an experiment with rhesus monkeys to test whether a reduction in food intake extends life span (as measured in years). The data are the life spans of 19 male and 15 female monkeys who were randomly assigned a normal nutritious diet or a similar diet reduced in amount by 30%. All monkeys were adults at the start of the study. Females—reduced: 16.5, 18.9, 22.6, 27.8, 30.2, 30.7, 35.9 Females—control: 23.7, 24.5, 24.7, 26.1, 28.1, 33.4, 33.7, 35.2 Males—reduced: 23.7, 28.1, 29.8, 31.1, 36.3, 37.7, 39.9, 39.9, 40.2, 40.2 Males—control: 24.9, 25.2, 29.6, 33.2, 34.1, 35.4, 38.1, 38.8, 40.7 a. Graph the results, using the most appropriate method and following the four principles of good graph design. b. According to your graph, which difference in life span is greater: that between the sexes, or that between diet groups? 36. The accompanying graph indicates the amount of time (latency) that female subjects were

willing to leave their hand in icy water while they were swearing (“words you might use after hitting yourself on the thumb with a hammer”) or while not swearing, using other words instead (“words to describe a table”). The data are from Stephens et al. (2009).

a. Identify the type of graph. b. Is any association apparent between the variables? Explain. c. What do the “whiskers” indicate in this graph? d. List two other types of graphs that would also be appropriate for showing these results. 37. Following is a list of all the named hurricanes in the Atlantic between 2001 and 2010, along with their category on the Saffir-Simpson Hurricane Scale,9 which categorizes each hurricane by a label from 1 to 5 depending on its power. 2001: Erin, 3; Feliz, 3; Gabrielle, 1; Humberto, 2; Iris, 4; Karen, 1; Michelle, 4; Noel, 1; Olga, 1. 2002: Gustav, 2; Isidore, 3; Kyle, 1; Lili, 4; 2003: Claudette, 1; Danny, 1; Erika, 1; Fabian, 4; Isabel, 5; Juan, 2; Kate, 3. 2004: Alex, 3; Charley, 4; Danielle, 2; Frances, 4; Gaston, 1; Ivan, 5; Jeanne, 3; Karl, 4; Lisa, 1. 2005: Cindy, 1; Dennis, 4; Emily, 5; Irene, 2; Katrina, 5; Maria, 3; Nate, 1; Ophelia, 1; Philippe, 1; Rita, 5; Stan, 1; Vince, 1; Wilma, 5; Beta, 3; Epsilon, 1. 2006: Ernesto, 1; Florence, 1; Gordon, 3; Helene, 3; Isaac, 1. 2007: Dean, 5; Felix, 5; Humberto, 1; Karen, 1; Lorenzo, 1; Noel, 1. 2008: Bertha, 3; Dolly, 2; Gustav, 4; Hanna, 1; Ike, 4; Kyle, 1; Omar, 4; Paloma, 4. 2009: Bill, 4; Fred, 3; Ida, 2. 2010: Alex, 2; Danielle, 4; Earl, 4; Igor, 4, Julia, 4; Karl, 3; Lisa, 1; Otto, 1; Paula, 2; Richard, 2; Shary, 1; Toas, 2. a. Make a frequency table showing the frequency of hurricanes in each severity category during the decade. b. Make a frequency table that shows the frequency of hurricanes in each year.

c. Explain how you chose to order the categories in your tables.

3

Describing data

Saiga

Descriptive statistics, or summary statistics, are quantities that capture important features of frequency distributions. Whereas graphs reveal shapes and patterns in the data, descriptive statistics provide hard numbers. The most important descriptive statistics for numerical data are those measuring the location of a frequency distribution and its spread. The location tells us something about the average or typical individual—where the observations are centered. The spread tells us how variable the measurements are from individual to individual—how widely scattered the observations are around the center. The proportion is the most important descriptive statistic for a categorical variable, measuring the fraction of observations in a given category. The importance of calculating the location of a distribution seems obvious. How else do we address questions like “Which species is larger?” or “Which drug yielded the greatest response?” The importance of describing distribution spread is less obvious but no less crucial,

at least in biology. In some fields of science, variability around a central value is instrument noise or measurement error, but in biology much of the variability signifies real differences among individuals. Different individuals respond differently to treatments, and this variability begs measurement. Measuring variability also gives us perspective. We can ask, “How large are the differences between groups compared with variations within groups?” Biologists also appreciate variation as the stuff of evolution—we wouldn’t be here without variation. In this chapter, we review the most common statistics to measure the location and spread of a frequency distribution and to calculate a proportion. We introduce the use of mathematical symbols to represent values of a variable, and we show formulas to calculate each summary statistic.

Arithmetic mean and standard deviation The arithmetic mean is the most common metric to describe the location of a frequency distribution. It is the average of a set of measurements. The standard deviation is the most commonly used measure of distribution spread. Example 3.1 illustrates the basic calculations for means and standard deviations. EXAMPLE 3.1 Gliding snakes When a paradise tree snake (Chrysopelea paradisi) flings itself from a treetop, it flattens its body everywhere except for the region around the heart. As it gains downward speed, the snake forms a tight horizontal S shape and then begins to undulate widely from side to side. This generates lift, causing the snake to glide away from the source tree. By orienting the head and anterior part of the body, the snake can change direction during a glide to avoid trees, reach a preferred landing site, and even chase aerial prey. To better understand how lift is generated, Socha (2002) videotaped the glides of eight snakes leaping from a 10-m tower.1 Among the measurements taken was the rate of side-to-side undulation on each snake. Undulation rates of the eight snakes, measured in hertz (cycles per second), were as follows:

0.9, 1.4, 1.2, 1.2, 1.3, 2.0, 1.4, 1.6

A histogram of these data is shown in Figure 3.1-1. The frequency distribution has a single peak between 1.2 and 1.4 Hz.

FIGURE 3.1-1 A histogram of the undulation rate of gliding paradise tree snakes. n =

8 snakes.

The sample mean The sample mean is the average of the measurements in the sample, the sum of all the observations divided by the number of observations. To show its calculation, we use the symbol Y to refer to the variable and Yi to represent the measurement of individual i. For the gliding snake data, i takes on values between 1 and 8, because there are eight snakes. Thus, Y1 = 0.9, Y2 = 1.4, Y3 = 1.2, Y4 = 1.2, and so on.2 The sample mean is the sum of all the observations in a sample divided by n, the number of observations. The sample mean, symbolized as Y¯ (and pronounced “Y-bar”), is calculated as

Y¯=∑i=1nYin, where n is the number of observations. The symbol Σ (uppercase Greek letter sigma) indicates a sum. The “i = 1” under the Σ and the “n” over it indicate that we are summing over all values of i between 1 and n, inclusive: ∑i=1nYi=Y1+Y2+Y3+…+Yn. When it is clear that i refers to individuals 1, 2, 3, . . . , n, the formula is often written more succinctly as Y¯=∑Yin. Applying this formula to the snake data yields the mean undulation rate: Y¯=0.9+1.2+1.2+2.0+1.6+1.3+1.4+1.48=1.375 Hz. Based on the histogram in Figure 3.1-1, we see that the value of the sample mean is close to the middle of the distribution. Note that the sample mean has the same units as the observations used to calculate it. In Section 3.6, we review how the sample mean is affected when the units of the observations are changed, such as by adding a constant or multiplying by a constant.

Variance and standard deviation The standard deviation is a commonly used measure of the spread of a distribution. It measures how far from the mean the observations typically are. The standard deviation is large if most observations are far from the mean, and it is small if most measurements lie close to the mean. The standard deviation is a common measure of the spread of a distribution. It indicates just how different measurements typically are from the mean. The standard deviation is calculated from the variance, another measure of spread. The standard deviation is simply the square root of the variance. The standard deviation is a more intuitive measure of the spread of a distribution (in part because it has the same units as the variable itself), but the variance has mathematical properties that make it useful sometimes as

well. The standard deviation from a sample is usually represented by the symbol s, and the sample variance is written as s2. To calculate the variance from a sample of data, we must first compute the deviations. A deviation from the mean is the difference between a measurement and the mean (Yi−Y¯). Deviations for the measurements of snake undulation rate are listed in Table 3.1-1. TABLE 3.1-1 Quantities needed to calculate the standard deviation and variance of snake undulation rate (Y¯=1.375 Hz). Squared deviations Observations Deviations (Yi) (Yi−Y¯) (Yi−Y¯)2 0.9 1.2

–0.475 –0.175

0.225625 0.030625

1.2 1.3 1.4 1.4 1.6 2.0

–0.175 –0.075 0.025 0.025 0.225 0.625

0.030625 0.005625 0.000625 0.000625 0.050625 0.390625

Sum

0.000

0.735

The best measure of the spread of this distribution isn’t just the average of the deviations (Yi−Y¯), because this average is always zero (the negative deviations cancel the positive deviations). Instead, we need to average the squared deviations (the third column in Table 3.1-1) to find the variance:

s2=∑i=1n(Yi-Y¯)2n-1. By squaring each number, deviations above and below the mean contribute equally3 to the variance. The summation in the numerator (top part) of the formula, ∑(Yi−Y¯)2, is called the sum of squares of Y. Note that the denominator (bottom part) is n − 1 instead of n, the total number of observations. Dividing by n − 1 gives a more accurate estimate of the population variance.4 We provide a shortcut formula for the variance in the Quick Formula Summary (Section 3.7). For the snake undulation data, the variance (rounded to hundredths) is s2=0.7357=0.11 Hz2. The variance has units equal to the square of the units of the original data. To obtain the standard deviation, we take the square root of the variance: s=∑(Yi-Y¯)2n-1. For the snake undulation data, s s=0.7357=0.324037 Hz. The standard deviation is never negative and has the same units as the observations from which it was calculated.

