Third Edition Statistical and Data Handling Skills in Biology Roland Ennos Statistical and Data Handling Skills in Biology Visit the Statistical and Data Handling Skills in Biology, Third Edition, Companion Website at www.pearsoned.co.uk/ennos to find valuable student learning material including: • • An Introduction to SPSS version 19 for Windows An Introduction to MINITAB version 16 for Windows Statistical and Data Handling Skills in Biology Third Edition Roland Ennos Faculty of Life Sciences, University of Manchester Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearson.com/uk First published 2000 Second edition published 2007 Third edition published 2012 © Pearson Education Limited 2012 The right of Roland Ennos to be identified as author of this Work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6‐10 Kirby Street, London EC1N 8TS All trademarks used therein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners Pearson Education is not responsible for the content of third‐party internet sites ISBN 978‐0‐273‐72949‐5 British Library Cataloguing‐in‐Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging‐in‐Publication Data A catalog record for this book is available from the Library of Congress 10 15 14 13 12 11 Typeset in 9/12.5 by 75 Printed by Ashford Colour Press Ltd., Gosport Dedication For my father Brief contents List of figures and tables Preface Publisher’s acknowledgements xiii xvii xix An introduction to statistics Dealing with variability 10 Testing for normality and transforming data 31 Testing for differences between one or two groups 43 Testing for difference between more than two groups: ANOVA and its non‐parametric equivalents Investigating relationships 132 Dealing with categorical data 164 Designing experiments 187 83 More complex statistical analysis 203 10 Dealing with measurements and units 213 Glossary Further reading Solutions Statistical tables Index 228 232 233 258 267 vii Contents List of figures and tables Preface Publisher’s acknowledgements An introduction to statistics 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1 3 Becoming a research biologist Awkward questions Why biologists have to repeat everything? Why biologists have to bother with statistics? Why is statistical logic so strange? Why are there are so many statistical tests? Using the decision chart Using this book Dealing with variability 10 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 10 10 13 16 17 19 21 22 26 30 Introduction Examining the distribution of data The normal distribution Describing the normal distribution The variability of samples Confidence limits Presenting descriptive statistics and confidence limits Introducing computer packages Calculating descriptive statistics Self‐assessment problems Testing for normality and transforming data 31 3.1 3.2 3.3 31 31 The importance of normality testing The Kolgomorov–Smirnov test What to if your data has a significantly different distribution from the normal Examining data in practice Transforming data The complete testing procedure Self‐assessment problems 34 35 37 42 42 Testing for differences between one or two groups 43 4.1 4.2 43 43 3.4 3.5 3.6 3.7 xiii xvii xix Introduction Why we need statistical tests for differences ix www.downloadslide.net Statistical tables Degrees of freedom Significance level 5% 262 1% 0.1% 38 39 40 235 249 264 194 207 220 150 161 172 41 42 43 44 45 279 294 310 327 343 233 247 261 276 291 183 195 207 220 233 46 47 48 49 50 361 378 396 415 434 307 322 339 355 373 246 