STATISTICAL METHODS FOR AGRICULTURAL WORKERS ST ATISTICAL METHODS FOR AGRICULTURAL WORKERS BY V. G. PANSE, B.Se., Ph.D. (Lond.), F.N.I. Statistical Adviser Indian Council of Agricultural Research, New Delhi AND \ P. V. SUKHATME, Ph.D., D.Se. (Lond.), F.N.1. Chief, Statistics Branch Food and Agriculture Organization of the United Nations, Rome Published by INDIAN COUNCIL OF AGRICULTURAL RESEARCH NEW DELHI Published 1954 Reprinted 1957 b r , ~ ,. CCihh k •. 0.ti'! .. PREFACE THIS book had its origin in the proposal to revise and reprint the' Handbook of Stati~tics for Use in Plant Breeding and Agricultural Problems' by F. J. F. Shaw published by the Indian (then Imperial) Council of Agricultural Research in 1936 but which has been out of print for some years. The present authors were invited by the Council to revise and bring it up to date in order that it should meet the normal requirements of the present-day agricultural research workers. In the course of the revision It became clear that not only would there have to be a change in the order and manner of the presentation of the subject-matter but also a considerable expansion in the scope of the material included in the book, even whde keeping within the limits set out by the objective of the book, namely, to provide an elementary text-book for' the use of agricultural research workers whose mathematical attainments are modest. As it turned out finally the revision amounted to an almost complete rewriting of the book and addition of several new chapters. The book now consists of sixteen chapters divided into two parts, the first with six chapters dealing with statistical methods and the second with the design of experiments. The elaboration of the second part is in keeping with the greater emphasis now rightly laid on the proper planning of experiments in order that they may supply efficiently and economically the information sought. The subject-matter has been illustrated throughout with appropriate examples and the appendices would provide material helpful in practical work. With the new arrangement the book commences with a chapter on frequency distribution, graphic representation, averages and measures of <iispersion, etc., which were dealt with in the second .and third. chapters of the previous publication. The material relating to nonna! c~;ve and probability integral is presented in the. second chapter ill'"a 'more concise and systematic manner than was done in the fourth and fifth chapters of the original. A note on binomial distribution is also included in this chapter on account of its utility in dealing with frequency data, proportions, etc. The third chapter introduces vi the idea of sampling and sampling errors and is practically new. Opportunity is taken for an early introduction of the analysis of variance by explaining the technique in an elementary form in the fourth chapter. The fifth chapter is devoted to methods of handling frequency data. Besides giving the various uses of the ;Y2 test, the more important methods· of estimating linkage including the one based on maximum likelihood are discussed. In presenting methods dealing with the relations between two variables consi derable changes have been made and the technique of regression has been given due prominence. It is presented in the sixth chapter. The topics of partial and multiple regression and corre lation and intrac1ass correlation which were not treated in the previous publication are incorporated in the same chapter. The remaining chapters of the book deal with design of experiments. Apart from randomized block and latin square designs and some ideas relating to soil heterogeneity and ana lysis of covariance, the entire· material in this part is new. Chapter VII introduces the principles of replication, randomization .and local control underlying the design of experiments as well as the idea of efficiency of an experiment. Randomized block and latin square designs are discussed in the eighth chapter with the addition of. material on size and shape of blocks, number of replications, the process of randomization, etc. Three new chapters, IX to XI, have been added in order to deal with factorial experimentation, confounding and split-plot and strip plot designs. Analysis of covariance and methods of dealing with incomplete data are presented in Chapter XII. Chapter XIIl should be of special interest to plant breeders as it discusses the designs and analysis of replicated progeny row and compact family block trials and introduces incomplete block designs. The important problem of the analysis of groups of experiments has. been discussed in Chapter XIV. A special chapter has been devoted to the consideration of practical aspects of field experi mentation in order to emphasize their importance in the efficient prosecution of experimental studies. The last chapter, XVI, deals with the problem of conducting experiments on farmers' own land, which is assuming a rapidly growing importance and in which interesting developments are expected. VII In view of the restriction on the scope of the book mentioned earlier and in due regard to its size it was not considered desirable to attempt a more exhaustive treatment of the subject-matter or to include more explanatory material concerning the principles discussed. For the benefit of the more enterprising student, who wishes to pursue the study of the subject in greater detail, references to further sources of information concerning various aspects of statistical methods and design of experiments have been given. It is to be hoped that in its present form the book will meet the day to day needs of the agricultural and other biological research workers and form a suitable text for teaching the subject to the students of these sciences. The authors wish to record their keen appreciation of the careful and painstaking work done by Mr. S. D. Bokil, who was associated throughout with the project of preparing the present text. Their thanks are also due to Mr. V. N. Amble for hIs help in preparing the drafts of certain new sections and in making a critical scrutiny of several chapters of the manuscript and to Messrs. K. S. Krishnan and K. S. Avadhani for checking the numerical calculations. Lastly, they wish to make a special mention of the help they received from Dr. P. N. Saxena in reading the proofs. The authors are indebted to Professor Sir Ronald A. Fisher, Cambridge, and to Messrs. Oliver and Boyd Ltd., Edinburgh, for their permission to reprint Tables I and III from their book 'Statistical Methods for Research Workers '. V. G. PANSE. New Delhi, January 1954. P. V. SUKHATME. CONTENTS PAGE v PREFACE PART 1. STATISTICAL METHODS CHAPTER I. FREQUENCY DISTRIBUTIONS la.l Introduction. 1 Ib.l Frequency Distribution . 2 Ib.2 Graphic Representation 9 Ib.3 Frequency Curve and its Characteristics 10 Ie .1 Measures of Central Value or Location 10 le.2 Measures of Dispersion 12 le.3 The Coefficient of Variation 14 Id.l Methods of Computation . 15 II. THE NORMAL AND BINOMIAL DISTRIBUTIONS 2a. 1 The Probability Integral 22 2a.2 The Normal Curve . 25 2a.3 Use of the Normal Probability Integral 27 2b. 1 Binomial Distribution . 32 III SAMPLING METHOD AND STANDARD ERRORS 3a. 1 Introduction. 36 3b. 1 Random Sampling . 36 3b.2 Use of Random Numbers 38 3b. 3 Expectation of Sample Estimates 40 3b.4 Requisites of Randomization . 40 3c. 1 Distribution of Sample Means 42 3e.2 Confidence Limits 44 PAGE 3d I Standard Error of Sum and Difference of Means 4S 3d.2 Standard Error of Linear Function of Independent Variates. 47 3d.3 Sample Estimate of Variance 48 3d.4 Standard Error of Estimate of Propor- tion . 48 IV. TESTS OF SIGNIFICANCE OF MEANS AND THEIR DIF FERENCES 4a.l Significance of Means in Large Samples SI 4a.2 Significance of Difference of Means in Large Samples S4 4b.l Small Samples: 't' Test S7 4b.2 't' Test in Paired Samples 62 4c.l Analysis of Variance 63 4c.2 The 'F' Test 66 4d.l' Fisher and Behren's 'd' Test 67 V. THE X2 TEST AND ESTIMATION OF LINKAGE Sa.l Introduction 70 Sa.2 Testing the Significance of Proportion 70 Sb.l The X2 Test of Goodness of Fit 72 Sb.2 Test of Independence: 2 x 2 Table 7S Sb.3 Test of Independence: 2 X r Table 79 Sb.4 Testing Heterogeneity 81 Sb.S Test of Independence: r X c Table 83 Sc.l Use of X2 Test in Genetic Experiments 83 Sc.2 Size of Experiments . 8S Sc.3 The Partition of X2 86 Sc.4 Detection of Linkage 88