The standard deviation has a straightforward connection to the frequency distribution. If the frequency distribution is bell shaped, like the example in Figure 2.2-4, then about two-thirds of the observations will lie within one standard deviation of the mean, and about 95% will lie within two standard deviations. In other words, about 67% of the data will fall between Y¯−s and Y¯+s, and about 95% will fall between Y¯−2s and Y¯+2s. For an in-depth discussion of standard deviation, see Chapter 10. This straightforward connection between the standard deviation and the frequency distribution diminishes when the frequency distribution deviates from the bell-shaped (normal) distribution. In such cases, the standard deviation is less informative about where the data lie in relation to the mean. This point is explored in greater detail in Section 3.3.

Rounding means, standard deviations, and other quantities To avoid rounding errors when carrying out calculations of means, standard deviations, and other descriptive statistics, always retain as many significant digits as your calculator or computer can provide. Intermediate results written down on a page should also retain as many digits as feasible. Final results, however, should be rounded before being presented. There are no strict rules on the number of significant digits that should be retained when rounding. A common strategy, which we adopt here, is to round descriptive statistics to one decimal place more than the measurements themselves. For example, the undulation rates in snakes were measured to a single decimal place (tenths). We therefore present descriptive statistics with two decimals (hundredths). The mean rate of undulation for the eight snakes, calculated as 1.375 Hz, would be communicated as Y¯=1.38 Hz. Similarly, the standard deviation, calculated as 0.324037 Hz, would be reported as s=0.32 Hz. Note that even though we report the rounded value of the mean as Y¯=1.38, we used the more exact value, Y¯=1.375, in the calculation of s to avoid rounding errors.

Coefficient of variation For many traits, standard deviation and mean change together when organisms of different sizes are compared. Elephants have greater mass than mice and also more variability in mass. For many purposes, we care more about the relative variation among individuals. A gain of 10 g for an elephant is inconsequential, but it would double the mass of a mouse. On the other hand, an elephant that is 10% larger than the elephant mean may have something in common with a mouse that is 10% larger than the mouse mean. For these reasons, it is sometimes useful to express the standard deviation relative to the mean. The coefficient of variation (CV) calculates the standard deviation as a percentage of the mean: CV=sY¯×100%. The coefficient of variation is the standard deviation expressed as a percentage of the mean. A higher CV means that there is more variability, whereas a lower CV means that individuals are more consistently the same. For the snake undulation data, the coefficient of variation is

CV=0.3241.375100%=24%. The coefficient of variation makes sense only when all of the measurements are greater than or equal to zero. The coefficient of variation can also be used to compare the variability of traits that do not have the same units. If we wanted to ask, “What is more variable in elephants, body mass or life span?” then the standard deviation is not very informative, because mass is measured in kilograms and life span is measured in years. The coefficient of variation would allow us to make this comparison.

Calculating mean and standard deviation from a frequency table Sometimes the data include many tied observations and are given in a frequency table. The frequency table in Table 3.1-2, for example, lists the number of criminal convictions of a cohort of 395 boys (Farrington 1994; see Assignment Problem 22 in Chapter 2). TABLE 3.1-2 Number of criminal convictions of a cohort of 395 boys. Number of convictions Frequency 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

265 49 21 19 10 10 2 2 4 2 1 4 3 1 2

Total

395

To calculate the mean and standard deviation of the number of convictions, notice first that the sample size is not 15, the number of rows in Table 3.1-2, but 395, the frequency total: n=265+49+21+19+…+2=395 Calculating the mean thus requires that the measurement of “0” be represented 265 times, the number “1” be represented 49 times, and so on. The sum of the measurements is thus ∑Yi=(265×0)+(49×1)+(21×2)+(19×3)+ …+(2×14)=445. The mean of these data is then Y¯=445395=1.126582,

which we round to Y¯=1.1 when presenting the results. The calculation of standard deviation must also take into account the number of individuals with each value. The sum of the squared deviations is ∑(Yi−Y¯)2=265(0−Y¯)2+49(1−Y¯)2+21(2−Y¯)2+…+2(14−Y¯)2=2377.671.

The standard deviation for these data is therefore s=2377.671395-1=2.4566, which we present as s = 2.5. These calculations assume that all the data are presented in the table. This approach would not work, however, for frequency tables in which the data are grouped into intervals, such as Table 2.2-3.

Effect of changing measurement scale Results may need to be converted to a different scale than the one in which they were originally measured. For example, if temperature measurements were made in °F, it may be necessary to convert results to °C. The snake data were measured in hertz (cycles per second), but in some cases hertz must be converted to angular velocity (radians per second) instead. The good news is that we don’t need to start over by converting the raw data. Instead, we can convert the descriptive statistics directly, as follows. Briefly, here are the rules (we summarize them in the Quick Formula Summary at the end of this chapter). If converting data to a new scale, Y′, involves multiplying the data, Y, by a constant, c, Y′=cY, then multiply the original mean Y¯ by the same constant to obtain the new mean, and multiply the original standard deviation s by the absolute value of c to get the new standard deviation: Y¯′=cY¯s′=|c|s. However, the variance s2 is converted by multiplying by c2: s′2=c2s2. If converting data to a new scale, Y′, involves adding a constant, c, then the mean is converted by adding the same constant, Y¯′=Y¯+c, whereas the standard deviation and variance are unchanged: s′=ss′2=s2. This makes sense. Adding a constant to the data changes the location of the frequency distribution by the same amount but does not alter its spread. For example, converting degrees Fahrenheit to degrees Celsius uses the transformation °C=(5/9)°F−17.8. Therefore, if the mean temperature in a data set is Y¯=80°F, with a standard deviation of s = 3°F, then the new mean temperature is Y¯′=(5/9)80−17.8=26.6°C and the new standard deviation is

s′=(5/9)3=1.7°C. The new variance is s′2=(5/9)232=2.8°C2.

Median and interquartile range After the sample mean, the median is the next most common metric used to describe the location of a frequency distribution. As we showed in Chapter 2, the median is often displayed in a box plot alongside the span between the first and third quartiles, or interquartile range, another measure of the spread of the distribution. We define and demonstrate these concepts with the help of Example 3.2. EXAMPLE 3.2 I’d give my right arm for a female Male spiders in the genus Tidarren are tiny, weighing only about 1% as much as females. They also have disproportionately large pedipalps, copulatory organs that make up about 10% of a male’s mass. (See the adjacent photo; the pedipalps are indicated by arrows.) Males load the pedipalps with sperm and then search for females to inseminate. Astonishingly, male Tidarren spiders voluntarily amputate one of their two organs, right or left, just before sexual maturity. Why do they do this? Perhaps speed is important to males searching for females, and amputation increases running performance. To test this hypothesis, Ramos et al. (2004) used video to measure the running speed of males on strands of spider silk. The data are presented in Table 3.2-1.

