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FUNDAMENTALS OFBIOSTATISTICS

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FUNDAMENTALS OFBIOSTATISTICS

P. HANMANTH RAOM.Sc., Ph.D.

Academic Advisor

Sujatha Degree & PG College for Women

Hyderabad, Andhra Pradesh

K. JANARDHANM.Sc., Ph.D., PG Dip.(Maths)

Head, PG Biotechnology

A.V. College, Hyderabad

I.K. International Publishing House Pvt. Ltd.

NEW DELHI BANGALORE

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Published by

I.K. International Publishing House Pvt. Ltd.S-25, Green Park ExtensionUphaar Cinema MarketNew Delhi - 110 016 (India)E-mail: info@ikinternational.com

ISBN: 9 7 8 - 9 3 - 8 0 5 7 8 - 7 1 - 2

c⃝ 2010 I.K. International Publishing House Pvt. Ltd.

10 9 8 7 6 5 4 3 2 1

All rights reserved. No part of this book may be reproduced or used in any form,electronic or mechanical, including photocopying, recording, or by any informationstorage and retrieval system, without written permission from the publisher.

Published by Krishan Makhijani for I.K.International Publishing House Pvt. Ltd.,

S-25, Green Park Extension, Uphaar Cinema Market, New Delhi-110 016. Printed

by Rekha Printers Pvt. Ltd., Okhla Industrial Area, Phase II, New Delhi-110 020.

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FOREWORD

Statistics and more so, mathematics are usual deterrents to students of Biologyand Agriculture. But developments in the field of Biology are vast and intense;many of the basic modes of inference are based on principles of statisticsexplicitly or implicitly. Modern science is no exception in this context. Forexample, studies using molecular techniques are not only advanced in termsof the laboratory tools employed but equally in the interpretation of resultsusing underlying statistical and mathematical logic. Investigators specialisedin molecular biology and genetic engineering is often restrained from compre-hending the emerging results in proper depth and perspective due to restrictedknowledge of fundamentals of statistics and their extension to biology. On theother hand, quantitative tools employed in the areas of biology and agricultureremains yet to be understood in depth. Consequently, classical interventions foragricultural and biological improvement remain more routine than innovative.

Textbooks on principles of statistics are many, written by Indian or foreignauthors. The basic knowledge assumed in foreign texts is most often beyondthat of Indian readers. Indian textbooks, on the other hand, reflect most often,the university or academic environment in which the authors spent a majorpart of their career. They thus do not serve the needs of students across otheruniversities. Any textbook therefore has to take into account the inherentdiversity of the readership.

This book by Hanmanth Rao and Janardhan sets to recognise such needs.Both the authors have taught the subject at various levels to undergraduateand graduate students for many years. They have therefore attempted to coverthe needs of biology students of most of the universities. They have keptmathematical exposition in a low profile and set to explain mathematical andstatistical concepts through varied examples.

I hope this book would serve as a companion to students and teachers ofbiology and statistics. As a timely venture, I wish this book every success.

V. Arunachalam,Formerly Head, Division of Genetics,

Indian Agriculture Research Institute,New Delhi.

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PREFACE

The textbook “Fundamentals of Biostatistics” caters to undergraduate, gradu-ate levels of Biological sciences, Agriculture, Pharmacy, Medicine andGenetics.

Chapter One “Mathematics” deals with the set theory, permutations andcombinations, functions, logarithms, limits and differentiation.

The Second Chapter deals with Frequency Distribution, measures of centralvalues and dispersion. Many worked out examples are included based onbiological concepts.

In Chapter Three and Four Probability Distributions are covered. Salientfeatures are Kolmogorov’s axioms, Bernoullian trials, concept of random vari-able, fitting of binomial and Poisson distributions, approximation of binomialto normal distribution.

Chapter Five covers Sampling and Test of Hypothesis with a clear notion onlevels of significance, null and alternate hypothesis, Type I and Type II errors.

Chapter Six deals with t-Test (small sample test).Chapter Seven is on The Chi-square Distribution and its application to

genetics.Chapter Eight deals with Linear Regression and Correlation. Significance of

Correlation Coefficient and Regression Coefficients are discussed with workedout examples.

Chapter Nine is on “Analysis of Variance (ANOVA)”. One-way ANOVA,two-way ANOVA are elaborated with worked out examples.

Technological changes have rapidly taken place during the past decade.Statistical software like SPSS(II) is available for data analysis. Students andteachers should note that statistical software packages are regularly reviewedand advertised in the periodicals such as The American Statistician (Quarterlypublication of the American Statistical Association). Students may also bebenefited with free interactive demos of Sigma plot (statistical package availableat www.systat.com)

Those who want to pursue advance studies in Statistical Genetics, ClinicalTrials, Epidemiology would find this text as the primer.

We thank Prof. V. Arunachalam, Former Head, Genetics Division IARI,New Delhi for reviewing the manuscript and for offering critical commentsduring the preparing of textbook. We acknowledge our family members fortheir patience during the preparation of the book which took more than onedecade. Author K. Janardhan expresses his gratitude to his daughter Prasoonafor her constant help.

The Authors take this opportunity to thank Shri. Krishan Makhijani,Managing Director, I.K. International Publishing House Pvt. Ltd., for hisco-operation in bringing out this textbook on time.

The Authors.

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CONTENTS

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1. Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-76

2. Frequency Distribution — Measures of Central Values and

Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77-124

3. Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125-150

4. Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . 151-190

5. Concept of Sampling and Tests of Hypotheses . . . . . . . . . . . . 191-220

6. t-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221-230

7. Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . 231-240

8. Linear Regression and Correlation . . . . . . . . . . . . . . . . . . 241-262

9. Analysis of Variance (ANOVA) . . . . . . . . . . . . . . . . . . . . 263-272

Annexure

Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274-275

Antilogarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 276-277

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