Introductory Statistics

Preface

Introduction


1 Presentation of Sets of Measurements


1.1 Sampling


1.2 Ordering the Sample


1.3 Grouping


1.4 Diagrammatic Representation


1.5 The Arithmetic Mean and Its Calculation


1.6 Inadequacy of Arithmetic Mean


1.7 Measures of Spread: Range and Mean Deviation


1.8 Measures of Spread: Standard Deviation and Variance


1.9 Sample Estimates


1.10 Calculation of Standard Deviation and Variance

Summary

Examples


1A.11 Other Measures: Median and Mode


1A.12 Further Measures: Quartiles and Coefficient of Variability


1A.13 Moments and Other Coefficients


1A.14 Formula for Calculation of Standard Deviation

Summary

Examples


2 Normal Distribution


2.1 Frequency Distributions


2.2 Frequency Curves


2.3 Normal Distribution


2.4 Applications of Normal Distribution


2.5 Alternative Form of Normal-Deviate Table

Summary

Examples


2A.6 Theoretical Distributions: The Binomial


2A.7 Application of Normal Approximation to Binomial Distribution


2A.8 Accuracy of an Estimated Proportion


2A.9 Theoretical Distributions: The Poisson

Summary

Examples


2A.10 A Rough Estimate of The Standard Deviation


3 Comparison of Two Sets of Measurements


3.1 Method of Comparison


3.2 Variance-Ratio Test


3.3 Examples of Use of Variance-Ratio Test


3.4 Pooling of Variances


3.5 Accuracy of Arithmetic Mean


3.6 Examples of Determination of Accuracy of Arithmetic Mean


3.7 Comparison of Arithmetic Means


3.8 Examples of Test of Difference Between Means


3.9 The t Test


3.Summary


3.Examples


3A.10 Calculation of Number of Observations Necessary for a Given Accuracy


3A.11 Relative Precision and Combination of Experimental Results


3A.12 Estimation of Variance From Observations of Differing Precision


3A.13 Proof of Formula for Variance of Sum of Independent Measurements

Summary

Examples


4 Comparison of Several Sets of Measurements


4.1 The Problem and Its Solution


4.2 Analysis of Variance


4.3 Further Examples of Analysis of Variance


4.4 Orthogonality and Interaction of Effects


4.5 Randomized Block Analysis


4.6 Examples of Randomized Block Analysis


4.7 Latin and Graeco-Latin Squares


4.8 Analysis of A Latin Square

Summary

Examples


4A.9 Testing Particular Comparisons


4A.10 Test for Interaction


4A.11. Examples of Test for Interaction


4A.12. Higher Order Interactions and Factorial Principle


4A.13. Confounding

Summary

Examples


5 Attributes and Comparison of Proportions


5.1 Measurement of Attributes


5-2 Comparison of Several Groups


5.3 The Chi-Squared Test


5.4 An Alternative Computational Procedure


5.5 Comparison of Several Proportions


5.6 Testing 2 x 2 Tables


5.7 Yates' Correction for Continuity


5.8 Testing Goodness of Fit


5.9 Goodness of Fit With Estimated Constants

Summary

Examples


5A.10 Components of Chi-Squared


5A.11 Combination of Tests of Significance


5A.12 Exact Testing of 2 x 2 Tables

Summary

Examples


6 Interrelations of Sets of Measurements


6.1 Associated Measurements


6.2 Diagrammatic Presentation


6.3 General Test for Association


6.4 Measures of Joint Variation


6.5 Fitting Straight Lines


6.6 Regression Lines


6.7 Correlation Coefficients

Summary

Examples


6A.8 Theory of Minimal Variance


6A.9 Example of Multiple Regression


6A.10 Significance of Particular Variates


6A.11 Partial Correlation Coefficients


6A.12 Curvilinear Regression

Summary

Examples


7 Concomitant Observations


7.1 Comparison of Regression Coefficients


7.2 Comparison of Regressions


7.3 Concomitant Observations


7.4 Analysis of Covariance


7.5 Use of Analysis of Covariance in Estimation of Associations


7.6 Regression of Group Means

Summary

Examples


7A.7 Analysis of Covariance With Two or More Sets of Concomitant Observations


7A.8 Dummy Variates


7A.9 Non-Orthogonality


7A.10 Non-Orthogonal Comparisons When Interaction Exists


7A.11 Discriminant Functions

Summary

Examples


8 Transformations and Non-Normal Distributions


8.1 Reasons for Transformations


8.2 Transformations To Equalize Variances

I Logarithmic Transformation

II Square Root Transformation

III Sin- 1 √P Transformation

Iv 1/ßsinh-1ß√x Transformation

V Reciprocal Transformation


8.3 Transformations To Achieve Normality

I Logarithmic, Square Root and Reciprocal Transformation

II Transformation of Ranks To Normal Scores


8.4 Numerical Presentation of Transformed Data


8.5 Graphical Presentation of Transformed Data

Summary

Examples


8A.6 Additive Transformations

I Logarithmic Transformation

II Probit Transformation

III -Log(1-p) Transformations


8A.7 Theoretical Variances of Transformed Data


8A.8 Transformations of Statistical Measures

I Estimated Variances and Standard Deviations

II Variance Ratios

III Correlation Coefficients

IV Estimated Deviate


8A.9 Test for Homogeneity of Variance


8A.10 Testing for Normality


8A.11 Serial Correlation

Summary

Examples


9 Sampling Methods


9.1 Random Selection


9.2 Use of Tables of Random Numbers


9.3 Randomization In Experimentation


9.4 Methods of Sampling


9.5 Analysis of Stratified Random Samples


9.6 Analysis of Systematic Samples


9.7 Sampling from Finite Populations


9.8 Sampling Efficiency

Summary

Examples


9.9 Multi-Stage Sampling


9.10 Ratio Method in Sampling


9.11 Regression Method in Sampling

Summary

Examples

Bibliography

Appendix of Statistical Tables

I Table of The Percentage of Observations Exceeding A Given Normal Deviate d

II Table of Limits for The Deviate d Corresponding to a Given Percentage

III Variance-Ratio Table 5 per cent Points

IV Variance-Ratio Table 1 per cent Points

V Table Giving The Percentage of Trials in Which a Given Estimated Deviate t is Exceeded

VI Table of χ2 Distribution

VII Table of Random Numbers

VIII Tables of Common Logarithms and Natural Logarithms

IX Values of Sin-1√p

X Values of Sinh-1√x

XI Table of Squares and Square Roots

Index