TABLE 3.2-1 Running speed (cm/s) of male Tidarren spiders before and after voluntary amputation of a pedipalp. Spider Speed before Speed after 1 2 3 4 5 6 7 8

1.25 2.94 2.38 3.09 3.41 3.00 2.31 2.93

2.40 3.50 4.49 3.17 5.26 3.22 2.32 3.31

9

2.98

3.70

10 11 12 13 14 15 16

3.55 2.84 1.64 3.22 2.87 2.37 1.91

4.70 4.94 5.06 3.22 3.52 5.45 3.40

The median The median is the middle observation in a set of data, the measurement that partitions the ordered measurements into two halves. To calculate the median, first sort the sample observations from smallest to largest. The sorted measurements of running speed of male spiders before amputation (Table 3.2-1) are 1.25, 1.64, 1.91, 2.31, 2.37, 2.38, 2.84, 2.87, 2.93, 2.94, 2.98, 3.00, 3.09, 3.22, 3.41, 3.55 in cm/s. Let Y(i) refer to the ith sorted observation, so Y(1) is 1.25, Y(2) is 1.64, Y(3) is 1.91, and so on. If the number of observations (n) is odd, then the median is the middle observation: Median=Y([n+1]/2). The median is the middle measurement of a set of observations. If the number of observations is even, as in the spider data, then the median is the average of the middle pair: Median=[Y(n/2)+Y(n/2+1)]/2. Thus, n/2 = 8, Y(8) = 2.87, and Y(9) = 2.93 for the spider data (before amputation). The median is the average of these two numbers: Median=(2.87+2.93)/2=2.90 cm/s.

The interquartile range Quartiles are values that partition the data into quarters. The first quartile is the middle value of the measurements lying below the median. The second quartile is the median. The third quartile is the middle value of the measurements larger than the median. The interquartile range (IQR) is the span of the middle half of the data, from the first quartile to the third quartile: Interquartile range=third quartile−first quartile. The interquartile range is the difference between the third and first quartiles of the data. It is the span of the middle 50% of the data. Figure 3.2-1 shows the meaning of the median, first quartile, third quartile, and interquartile range for the spider data set (before amputation).

FIGURE 3.2-1 The first quartile, median, and third quartile break the data set into four equal portions. The median is the middle value, and the first and third quartiles are the middles of the first and second halves of the data. The interquartile range is the span of the middle half of the data.

The first step in calculating the interquartile range is to compute the first and third quartiles, as follows.5 For the first quartile, calculate j=0.25n, where n is the number of observations. If j is an integer then the first quartile is the average of Y(j) and Y(j+1): First quartile=(Y(j)+Y(j+1))/2, where Y(j) is the jth sorted observation. For the sorted spider data, j=(0.25)(16)=4, which is an integer. Therefore, the first quartile is the average of Y(4) and Y(5): First quartile=(2.31+2.37)/2=2.34. If j is not an integer, then convert j to an integer by replacing it with the next integer that exceeds it (i.e., round j up to the nearest integer). The first quartile is then First quartile=Y(j), where j is now the integer you rounded to. The third quartile is computed similarly. Calculate k=0.75n. If k is an integer, then the third quartile is the average of Y(k) and Y(k+1): Third quartile=(Y(k)+Y(k+1))/2, where Y(k) is the kth sorted observation. For the sorted spider data, j=(0.75)(16)=12, which is an integer. Therefore, the third quartile is the average of Y(12) and Y(13): Third quartile=(3.00+3.09)/2=3.045. If k is not an integer, then convert k to an integer by replacing it with the next integer that exceeds it (i.e., round k up to the nearest integer). The third quartile is then Third quartile=Y(k), where k is the integer you rounded to. The interquartile range is then Interquartile range=3.045−2.34−0.705 cm/s.

The box plot A box plot displays the median and interquartile range, along with other quantities of the

frequency distribution. We introduced the box plot in Chapter 2. Figure 3.2-2 shows a box plot for the spider running speeds, with data before and after amputation plotted separately. The lower and upper edges of the box are the first and third quartiles. Thus, the interquartile range is visualized by the span of the box. The horizontal line dividing each box is the median. The whiskers extend outward from the box at each end, stopping at the smallest and largest “nonextreme” values in the data. “Extreme” values are defined as those lying farther from the box edge than 1.5 times the interquartile range. Extreme values are plotted as isolated dots past the ends of the whiskers.6 There is one extreme value in the box plots shown in Figure 3.2-2, the smallest measurement for running speed before amputation.

FIGURE 3.2-2 Box plot of the running speeds of 16 male spiders before and after self-amputation of a pedipalp.

How measures of location and spread compare Which measure of location, the sample mean or the median, is most revealing about the center of a distribution of measurements? And which measure of spread, the standard deviation or the interquartile range, best describes how widely the observations are scattered about the center? The answer depends on the shape of the frequency distribution. These alternative measures of location and of spread yield similar information when the frequency distribution is symmetric and unimodal. The mean and standard deviation become less informative than the median and interquartile range when the data are strongly skewed or include extreme observations. We compare these measures using Example 3.3. EXAMPLE 3.3 Disarming fish The marine threespine stickleback is a small coastal fish named for its defensive armor. It has three sharp spines down its back, two pelvic spines under the belly, and a series of lateral bony plates down both sides. The armor seems to reduce mortality from predatory fish and diving birds. In contrast, in lakes and streams, where predators are fewer, stickleback populations have reduced armor. (See the photo at the right for examples of different types. Bony tissue has been stained red to make it more visible.) Colosimo et al. (2004) measured the grandchildren of a cross made between a marine and a freshwater stickleback. The study found that much of the difference in number of plates is caused by a single gene, Ectodysplasin.7 Fish inheriting two copies of the gene from the marine grandparent, called MM fish, had many plates (the top histogram in Figure 3.3-1). Fish inheriting both copies of the gene from the freshwater grandparent (mm) had few plates (the bottom histogram in Figure 3.3-1). Fish having one copy from each grandparent (Mm) had any of a wide range of plate numbers (the middle histogram in Figure 3.3-1).

FIGURE 3.3-1 Frequency distributions of lateral plate number in three genotypes of stickleback, MM, Mm, and mm, descended from a cross between marine and freshwater grandparents. Plates are counted as the total number down the left and right sides of the fish. The total number of fish: 82 (MM), 174 (Mm), and 88 (mm).

Mean versus median The mean and median of the three distributions in Figure 3.3-1 are compared in Table 3.3-1. The two measures of location give similar values in the case of the MM and mm genotypes, whose distributions are fairly symmetric, although one or two outliers are present. The mean is smaller than the median in the case of the Mm fish, whose distribution is strongly asymmetric. TABLE 3.3-1 Descriptive statistics for the number of lateral plates of the three genotypes of threespine sticklebacks8 discussed in Example 3.3. Genotype n Mean Median Standard deviation Interquartile range MM Mm mm

82 174 88

62.8 50.4 11.7

63 59 11

3.4 15.1 3.6

2 21 3

Why are the median and mean different from one another when the distribution is asymmetric? The answer, shown in Figure 3.3-2, is that the median is the middle measurement of a distribution, whereas the mean is the “center of gravity.”

FIGURE 3.3-2 Comparison between the median and the mean using the frequency distribution for the Mm genotype (middle panel of Figure 3.3-1). The median is the middle measurement of the distribution (different colors represent the two halves of the distribution). The mean is the center of gravity, the point at which the frequency distribution would be balanced (if observations had weight).

The balancing act illustrated in Figure 3.3-2 suggests that the mean is sensitive to extreme observations. To demonstrate, imagine taking the four smallest observations of the MM genotype (top panel in Figure 3.3-1) and moving them far to the left. The median would be completely unaffected, but the mean would shift leftward to a point near the edge of the range of most observations (Figure 3.3-3).

FIGURE 3.3-3 Sensitivity of the mean to extreme observations using the frequency distribution of the MM genotypes (see the upper panel in Figure 3.3-1). The two different colors represent the two halves of the distribution. When the four smallest observations of the MM genotype are shifted far to the left (lower panel), the mean is displaced downward, to the edge of the range of the bulk of the observations. The median, on the other hand, which is located where the two colors meet, is unaffected by the shift.

Median and mean measure different aspects of the location of a distribution. The median is the middle value of the data, whereas the mean is its center of gravity. Thus, the mean is displaced from the location of the “typical” measurement when the frequency distribution is strongly skewed, particularly when there are extreme observations. The mean is still useful as a description of the data as a whole, but it no longer indicates where most of the observations are located. The median is less sensitive to extreme observations, and hence the median is the more informative descriptor of the typical observation in such instances. However, the mean has better mathematical properties, and it is easier to calculate measures of the reliability of estimates of the mean.

Standard deviation versus interquartile range Because it is calculated from the square of the deviations, the standard deviation is even more sensitive to extreme observations than is the mean. When the four smallest observations of the MM genotype are shifted far to the left, such that the smallest is set to zero (Figure 3.3-3), the standard deviation jumps from 3.4 to 12.0, whereas the interquartile range is not affected. For this reason, the interquartile range is a better indicator of the spread of the main part of a distribution than the standard deviation when the data are strongly skewed to one side or the other, especially when there are extreme observations. On the other hand, the standard deviation reflects the variation among all of the data points.