260 274 289 304 51 52 53 54 55 453 473 494 514 536 390 408 427 445 465 319 335 351 368 385 56 57 58 59 60 557 579 602 625 648 484 504 525 546 567 402 420 438 457 476 www.downloadslide.net Statistical tables Table S5 Critical values for the Mann–Whitney U distribution Critical values of U at the 5% significance level Reject the null hypothesis if your value of U is less than or equal to than the tabulated value, for the sizes of the two samples, u1 and u2 n2 n1 10 11 12 13 14 15 16 17 18 19 20 – – – – – – – – – – – – – 0 – – 0 – – – – – – – – – – – 11 – 12 – 13 – 10 14 – 11 15 – 11 17 – 12 18 – 13 19 – 14 20 10 – – – – – – – 0 1 2 3 4 5 8 10 11 10 12 14 10 13 15 17 10 12 15 17 20 11 14 17 20 23 13 16 19 23 26 14 18 22 26 29 16 20 24 28 33 17 22 26 31 36 19 24 29 34 39 21 26 31 37 42 22 28 34 39 45 24 30 36 42 48 25 32 38 45 52 27 34 41 48 55 11 12 13 14 15 – – – – – 1 1 4 5 10 11 12 13 14 13 14 16 17 19 16 18 20 22 24 19 22 24 26 29 23 26 28 31 34 26 29 33 36 39 30 33 37 40 44 33 37 41 45 49 37 41 45 50 54 40 45 50 55 59 44 49 54 59 64 47 53 59 64 70 51 57 63 67 75 55 61 67 74 80 58 65 72 78 85 62 69 76 83 90 16 17 18 19 20 – – – – – 2 2 6 7 11 11 12 13 14 15 17 18 19 20 21 22 24 25 27 26 28 30 32 34 31 34 36 38 41 37 39 42 45 48 42 45 48 52 55 47 51 55 58 62 53 57 61 65 69 59 63 67 72 76 64 67 74 78 83 70 75 80 85 90 75 81 86 92 98 81 86 92 87 93 99 93 99 106 99 106 113 105 112 119 98 105 112 119 127 263 www.downloadslide.net Statistical tables Table S6 Critical values for the Friedman x2 distribution Critical values of x2 at the 5%, 1% and 0.1% significance levels Reject the null hypothesis if your value of x2 is greater than or equal to than the tabulated value, for a groups and b blocks 264 Groups Blocks Significance level a b 5% 1% 0.1% 3 3 3 3 3 3 3 10 11 12 13 14 15 – 6.000 6.500 6.400 7.000 7.143 6.250 6.222 6.200 6.545 6.167 6.000 6.143 6.400 – – 8.000 8.400 9.000 8.857 9.000 9.556 9.600 9.455 9.500 9.385 9.000 8.933 – – – 10.000 12.000 12.286 12.250 12.667 12.600 13.273 12.500 12.923 13.286 12.933 4 4 4 4 4 4 4 10 11 12 13 14 15 6.000 7.400 7.800 7.800 7.600 7.800 7.650 7.800 7.800 7.909 7.900 7.985 7.886 8.040 – 9.000 9.600 9.960 10.200 10.371 10.350 10.867 10.800 11.073 11.100 11.123 11.143 11.240 – – 11.100 12.600 12.800 13.800 13.800 14.467 14.640 14.891 15.000 15.277 15.257 15.400 5 5 5 5 5 10 7.600 8.533 8.800 8.960 9.067 9.143 9.300 9.244 9.280 8.000 10.133 11.200 11.680 11.867 12.114 12.300 12.444 12.480 – 11.467 13.200 14.400 15.200 15.657 16.000 16.356 16.480 www.downloadslide.net Statistical tables Groups Blocks Significance level a b 5% 1% 0.1% 6 6 6 6 6 10 9.143 9.857 10.286 10.486 10.571 10.674 10.714 10.778 10.800 9.714 11.762 12.714 13.229 13.619 13.857 14.000 14.143 14.299 – 13.286 15.286 16.429 17.048 17.612 18.000 18.270 18.514 265 www.downloadslide.net Statistical tables Table S7 Critical values for the Spearman rank correlation coefficient R Critical values of the correlation coefficient r at the 5%, 1% and 0.1% significance levels Reject the null hypothesis if your absolute value of r is greater than or equal to the tabulated value at the chosen significance level, for the calculated number of degrees of freedom Degrees of freedom Significance level 5% 266 1% 0.1% 1.000 10 0.886 0.786 0.738 0.700 0.648 1.000 0.929 0.881 0.833 0.794 1.000 0.976 0.933 0.903 11 12 13 14 15 0.618 0.587 0.560 0.538 0.521 0.755 0.727 0.703 0.679 0.654 0.873 0.846 0.824 0.802 0.779 16 17 18 19 20 0.503 0.485 0.472 0.460 0.447 0.635 0.615 0.600 0.584 0.570 0.762 0.748 