Cumulative frequency distribution The median and quartiles are examples of percentiles, or quantiles, of the frequency distribution for a numerical variable. Plotting all the quantiles using the cumulative frequency distribution is another way to compare the shapes and positions of two or more frequency distributions.

Percentiles and quantiles The Xth percentile of a sample is the value below which X percent of the individuals lie. For example, the median, the measurement that splits a frequency distribution into equal halves, is the 50th percentile. Ten percent of the observations lie below the 10th percentile, and the other 90% of observations exceed it. The first and third quartiles are the 25th and 75th percentiles, respectively. The percentile of a measurement specifies the percentage of observations less than or equal to it; the remaining observations exceed it. The quantile of a measurement specifies the fraction of observations less than or equal to it. The same information in a percentile is sometimes represented as a quantile. This only means that the proportion less than or equal to the given value is represented as a decimal rather than as a percentage. For example, the 10th percentile is the 0.10 quantile, and the median is the 0.50 quantile. Be careful not to mix up the words quantile and quartile (note the difference in the fourth letters). The first and third quartiles are the 0.25 and 0.75 quantiles.

Displaying cumulative relative frequencies All the quantiles of a numerical variable can be displayed by graphing the cumulative frequency distribution. Cumulative relative frequency at a given measurement is the fraction of observations less than or equal to that measurement. Figure 3.4-1 shows the cumulative frequency distribution of the running speeds of male spiders before amputation. The raw data are from Table 3.2-1. To make this graph, all the measurements of running speed (before amputation) were sorted from smallest to largest. Next, the fraction of observations less than or equal to each data value was calculated. This fraction, which is called the cumulative relative frequency, is indicated by the height of the curve in Figure 3.4-1 at the corresponding data value. Finally, these points were connected with straight lines to form an ascending curve. The result is an irregular sequence of “steps” from the smallest data value to the largest data value. Each step is flat, but the curve jumps up by 1/n at every observed measurement, where n is the total number of observations (here, 16 spiders), to a maximum of 1. There may be multiple jumps at one measurement if multiple data points have the same measurement.

FIGURE 3.4-1 The cumulative frequency distribution of male spiders before amputation (solid curve). Horizontal dotted lines indicate the cumulative relative frequencies 0.25 (lower) and 0.75 (upper); vertical lines indicate corresponding 0.25 and 0.75 quantiles of running speed (2.34 and 3.045). The data are from Table 3.2-1. n = 16 spiders.

The curve in Figure 3.4-1 shows a lot of information because all the data points are represented. We can see that one-fourth of the observations (corresponding to a cumulative relative frequency of 0.25) had running speeds below 2.34, which is the value of the first quartile calculated earlier. Three-fourths of all observations lie below 3.045, which is the value of the third quartile calculated earlier. Both these values are indicated in Figure 3.4-1 with the dashed lines. Because of their simplicity and ease of interpretation, the histogram and box plot are usually superior to the cumulative frequency distribution for showing the data. However, with practice, the cumulative frequency distribution can be very useful, especially to compare frequency distributions of multiple groups.

Proportions The proportion is the most important descriptive statistic for a categorical variable.

Calculating a proportion The proportion of observations in a given category, symbolized pˆ is calculated as pˆ=Number in categoryn, where the numerator is the number of observations in the category of interest, and n is the total number of observations in all categories combined.9 For example, of the 344 individual sticklebacks in Example 3.3, 82 had genotype MM, 88 were mm, and 174 were Mm (Table 3.3-1). The proportion of MM fish is pˆ=82344=0.238. The other proportions are calculated similarly, and all three proportions are listed in Table 3.5-1. TABLE 3.5-1 The number of fish of each genotype from a cross between a marine stickleback and a freshwater stickleback (Example 3.3). As written, the sum of the proportions does not add precisely to one because of rounding. Genotype Frequency Proportion MM Mm mm

82 174 88

0.24 0.51 0.26

Total

344

1.00

The proportion is like a sample mean The proportion pˆ has properties in common with the arithmetic mean. To see this, let’s create a new numerical variable Y for the stickleback study. Give individual fish i the value Yi = 1 if it has the MM genotype, and give it the value Yi = 0 otherwise. The sum of all the ones and zeroes, ∑Yi, is the frequency of fish having genotype MM. The mean of the ones and zeroes is Y¯=∑Yin=82344=0.238, which is just pˆ, the proportion of observations in the first category. If we imagine the Ymeasurements to have weight, then the proportion is their center of gravity (Figure 3.5-1).

FIGURE 3.5-1 The distribution of Y, where Y = 1 if a stickleback is genotype MM and 0 otherwise. The mean of Y is the proportion of MM individuals in the sample (0.238).

Summary ■ The location of a distribution for a numerical variable can be measured by its mean or by its median. The mean gives the center of gravity of the distribution and is calculated as the sum of all measurements divided by the number of measurements. The median gives the middle value. ■ The standard deviation measures the spread of a distribution for a numerical variable. It is a measure of the typical distance between observations and the mean. The variance is the square of the standard deviation. ■ The quartiles break the ordered observations into four equal parts. The inter-quartile range, the difference between the first and third quartiles, is another measure of the spread of a frequency distribution. ■ The mean and median yield similar information when the frequency distribution of the measurements is symmetric and unimodal. The mean and standard deviation become less informative about the location and spread of typical observations than the median and interquartile range when the data include extreme observations. ■ The percentile of a measurement specifies the percentage of observations less than or equal to it. The quantile of a measurement specifies the fraction of observations less than or equal to it. ■ All the quantiles of a sample of data can be shown using a graph of the cumulative frequency distribution. ■ The proportion is the most important descriptive statistic for a categorical variable. It is calculated by dividing the number of observations in the category of interest by n, the total number of observations in all categories combined.

Quick Formula Summary Table of formulas for descriptive statistics Quantity Sample size

Formula n

Mean

Y¯=∑Yn

Variance

s2=∑(Yi-Y¯)2n-1

shortcut s2=∑(Yi2)-nY¯2n-1 formula: Standard deviation s=∑(Yi-Y¯)2n-1 shortcut formula: s=∑(Yi2)-nY¯2n-1 Sum of squares ∑(Yi-Y¯)2=∑(Yi2)-nY¯2 Coefficient of CV=sY¯×100% variation Y([n+1]/2)  (if n is odd)[Y(n/2)+Y(n/2+1)]/2   (if n is odd)where Y(1),Y(2), Median …,Y(n) are the ordered observations Proportion pˆ=Number in categoryn

Effect of arithmetic operations on descriptive statistics The table below lists the effect on the descriptive statistics of adding or multiplying all the measurements by a constant. The rules listed in the table are useful when converting measurements from one system of units to another, such as English to metric or degrees Fahrenheit to degrees Celsius.

Statistic Mean Standard deviation Variance Median Interquartile range

Value Y¯ s s2 M IQR

Adding a constant c to Multiplying all the all the measurements, measurements by a Y′=Y+c constant c, Y′=cY Y¯′=Y¯+c s' = s s'2 = s2 M' = M + c IQR' = IQR

Y¯′=cY¯ s' = |c|s s'2 = c2s2 M' = cM IQR' = |c|IQR

PRACTICE PROBLEMS 1. Calculation practice: Basic descriptive stats. Systolic blood pressure was measured (in units of mm Hg) during preventative health examinations on people in Dallas, Texas. Here are the measurements for a subset of these patients. 112, 128, 108, 129, 125, 153, 155, 132, 137 a. How many individuals are in the sample (i.e., what is the sample size, n)? b. What is the sum of all of the observations? c. What is the mean of this sample? Here and forever after, provide units with your answer. d. What is the sum of the squares of the measurements? e. What is the variance of this sample? f. What is the standard deviation of this sample? g. What is the coefficient of variation for this sample? 2. Calculation practice: Box plots. Here is another sample of systolic blood pressure (in units of mm Hg), this time with all 101 data points. The mean is 122.73 and the standard deviation is 13.83. 88, 88, 92, 96, 96, 100, 102, 102, 104, 104, 105, 105, 105, 107, 107, 108, 110, 110, 110, 111, 111, 112, 113, 114, 114, 115, 115, 116, 116, 117, 117, 117, 117, 117, 117, 119, 119, 120, 121, 121, 121, 121, 121, 121, 122, 122, 123, 123, 123, 123, 123, 124, 124, 124, 124, 125, 125, 125, 126, 126, 126, 126, 126, 127, 127, 128, 128, 128, 128, 129, 129, 130, 131, 131, 131, 131, 131, 131, 133, 133, 133, 134, 135, 136, 136, 136, 138, 138, 139, 139, 141, 142, 142, 142, 143, 144, 146, 146, 147, 155, 156 a. What is the median of this sample? b. What is the upper (third) quartile (or 75th percentile)? c. What is the lower (first) quartile (or 25th percentile)? d. What is the interquartile range (IQR)? e. Calculate the upper quartile plus 1.5 times the IQR. Is this greater than the largest value in the data set? f. Calculate the lower quartile minus 1.5 times the IQR. Is this less than the smallest value in the data set? g. Plot the data in a box plot. (A rough sketch by hand is appropriate, as long as the correct values are shown for each critical point.) 3. A review of the performance of hospital gynecologists in two regions of England measured the outcomes of patient admissions under each doctor’s care (Harley et al. 2005). One measurement taken was the percentage of patient admissions made up of women under 25 years old who were sterilized. We are interested in describing what constitutes a typical rate of sterilization, so that the behavior of atypical doctors can be better scrutinized. The frequency distribution of this measurement for all doctors is plotted in the following graph.

a. Explain what the vertical axis measures. b. What would be the best choice to describe the location of this frequency distribution, the mean or the median, if our goal was to describe the typical individual? Why? c. Do you see any evidence that might lead to further investigation of any of the doctors? 4. The data displayed in the plot below are from a nearly complete record of body masses of the world’s native mammals (in grams, then converted to log base 10; Smith et al. 2003). The data were divided into three groups: those surviving from the last ice age to the present day (n = 4061), those who went extinct around the end of the last ice age (n = 227), and those driven extinct within the last 300 years (recent; n = 44). a. What type of graph is this?

b. What does the horizontal line in the center of each rectangle represent? c. What are the horizontal lines at the top and bottom edges of each rectangle supposed to represent? d. What are the data points (indicated by “—”) lying outside the rectangle? e. What are the vertical lines extending above and below each rectangle? f. Compare the locations of the three body-size distributions. How do they differ? g. Compare the shapes of the three frequency distributions. Which are relatively symmetric and which are asymmetric? Explain your reasoning. h. Which group’s frequency distribution has the lowest spread? Explain your reasoning. i. What has been the likely effect of ice-age and recent extinctions on the median body size of mammals? 5. Mehl et al. (2007) wired 396 volunteers with electronically activated recorders that allowed

the researchers to calculate the number of words each individual spoke, on average, per 17hour waking day. They found that the mean number of words spoken was only slightly higher for the 210 women (16,215 words) than for the 186 men (15,669) in the sample. The frequency distribution of the number of words spoken by all individuals of each sex is shown in the accompanying graphs (modified from Mehl et al. 2007).

a. What type of graph is shown? b. What are the explanatory and response variables in the figure? c. What is the mode of the frequency distribution of each sex? d. Which sex likely has the higher median number of words spoken per day? e. Which sex had the highest variance in number of words spoken per day? 6. The following data are measurements of body mass, in grams, of finches captured in mist nets during a survey of various habitats in Kenya, East Africa (Schluter 1988). Crimson-rumped waxbill Cutthroat finch White-browed sparrow weaver

8, 8, 8, 8, 8, 8, 8, 6, 7, 7, 7, 8, 8, 8, 7, 7, 7 16, 16, 16, 12, 16, 15, 15, 17, 15, 16, 15, 16 40, 43, 37, 38, 43, 33, 35, 37, 36, 42, 36, 36, 39, 37, 34, 41

a. Calculate the mean body mass of each of these three finch species. Which species is largest, and which is smallest? b. Which species has the greatest standard deviation in body mass? Which has the least? c. Calculate the coefficient of variation (CV) in mass for each finch species. How different are the coefficients between the species? Compare the difference in CVs with the differences in standard deviation calculated in part (b).

A photo of a finch perched on a tree is shown.

d. The following measurements are of another trait, beak length, in mm, of the 16 whitebrowed sparrow weavers. Which measurement is more variable in this species (relative to the mean), body mass or beak length? 10.6, 10.8, 10.9, 11.3, 10.9, 10.1, 10.7, 10.7, 10.9, 11.4, 10.8, 11.2, 10.7, 10.0, 10.1, 10.7 7. The spider data in Example 3.2 consist of pairs of measurements made on the same subjects. One measurement is running speed before amputation and the second is running speed after amputation. Calculate a new variable called “change in speed,” defined as the speed of each spider after amputation minus its speed before amputation. a. What are the units of the new variable? b. Draw a box plot for the change in running speed. Use the method outlined in Section 3.2 to calculate the quartiles. c. Based on your drawing in part (b), is the frequency distribution of the change in running speed symmetric or asymmetric? Explain how you decided this. d. What is the quantity measured by the span of the box in part (b)? e. Calculate the mean change in running speed. Is it the same as the median? Why or why not? f. Calculate the variance of the change in running speed.

g. What fraction of observations fall within one standard deviation above and below the mean? 8. Refer to the previous problem. If you were to convert all of the observations of change in running speed from cm/s into mm/s, how would this change a. the mean? b. the standard deviation? c. the median? d. the interquartile range? e. the coefficient of variation? f. the variance? 9. Niderkorn’s (1872; from Pounder 1995) measurements on 114 human corpses provided the first quantitative study on the development of rigor mortis.10 The data in the following table give the number of bodies achieving rigor mortis in each hour after death, recorded in onehour intervals. Hours

Number of bodies

1 2 3 4 5 6 7 8 9 10 11 12 13

0 2 14 31 14 20 11 7 4 7 1 1 2

Total

114

a. Calculate the mean number of hours after death that it took for rigor mortis to set in. b. Calculate the standard deviation in the number of hours until rigor mortis. c. What fraction of observations lie within one standard deviation of the mean (i.e., between the value Y¯−s

and the value Y¯+s )? d. Calculate the median number of hours until rigor mortis sets in. What is the likely explanation for the difference between the median and the mean? 10. The following graph shows the population growth rates of the 204 countries recognized by the United Nations. Growth rate is measured as the average annual percent change in the total human population between 2000 and 2004 (United Nations Statistics Division 2004).

a. Identify the type of graph depicted. b. Explain the quantity along the y-axis. c. Approximately what percentage of countries had a negative change in population? d. Identify by eye the 0.10, 0.50, and 0.90 quantiles of change in population size. e. Identify by eye the 60th percentile of change in population size. 11. Refer to the previous problem. a. Draw a box plot using the information provided in the graph in that problem. b. Label three features of this box plot. 12. Spot the flaw. The accompanying table shows means and standard deviations for the length

of migration on a microgel of 20 lymphocyte cells exposed to X-irradiation. The length of migration is an indication of DNA damage suffered by the cells. The data are from Singh et al. (1988). a. Identify the main flaw in the construction of this table. b. Redraw the table following the principles recommended in this chapter and Chapter 2. TABLE FOR PROBLEM 12

X-ray dose Mean Standard deviation

Control 25 rads 50 rads 100 rads 200 rads 3.70 1.10

5.27 1.19

12.37 4.69

23.30 3.27

29.80 2.99

13. The following graph illustrates an association between two variables. It shows percent changes in the range sizes of different species of native butterflies (red), birds (blue), and plants (black) of Britain over the past two to four decades (modified from Thomas et al. 2004). Identify (a) the type of graph, (b) the explanatory and response variables, and (c) the type of data (whether numerical or categorical) for each variable.