0.728 0.712 0.696 25 30 35 40 45 0.398 0.362 0.335 0.313 0.294 0.511 0.467 0.433 0.405 0.382 0.630 0.580 0.539 0.507 0.479 50 60 70 80 90 0.279 0.255 0.235 0.220 0.207 0.363 0.331 0.307 0.287 0.271 0.456 0.418 0.388 0.363 0.343 100 0.197 0.257 0.326 www.downloadslide.net Index Note: definitions in the Glossary are indicated by emboldened numbers allometry 154–5 analysis of covariance (ANCOVA) 205, 228 univariate ANOVA 204 analysis of variance (ANOVA) 83, 146, 228 multivariate analysis of variance (MANOVA) 204 nested ANOVA 122–9 nomenclature 87 one‐way ANOVA 84–96 post hoc tests 91–4 repeated measures ANOVA 96–101 two‐way ANOVA 112–17 univariate ANOVA 204 Anderson–Darling test 31 association, tests for 228 correlation 135–44, 208 regression 145–51 bar chart 10 see also histogram between‐group variability 85 binomial distributions 15, 165, 228 biological replicates 189 blocking 190, 228 calculations 221 canonical variate analysis (CVA) 211 categorical data problem of variation 164–6 quantification 164 category 228 causal relationship 145, 228 chi‐squared (x2) test 166, 228 for association 171–8 critical values 260 for differences 166–71 validity 178 cluster analysis 211 complex statistical analysis experiments in which no control of variables 204–8 experiments investigating several factors 204 exploring data to investigate groupings 211–12 investigate the relationships between two sets of variables 208–10 legitimate cases 203 computer packages, for statistical calculations see MINITAB; SPSS confidence limits 19–21, 228 calculating 45–6 confounding variables 188–9, 228 contingency table 228 continuity correction 178 continuous data controls correlation 135–45, 208, 228 Pearson correlation coefficient 136–8 Spearman rank correlation 156‐161 correlation coefficient 136–8 critical values 259 uses 145 critical values chi‐squared (x2) test 260 correlation coefficient 259 for Friedman x2 distribution 264–5 for Mann–Whitney U distribution 263 for Spearman rank correlation coefficient 266 of t statistic 19, 45, 228, 258 for Wilcoxon T distribution 261–2 cumulative standard deviation 18–19 data 228 continuous 10–11 discrete 10–11 distribution of 10–13 MINITAB analysis in distribution of 25–6 267 www.downloadslide.net Index data (continued ) non‐normally distributed 156 normally distributed see normally distributed data proportional 34–5 ranked 34 relationship 132–3 skewed 12, 34 SPSS analysis in distribution of 23–5 symmetrically distributed 34 decision chart for statistical tests 6–8 degrees of freedom 16, 19, 87, 229 dependent variable 132, 229 descriptive statistics 3, 229 graphical representation 22 measuring variability 21–2, 26–9 in text/tables 21–2 using MINITAB 28–9 using SPSS 26–8 discrete data 10, 34 discrimination function analysis 211 distribution of data 14, 229 normal 13–17 representation 11–13 types of 10–11 distribution of sample double blind experiment 188 Dunnett test 91 error bars 22, 229 estimate 16, 229 experiment experimental design dealing with results 200 excluding confounding variables 188 power calculations 194–200 preparation 188 pseudoreplication 188–9 replication 188–9 role of 187 exponent 229 exponential relationship 153–4, 229 F statistic 86–7 Fisher’s exact test 166 frequency 5, 164, 229 Friedman’s test 106–12 G test 166 general linear model (GLM) 205, 229 268 graphs associations in 133 correlation analysis 144 plotting of 133 histogram 10, 31 see also bar chart hypothesis testing imperial scale 218, 229 independent sample t test 46 independent variable 