ASSIGNMENT PROBLEMS 14. The gene for the vasopressin receptor V1a is expressed at higher levels in the forebrain of monogamous vole species than in promiscuous vole species.11 Can expression of this gene influence monogamy? To test this, Lim et al. (2004) experimentally enhanced V1a expression in the forebrain of 11 males of the meadow vole, a solitary promiscuous species. The percentage of time each male spent huddling with the female provided to him (an index of monogamy) was recorded. The same measurements were taken in 20 control males left untreated. Control males: 98, 96, 94, 88, 86, 82, 77, 74, 70, 60, 59, 52, 50, 47, 40, 35, 29, 13, 6, 5 V1a-enhanced males: 100, 97, 96, 97, 93, 89, 88, 84, 77, 67, 61 a. Display these data in a graph. Explain your choice of graph. b. Which group has the higher mean percentage of time spent huddling with females? c. Which group has the higher standard deviation in percentage of time spent huddling with females? 15. The data in the accompanying table are from an ecological study of the entire rainforest community at El Verde in Puerto Rico (Waide and Reagan 1996). Diet breadth is the number of types of food eaten by an animal species. The number of animal species having each diet breadth is shown in the second column. The total number of species listed is n = 127. Diet breadth (number of prey types eaten) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Frequency (number of species) 21 8 9 10 8 3 4 8 4 4 4 2 5 2 1 1 2 1 3 2

>20 Total

25 127

a. Calculate the median number of prey types consumed by animal species in the community. b. What is the interquartile range in the number of prey types? Use the method outlined in Section 3.2 to calculate the quartiles. c. Can you calculate the mean number of prey types in the diet? Explain. 16. Francis Galton (1894) presented the following data on the flight speeds of 3207 “old” homing pigeons traveling at least 90 miles.

a. What type of graph is this? b. Examine the graph and visually determine the approximate value of the mean (to the nearest 100 yards per minute). Explain how you obtained your estimate. c. Examine the graph and visually determine the approximate value of the median (to the nearest 100 yards per minute). Explain how you obtained your estimate. d. Examine the graph and visually determine the approximate value of the mode (to the nearest 100 yards per minute). Explain how you obtained your estimate. e. Examine the graph and visually determine the approximate value of the standard deviation (to the nearest 50 yards per minute). Explain how you obtained your estimate. 17. A study of the endangered saiga antelope (pictured at the beginning of the chapter) recorded the fraction of females in the population that were fecund in each year between 1993 and 2001 (Milner-Gulland et al. 2003). A graph of the data is as follows:

a. Assume that you want to describe the “typical” fraction of females that are fecund in a typical year, based on these data. What would be the better choice to describe this typical fraction, the mean or the median of the measurements? Why? b. With the same goal in mind, what would be the better choice to describe the spread of measurements around their center, the standard deviation or the interquartile range? Why? 18. Accurate prediction of the timing of death in patients with a terminal illness is important for their care. The following graph compares the survival times of terminally ill cancer patients with the clinical prediction of their survival times (modified from Glare et al. 2003).

a. Describe in words what features most of the frequency distributions of actual survival times have in common, based on the box plots for each group. b. Describe the differences in shape of actual survival time distributions between those for one to five months predicted survival times and those for six to 24 months. c. Describe the trend in median actual survival time with increasing predicted number. d. The predicted survival times of terminally ill cancer patients tend to overestimate the medians of actual survival times. Are the means of actual survival times likely to be closer to, further from, or no different from the predicted times than the medians? Explain. 19. Measurements of lifetime reproductive success (LRS) of individual wild animals reveal the

disparate contributions they make to the next generation. Jensen et al. (2004) estimated LRS of male and female house sparrows in an island population in Norway. They measured LRS of an individual as the total number of “recruits” produced in its lifetime, where a recruit is an offspring that survives to breed one year after birth. Parentage of recruits was determined from blood samples using DNA techniques. Their results are tabulated as follows: Lifetime reproductive success

Frequency Females

Males

0 1 2 3 4 5 6 7 8 >8

30 25 3 6 8 4 0 4 1 0

38 17 7 6 4 10 2 0 0 0

Total

81

84

a. Which sex has the higher mean lifetime reproductive success? b. Every recruit must have both a father and a mother, so it is not easy to see why male and female LRS should differ. Can you think of a biological explanation? c. Which sex has the higher variance in reproductive success? 20. If all the measurements in a sample of data are equal, what is the variance of the measurements in the sample? 21. Researchers have created every possible “knockout” line in yeast. Each line has exactly one gene deleted and all the other genes present (Steinmetz et al. 2002). The growth rate—how fast the number of cells increases per hour—of each of these yeast lines has also been measured, expressed as a multiple of the growth rate of the wild type that has all the genes present. In other words, a growth rate greater than 1 means that a given knockout line grows faster than the wild type, whereas a growth rate less than 1 means it grows more slowly. Below is the growth rate of a random sample of knockout lines: 0.86, 1.02, 1.02, 1.01, 1.02, 1, 0.99, 1.01, 0.91, 0.83, 1.01

a. What is the mean growth rate of this sample of yeast lines? b. What is the median growth rate of this sample? c. What is the variance of growth rate of the sample? d. What is the standard deviation of growth rate of the sample? 22. As in other vertebrates, individual zebrafish differ from one another along the shy–bold behavioral spectrum. In addition to other differences, bolder individuals tend to be more aggressive, whereas shy individuals tend to be less aggressive. Norton et al. (2011) compared several behaviors associated with this syndrome between zebrafish that had the spiegeldanio (spd) mutant at the Fgfr1a gene (reduced fibro-blast growth factor receptor 1a)

and the “wild type” lacking the mutation. The data below are measurements of the amount of time, in seconds, that individual zebrafish with and without this mutation spent in aggressive activity over 5 minutes when presented with a mirror image. Wild type: 0, 21, 22, 28, 60, 80, 99, 101, 106, 129, 168 Spd mutant: 96, 97, 100, 127, 128, 156, 162, 170, 190, 195 a. Draw a boxplot to compare the frequency distributions of aggression score in the two groups of zebrafish. According to the box plot, which genotype has the higher aggression scores? b. According to the box plot, which sample spans the higher range of values for aggression scores? c. Which sample has the larger interquartile range? d. What are the vertical lines projecting outward above and below each box? 23. A eunuch is a castrated human male. Eunuchs were often used as servants and guards in harems in Asia and the Middle East. In males of some mammal species, castration increases life span. Do male eunuchs also have long lives compared to other men? The accompanying graph shows data on life spans of 81 male eunuchs from the Korean Chosun Dynasty between about 1400 and 1900, according to historical records. These data are compared with life spans of non-eunuch males who lived at the same time, and who belonged to families of similar social status (n = 1126, 1414, and 49 for the three families shown). Modified from Min et al. (2012), with permission.

a. What type of graph is this? b. What do the upper and lower margins of the boxes indicate? c. Which male group had the highest median longevity? d. Although the mean is not indicated on the graph, which sample of men probably had the highest mean longevity? Explain your reasoning. 24. As the Arctic warms and winters become shorter, hibernation patterns of arctic mammals are expected to change. Sheriff et al. (2011) investigated emergence dates from hibernation of

arctic ground squirrels at sites in the Brooks Range of northern Alaska. The measurements shown in the following figure are emergence dates in a sample of male and female ground squirrels at one of their study sites.

a. What type of graph is this? b. Which sex, males or females, has the earliest median emergence date? Explain how you obtained your answer. c. Which sex, male or female, has the greater interquartile range in emergence date? Explain how you obtained your answer. 25. Convert the following statistics, calculated on samples in English units, to the metric equivalents. (Conversion factors are given as well.) a. Mean: 100 miles (1 km = 0.62 miles) b. Standard deviation: 17 miles (1 km = 0.62 miles) c. Variance: 289 miles2 (1 km = 0.62 miles) d. Coefficient of variation: 17% (1 km = 0.62 miles) e. Mean: 23 pounds (1 kg = 2.2 pounds) f. Standard deviation: 1.2 ounces (1 g = 0.032 ounces) g. Variance: 550 gallons2 (1 liter = 0.227 gallons) 26. The snake undulation data of Example 3.1 were measured in Hz, which has units of 1/s (cycles per second). Often frequency measurements are expressed instead as angular velocity, which is measured in radians per second. To convert measurements from Hz to angular velocity (rad/s), multiply by 2p, where p = 3.14159. a. The sample mean undulation rate in the snake sample was 1.375 Hz. Calculate the sample mean in units of angular velocity. b. The sample variance of undulation rate in the snake sample was 0.105 Hz2. Calculate the sample variance if the data were in units of angular velocity. c. The sample standard deviation of undulation rate in the snake sample was 0.324 Hz. Calculate the sample standard deviation in units of angular velocity. Provide the appropriate units with your answer. 27. Spot the flaw. Crohn’s disease is an autoimmune inflammatory disorder. The accompanying

table shows medians and interquartile ranges for three response variables in 62 Crohn’s disease patients randomly assigned either the immuno-suppressant drug azathioprine (n = 32) or a placebo (n = 30) in a clinical trial. Response variables are measured as change from baseline. IQR is interquartile range. The data are from Candy et al. 1995. a. Identify the main flaw in the construction of this table. b. Redraw the table following the principles recommended in this book. TABLE FOR PROBLEM 27

Azathioprine Response Variable Crohn’s Disease Activity Index Erythrocyte sedimentation rate (mm/hr) Serum C reactive protein (%)

Placebo

Median IQR Median IQR 191.5 15.5 30.0

211 30 53

50.0 –6.5 0.0

230 26 27

28. Reproduction in sea urchins involves the release of sperm and eggs in the open ocean. Fertilization begins when a sperm bumps into an egg and the sperm protein bindin attaches to recognition sites on the egg surface. Gene sequences of bindin and egg-surface proteins vary greatly between closely related urchin species, and eggs can identify and discriminate between different sperm. In the burrowing sea urchin, Echinometra mathaei, the protein sequence for bindin varies even between populations within the same species. Do these differences affect fertilization? To test this, Palumbi (1999) carried out trials in which a mixture of sperm from AA and BB males, referring to two populations differing in bindin gene sequence, were added to dishes containing eggs from a female from either the AA or the BB population. The results below indicate the fraction of fertilizations of eggs of each of the two types by AA sperm (remaining eggs were fertilized by BB sperm). AA females: 0.58, 0.59, 0.69, 0.72, 0.78, 0.78, 0.81, 0.85, 0.85, 0.92, 0.93, 0.95 BB females: 0.15, 0.22, 0.30, 0.37, 0.38, 0.50, 0.95

a. Plot the data using a method other than the box plot. Is there an association in these data between female type and fertilizations by AA sperm? b. Inspect the plot. On this basis, which method from this chapter (mean or median) would be best to compare the locations of the frequency distributions for the two groups? Explain your reasoning. Calculate and compare locations using this method.

c. Which method would be best to compare the spread of the frequency distributions for the two groups? Explain your reasoning. Calculate and compare spread using this method. 29. The following graph illustrates an association between two variables. The graph shows density of fine roots in Monterey pines (Pinus radiata) planted in three different years of study (redrawn from Moir and Bachelard 1969, with permission). Identify (a) the type of graph, (b) the explanatory and response variables, and (c) the type of data (whether numerical or categorical) for each variable.

The sampling distribution of an estimate Estimation is the process of inferring a population parameter from sample data. The value of an estimate calculated from data is almost never exactly the same as the value of the population parameter being estimated, because sampling is influenced by chance. The crucial question is, “In the face of chance, how much can we trust an estimate?” In other words, what is its precision? To answer this question, we need to know something about how the sampling process might affect the estimates we get. We use the sampling distribution of the estimate, which is the probability distribution of all the values for an estimate that we might have obtained when we sampled the population. We illustrate the concept of a sampling distribution using samples from a known population, the genes of the human genome. EXAMPLE 4.1 The length of human genes The international Human Genome Project was the largest coordinated research effort in the history of biology. It yielded the DNA sequence of all 23 human chromosomes, each consisting of millions of nucleotides chained end to end.2 These encode the genes whose products—RNA and proteins—shape the growth and development of each individual. We obtained the lengths of all 20,290 known and predicted genes of the published genome sequence (Hubbard et al. 2005).3 The length of a gene refers to the total number of nucleotides comprising the coding regions. The frequency distribution of gene lengths in the population of genes is shown in Figure 4.1-1. The figure includes only genes up to 15,000 nucleotides long; in addition, there are 26 longer genes. 4

FIGURE 4.1-1 Distribution of gene lengths in the known human genome. The graph is truncated at 15,000 nucleotides; 26 larger genes are too rare to be visible in this histogram.

The histogram in Figure 4.1-1 is like those we have seen before, except that it shows the distribution of lengths in the population of genes, not simply those in a sample of genes.

Because it is the population distribution, the relative frequency of genes of a given length interval in Figure 4.1-1 represents the probability of obtaining a gene of that length when sampling a single gene at random. The probability distribution of gene lengths is positively skewed, having a long tail extending to the right. The population mean and standard deviation of gene length in the human genome are listed in Table 4.1-1. These quantities are referred to as parameters because they are quantities that describe the population. TABLE 4.1-1 Population mean and standard deviation of gene length in the known human genome. Name Parameter Value (nucleotides) Mean

μ

2622.0

Standard deviation

σ

2036.9

In real life, we would not usually know the parameter values of the study population, but in this case we do. We’ll take advantage of this to illustrate the process of sampling.

Estimating mean gene length with a random sample To begin, we collected a single random sample of n = 100 genes from the known human genome.5 A histogram of the lengths of the resulting sample of genes is shown in Figure 4.1-2.

FIGURE 4.1-2 Frequency distribution of gene lengths in a unique random sample of n = 100 genes from the human genome.

The frequency distribution of the random sample (Figure 4.1-2) is not an exact replica of the population distribution (Figure 4.1-1), because of chance. The two distributions nevertheless share important features, including approximate location, spread, and shape. For example, the sample frequency distribution is skewed to the right like the true population distribution. The sample mean and standard deviation of gene length from the sample of 100 genes are listed in Table 4.1-2. How close are these estimates to the population mean and standard

deviation listed in Table 4.1-1? The sample mean is Y¯=2411.8, which is about 200 nucleotides shorter than the true value, the population mean of μ = 2622.0. The sample standard deviation (s = 1463.5) is also different from the standard deviation of gene length in the population (σ = 2036.9). We shouldn’t be surprised that the sample estimates differ from the parameter (population) values. Such differences are virtually inevitable because of chance in the random sampling process. TABLE 4.1-2 Mean and standard deviation of gene length Y in our unique random sample of n = 100 genes from the human genome. Name Statistic Sample value (number of nucleotides) Mean Standard deviation

Y¯ s

2411.8 1463.5

The sampling distribution of Y¯ We obtained Y¯=2411.8 nucleotides in our single sample, but by chance we might have obtained a different value. When we took a second random sample of 100 genes, we found Y¯=2643.5. Each new sample will usually generate a different estimate of the same parameter. If we were able to repeat this sampling an infinite number of times, we could create the probability distribution of our estimate. The probability distribution of values we might obtain for an estimate make up the estimate’s sampling distribution. The sampling distribution is the probability distribution of all values for an estimate that we might obtain when we sample a population. The sampling distribution represents the “population” of values for an estimate. It is not a real population, like the squirrels in Muir Woods or all the retirees basking in the Florida sunshine. Rather, the sampling distribution is an imaginary population of values for an estimate. Taking a random sample of n observations from a population and calculating Y¯ is equivalent to randomly sampling a single value of Y¯ from its sampling distribution. To visualize the sampling distribution for mean gene length, we used the computer to take a vast number of random samples of n = 100 genes from the human genome. We calculated the sample mean Y¯ each time. The resulting histogram in Figure 4.1-3 shows the values of Y¯ that might be obtained when randomly sampling 100 genes, together with their probabilities.

FIGURE 4.1-3 The sampling distribution of mean gene length, Y, when n = 100. Note the change in scale from Figure 4.1-2.

Figure 4.1-3 makes plain that, although the population mean μ is a constant (2622.0), its estimate Y¯ is a variable. Each new sample yields a different Y¯ value from the one before. We don’t ever see the sampling distribution of Y¯ because ordinarily we have only one sample, and therefore only one Y¯. Notice that the sampling distribution for Y¯ is centered exactly on the true mean, μ. This means that Y¯ is an unbiased estimate of μ.6 The spread of the sampling distribution of an estimate depends on the sample size. The sampling distribution of Y¯ based on n = 100 is narrower than that based on n = 20, and that based on n = 500 is narrower still (Figure 4.1-4). The larger the sample size, the narrower the sampling distribution. And the narrower the sampling distribution, the more precise the estimate. Thus, larger samples are desirable whenever possible because they yield more precise estimates. The same is true for the sampling distributions of estimates of other population quantities, not just Y¯.

FIGURE 4.1-4 Comparison of the sampling distributions of mean gene length, Y¯, when n = 20, 100, and 500.

Increasing sample size reduces the spread of the sampling distribution of an estimate, increasing precision.

Measuring the uncertainty of an estimate In this section we show how the sampling distribution is used to measure the uncertainty of an estimate.

Standard error The standard deviation of the sampling distribution of an estimate is called the standard error. Because it reflects the differences between an estimate and the target parameter, the standard error reflects the precision of an estimate. Estimates with smaller standard errors are more precise than those with larger standard errors. The smaller the standard error, the less uncertainty there is about the target parameter in the population. The standard error of an estimate is the standard deviation of the estimate’s sampling distribution.

The standard error of Y¯ The standard error of the sample mean is particularly simple to calculate, so we show it here. We can represent the standard error of the mean with the symbol σY¯. It has a remarkably straightforward relationship with s, the population standard deviation of the variable Y: σY¯=σn. The standard error decreases with increasing sample size. Table 4.2-1 lists the standard error of the sample mean based on random samples of n = 20, 100, and 500 from the known human genome. TABLE 4.2-1 Standard error of the sampling distributions of mean gene length Y¯ according to sample size. These measure the spread of the three sampling distributions in Figure 4.1-4. Sample size, n Standard error, σY¯ (nucleotides) 20 100 500

455.5 203.7 91.1

The standard error of Y¯ from data The trouble with the formula for the standard error of the mean (σY¯) is that we almost never know the value of the population standard deviation (σ), and so we cannot calculate σY¯. The next best thing is to approximate the standard error of the mean by using the sample standard deviation (s) as an estimate of σ. To show that it is approximate, we will use the symbol SEY¯. The approximate standard error of the mean is SEY¯=sn.