132, 229 interaction 204, 229 intercept 134, 229 interquartile range 12 interval data irregularly distributed samples 34 Kolgomorov–Smirnov test 31, 42, 166 Kruskall–Wallis test 101–6 Latin square design 190 line of best fit 146 linear associations 145 linear relationships 133–5 statistical tests 135 logarithms base 10 (log10) 222–3, 229 for graphs 224 natural 224 for pH 224 properties and uses 223 logistic regression 180–4, 210, 229 results of 184–5 vs chi‐squared tests 184 Mann–Whitney U test 46, 75–9 mean 12, 44, 165, 229 mean squares (MS) 87, 229 measurements 5, 213, 229 median 12, 34, 64, 69, 229 metric 229 MINITAB 25–6, 31 ANCOVA analysis 207–8 calculating descriptive statistics 28–9 Friedman’s test 109–11 Kruskall–Wallis test 104–5 logistic regression 182–4 Mann–Whitney U test 77–9 nested ANOVA 126 www.downloadslide.net Index one‐sample sign test 67–9 one‐sample t test 48–51 one‐way ANOVA 89–90 paired t test 54–6 Pearson correlation coefficient 142–3 plotting relationship data 140–1 post hoc tests 94 power calculations for ANOVA 198 in one‐sample and paired t tests 196–7 in two‐sample t tests 198 rank correlation 159–60 regression analysis 155–6 repeated measures ANOVA 100 Scheirer–Ray–Hare test 120 two‐sample t test 60–2 and two‐tailed tests 46 two‐way ANOVA 115–17 Wilcoxon matched pairs test 72–4 mode 11 molarity M of a solution 217 multiple regression 210 multivariate analysis of variance (MANOVA) 204 natural logarithms 224, 230 negative association 133–4 negatively skewed data 11, 34 nested ANOVA 122–9, 189 non‐linear relationships 152–6 non‐metric units 218 non‐parametric tests 5, 34, 46, 230 Friedman’s test 106–12 Kruskall–Wallis test 101–6 Mann–Whitney U test 75–9 one‐sample sign test 64–9 Scheirer–Ray–Hare test 117–22 Spearman rank correlation 156‐161 Wilcoxon matched pairs test 69–74 normal distribution 5, 13–17, 20, 230 characteristics 14–16 description 16–17 normally distributed data 204 characteristics 14–16, 31 correlation analysis 138, 208 in Kolgomorov–Smirnov tests 42 symmetrically distributed 34–5 t tests 46–7 two‐sample t test 57 null hypothesis 4, 45, 230 rejection of one‐sample sign test 64–9 one‐sample t test 46–51, 150 one‐way ANOVA 84–96 ordinal data paired measurements 51 relationship between two sets of 132, 135, 145, 156 two 69 paired t test 46, 51–6 parameters 16, 230 parametric tests 5, 34, 46, 230 partial correlation 209 Pearson correlation 156, 208 Pearson correlation coefficient 136–8 pH 224 placebo 188, 230 population 3, 230 population means 44 population standard deviation 16 positive association 133–4, 136–7, 144 positively skewed data 11, 34 post hoc tests 91, 230 Friedman’s test 111 Kruskall–Wallis test 105–6 power calculations 194–6 for ANOVA 198 for chi‐squared tests for differences 199 in one‐sample and paired t tests 196–7 for test for relationships 199–200 in two‐sample t tests 197–8 power relationships 152–3, 230 prefixes 214–15, 230 proportional data 34–5 pseudoreplication of experiment 188–9 quartiles 12, 230 randomisation 189, 190 rank, of data point 64, 230 rank correlation 156–61, 208 ranked data 5, 34, 156 ranks reduced major axis regression 147 regression analysis 145–51, 230 difference from an expected value 150 presenting results 151 269 www.downloadslide.net Index regression analysis (continued ) procedure 147–50 validity 147 related measurements, tests for 6, 96, 106 relationship between data 132–3 linear 133–5 repeated measures ANOVA 96–101 replicated observations, of a sample replicates 230 replication of