According to this simple relationship, all we need is one random sample to approximate the spread of the entire sampling distribution for Y¯. The quantity SEY¯ is usually called the “standard error of the mean.” The standard error of the mean is estimated from data as the sample standard deviation (s) divided by the square root of the sample size (n). Calculating SEY¯ is so routine in biology that a sample mean should never be reported without it. For example, if we were submitting the results of our unique random sample of 100 genes in Figure 4.1-2 for publication, we would calculate SEY¯ from the results in Table 4.1-2 as follows: SEY¯=sn=1463.5100=146.3. We would then report the sample mean in the text of the paper as 2411.8 ± 146.3(SE). Every estimate, not just the mean, has a sampling distribution with a standard error, including the proportion, median, correlation, difference between means, and so on. In the rest of this book, we will give formulas to calculate standard errors of many kinds of estimates. The standard error is the usual way to indicate uncertainty of an estimate.

Confidence intervals The confidence interval is another common way to quantify uncertainty about the value of a parameter. It is a range of numbers, calculated from the data, that is likely to contain within its span the unknown value of the target parameter. In this section, we introduce the concept without showing exact calculations. Confidence intervals can be calculated for means, proportions, correlations, differences between means, and other population parameters, as later chapters will demonstrate. A confidence interval is a range of values surrounding the sample estimate that is likely to contain the population parameter. An example is the 95% confidence interval for the mean. This confidence interval is a range likely to contain the value of the true population mean μ. It is calculated from the data and extends above and below the sample estimate Y¯. You will encounter confidence intervals frequently in the biological literature. We’ll show you in Chapter 11 how to calculate an exact confidence interval for the mean, but for now we give you the result and its interpretation. The 95% confidence interval for the mean calculated from the unique sample of 100 genes (Table 4.1-2) is The 95% confidence interval provides a most-plausible range for a parameter. Values lying within the interval are most plausible, whereas those outside are less plausible, based on the data. 2121.4Y]

n=10:Pr[Y¯>Y]

ASSIGNMENT PROBLEMS 14. Assume that Z is a number randomly chosen from a standard normal distribution. Use the standard normal table to calculate each of the following probabilities: a. Pr[Z > 0.24] b. Pr[Z < 0.24] c. Pr[Z > 2.01] d. Pr[Z < 1.02] e. Pr[0.60 < Z < 1.4] f. Pr[−2 < Z < 2] g. Pr[Z < 0.45] h. Pr[−0.2 < Z < 0.37] 15. The highest recorded temperature during the month of July for a given year in Death Valley, in California, has an approximately normal distribution with a mean of 123.8°F (!) and standard deviation of 3.1°F (Weather Source 2009). a. What is the probability for a given year that the temperature never exceeds 120°F in a given July in Death Valley? b. What is the probability that the temperature in Death Valley goes above 128°F during July in a randomly chosen year? 16. Draw a distribution that is approximately normal, with mean equal to 10 cm and variance equal to 360. On the same graph, draw the distribution of the means of samples taken from this distribution, if each sample was a random sample of 10 individuals. 17. The following data are 40 measurements of diameter growth rate of the tropical tree Dipteryx panamensis from a long-term study at La Selva, Costa Rica (Clark and Clark 2012). The data are log-transformed, with the original units in millimeters. 0.0, 0.1, 0.1, 0.3, 0.4, 0.4, 0.4, 0.5, 0.5, 0.5, 0.6, 0.6, 0.7, 0.7, 0.7, 0.7, 0.8, 0.8, 0.8, 1.2, 1.2, 1.3, 1.4, 1.6, 1.9, 2.0, 2.1, 2.1, 2.2, 2.2, 2.3, 2.4, 2.5, 2.5, 2.7, 2.7, 2.7, 2.7, 2.8, 3.1 a. Make an appropriate graph of the data. b. Examine the graph. Do the data appear to be sampled from a population having a normal distribution? Why or why not? Identify all the features on which you based your conclusion. 18. In Europe, 53% of flowers of the rewardless orchid, Dactylorhiza sambucina, are yellow, whereas the remaining flowers are purple (Gigord et al. 2001). For this problem, you may use the normal approximation only if it is appropriate to do so.

a. If we took a random sample of a single individual from this population, what is the probability that it would be purple? b. If we took a random sample of five individuals, what is the probability that at least three are yellow? c. If we took many samples of n = 5 individuals, what is the expected standard deviation of the sampling distribution for the proportion of yellow flowers? d. If we took a random sample of 263 individuals, what is the probability that no more than 150 are yellow? 19. The amount of money spent on health care per person varies enormously among countries (The World Bank 2013). In 2010, this expense ranged from 11.9 U.S. dollars per person (in Eritrea) to $8361 (in the United States). The distribution of this per capita health care expenditure is very skewed, with a long tail corresponding to countries that spend a lot on health care per capita (see the top histogram in the accompanying graph). However, if we look at the log (base 10) of each country’s per capita health expenditure, it has a distribution that can be approximated by a normal distribution (see bottom histogram). On the log scale, the mean of the log expenditure is 2.47, with standard deviation equal to 0.72.

a. Assuming that log health expenditure is normal, calculate the proportion of countries that spend less than $100 per capita on health care. (This corresponds to a log expenditure of 2.) b. Assuming that log health expenditure is normally distributed, calculate the proportion of countries that spend more than $1000 per capita on health care. (This corresponds to a log expenditure equal to 3.) c. The true proportions of countries with per capita health expenditure less than $100 or more than $1000 are 0.30 and 0.21, respectively. Comment on why your answers from parts (a) and (b) above do not exactly match these values. 20. Recall from Example 10.4 that more women (29.1%) than men (11.1%) are excluded from the astronaut pilot program by the minimum and maximum height restrictions. What value of minimum height for women would exclude the same total proportion of women as men, given that 0.3% of women are too tall? 21. Draw a probability distribution that isn’t normal. Describe the features of your distribution that identify it as non-normal. 22. In the accompanying graph of a normal distribution, each of the two red areas represents one-sixth of the area under the curve. Estimate the following quantities from this graph: a. The mean b. The mode c. The median d. The standard deviation e. The variance

23. The proportion of traffic fatalities resulting from drivers with high alcohol blood levels in 1982 was approximately normally distributed among U.S. states, with mean 0.569 and standard deviation 0.068 (U.S. Department of Transportation Traffic Safety Facts 1999). a. What proportion of states would you expect to have more than 65% of their traffic fatalities from drunk driving? b. What proportion of deaths due to drunk driving would you expect to be at the 25th percentile of this distribution? 24. The table at the bottom of the page lists the means and standard deviations of several different normal distributions. For each distribution, calculate the probability of drawing a single Y value greater than the given threshold and the probability of drawing a value less than that threshold. 25. In European earwigs, the males sometimes have long pincers protruding from the end of their abdomens, as shown in the accompanying photo.

In graphs (a)−(c) we have plotted three frequency distributions of sample means of pincer lengths (in millimeters) based on random samples from an earwig population. One of the distributions shows means of samples based on n = 1, one shows means of samples based on n = 2, and one shows means of samples based on n = 8. Identify which frequency distribution corresponds to each sample size. Explain the basis for your decisions.

TABLE FOR PROBLEM 24

Mean Standard deviationThreshold Pr[Y > threshold] Pr[Y < threshold] 14

5

15 −23 14,000

3 4 5000

9 18.5 −16 9000

26. The crab spider, Thomisus spectabilis, sits on flowers and preys upon visiting honeybees, as shown in the photo at the beginning of the chapter. (Remember this the next time you sniff a wild flower.) Do honeybees distinguish between flowers that have crab spiders and flowers that do not? To test this, Heiling et al. (2003) gave 33 bees a choice between two flowers: one had a crab spider and the other did not. In 24 of the 33 trials, the bees picked the flower that had the spider. In the remaining nine trials, the bees chose the spiderless flower. With these data, carry out the appropriate hypothesis test, using an appropriate approximation to calculate P. 27. The following table lists the mean and standard deviation of several different normal distributions. In each case, a sample of 20 individuals was taken, as well as a sample of 50 individuals. For each sample, calculate the probability that the mean of the sample was less than the given value. TABLE FOR PROBLEM 27

Standard Meandeviation Value n=20: Pr[Y¯