experiment 188–9, 230 research biologist becoming a questions related to biological statistics 1–2 and repeating experiments 1–2 significance of statistics Ryan–Joiner test 31 sample 16, 230 sample means 44 different from an expected value 44 scatter plot 132, 230 Scheffe test 91 Scheirer–Ray–Hare test 117–22 scientific notation 215–16, 230 Shapiro–Wilk test 31 SI units 213–14, 230 combining values 218–19 constants and formulae 220–1 dealing with large and small numbers 214–16 derived 215 expressing answers 219 non 216–18 prefixes 214–15 scientific notation 215–16 steps in converting to 219–20 sign test 46 significance probability 3–4, 231 calculating 45 significant difference 44, 45, 231 skewed data 12, 34, 231 slope of line 134, 231 Spearman rank correlation 156 critical values 266 SPSS 23–5, 31 ANCOVA analysis 206–7 calculating descriptive statistics 26–8 Friedman’s test 108–11 Kruskall–Wallis test 102–3 logistic regression 181–2 270 Mann–Whitney U test 76–9 nested ANOVA 124–6 one‐sample sign test 65–9 one‐sample t test 47–51 one‐way ANOVA 88–9 paired t test 53–6 partial correlation 209–10 Pearson correlation coefficient 141–2 plotting relationship data 139–40 post hoc tests 93–4 rank correlation 158, 160 regression analysis 154–5 repeated measures ANOVA 96–9 Scheirer–Ray–Hare test 118–20 two‐sample t test 58–62 and two‐tailed tests 46 two‐way ANOVA 114–15 Wilcoxon matched pairs test 71–4 standard deviation 16, 146, 165, 231 standard error 19, 44, 165, 231 standard error bars 22 standard error of the difference 51, 57, 231 statistical logic 3–4 statistical tests choosing 192–4 decision chart for 6–8 for differences 43–4 for differences and relationships between categorical data for differences between sets of measurements 5–6 for relationships between measurements stages types of data types of questions 5–6 statistics 16, 231 sums of squares (SS) 86 survey symmetrically distributed data 10–11, 34 t distribution 19, 231 t tests 43, 146, 231 stages 44–5 unsuitability 83–4 technical replicates 189 test statistic in ANOVA tests 86 calculating 45 three‐way ANOVA 204 www.downloadslide.net Index transformation 34, 135, 231 using MINITAB 38–40 using SPSS 37–8 Tukey test 91 two paired measurements 69 two‐sample t test 46, 56–64 two‐tailed tests 46, 231 two‐way ANOVA 112–17, 190 type error 4, 84, 231 type error 4, 231 univariate ANOVA 204 univariate tests 204 unpaired measurements 56 unrelated measurements, tests for 84, 101 in data distribution 10–13 normal distribution 13–17 one‐way ANOVA 84–7 of samples 17–19 using descriptive statistics 21–2, 26–9 using statistical packages 22–6 variance 16, 87, 231 Wilcoxon matched pairs test 69–74 within‐group variability 86 Yates continuity correction 178 Z distribution see normal distribution variability confidence limits for the population 19–21 271 www.downloadslide.net www.downloadslide.net www.downloadslide.net www.downloadslide.net www.downloadslide.net .. .Statistical and Data Handling Skills in Biology Visit the Statistical and Data Handling Skills in Biology, Third Edition, Companion Website at www.pearsoned.co.uk /ennos to find valuable... learning material including: • • An Introduction to SPSS version 19 for Windows An Introduction to MINITAB version 16 for Windows Statistical and Data Handling Skills in Biology Third Edition Roland... years since the second edition of Statistical and Data Handling Skills in Biology was first published and I am grateful to Person Education for allowing me the opportunity to update and expand the