Experimental Statistics

By Mary Gibbons Natrella

Formulated to assist scientists and engineers engaged in army ordnance research and development programs, this well-known and highly regarded handbook is a ready reference for advanced undergraduate and graduate students as well as for professionals seeking engineering information and quantitative data for designing, developing, constructing, and testing equipment. Topics include characterizing and comparing the measured performance of a material, product, or process; general considerations in planning experiments; statistical techniques for analyzing extreme-value data; use of transformations; and many other practical methods. 1966 edition. Index. 52 figures. 76 tables.

• Language: English
• Publisher: Dover Publications
• Release date: Mar 13, 2013
• ISBN: 9780486154558

Book preview

EXPERIMENTAL STATISTICS

EXPERIMENTAL STATISTICS

Mary Gibbons Natrella

National Bureau of Standards

DOVER PUBLICATIONS, INC.

Mineola, New York

Bibliographical Note

This Dover edition, first published in 2005, is an unabridged republication of the work originally published by the National Bureau of Standards, Washington, D. C., in 1963. That work, published as National Bureau of Standards Handbook 91, was a reprint of the Experimental Statistics portion of the AMC Handbook by permission of the Army Materiel Command. (See the Preface and Foreword for further bibliographical details).

Library of Congress Cataloging-in-Publication Data

Natrella, Mary Gibbons.

Experimental statistics / Mary Gibbons Natrella.

p. cm.

Originally published: Washington, D.C. : U.S. Dept. of Commerce, National Bureau of Standards, 1963, in series: National Bureau of Standards handbook; 91.

Includes bibliographical references and index.

ISBN 0-486-43937-2 (pbk.)

1. Mathematical statistics—Handbooks, manuals, etc. 2. Experimental design—Handbooks, manuals, etc. I. Title.

QA276.25N38 2005

519.5′7—dc22

2004058243

Manufactured in the United States of America

Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

Preface

This Handbook brings together in a single volume material on experimental statistics that was previously printed for limited distribution as U.S. Army Ordnance Pamphlets ORDP 20–110, 20–111, 20–112, 20–113, and 20–114. These pamphlets are parts of the AMC Engineering Design Handbook series now under the jurisdiction of the Army Materiel Command. Future issues by the Army Materiel Command for its own use will be in the AMCP–706 series.

The material contained in the present publication was prepared in the Statistical Engineering Laboratory, National Bureau of Standards, under a contract with the former Office of Ordnance Research (now Army Research Office—Durham). Although originally developed with the needs of the Army in mind, it promises to be equally useful to other groups concerned with research and development, both within and outside the Government. To make this material more widely available to such groups, Experimental Statistics is now being published as a National Bureau of Standards Handbook for sale to the public through the Superintendent of Documents, U.S. Government Printing Office.

ERRATA NOTICE

The original printing of this Handbook (August 1963) contained a few errors that have been corrected in the reprinted editions. These corrections are marked with an asterisk (*) for identification. The errors occurred on the following pages: 4–10, 6–3, 6–14, 6–18, 6–27, 6–29, 6–30, 6–36, 12–6, T–14 and T–15 (Table A–7), and T–80 and T–81 (Table A–35).

FOREWORD

INTRODUCTION

This is one of a group of handbooks covering the engineering information and quantitative data needed in the design, development, construction, and test of ordnance equipment which (as a group) constitute the Ordnance Engineering Design Handbook.

PURPOSE OF HANDBOOK

The Handbook on Experimental Statistics has been prepared as an aid to scientists and engineers engaged in Army Ordnance research and development programs, and especially as a guide and ready reference for military and civilian personnel who have responsibility for the planning and interpretation of experiments and tests relating to the performance of Army Ordnance equipment in the design and developmental stages of production.

SCOPE AND USE OF HANDBOOK

This Handbook is a collection of statistical procedures useful in ordnance applications. It is presented in five sections, viz:

ORDP 20-110, Section 1, Basic Concepts and Analysis of Measurement Data (Chapters 1-6)

ORDP 20-111, Section 2, Analysis of Enumerative and Classificatory Data (Chapters 7-10)

ORDP 20-112, Section 3, Planning and Analysis of Comparative Experiments (Chapters 11-14)

ORDP 20-113, Section 4, Special Topics (Chapters 15-23)

ORDP 20-114, Section 5, Tables

Section 1 provides an elementary introduction to basic statistical concepts and furnishes full details on standard statistical techniques for the analysis and interpretation of measurement data. Section 2 provides detailed procedures for the analysis and interpretation of enumerative and classificatory data. Section 3 has to do with the planning and analysis of comparative experiments. Section 4 is devoted to consideration and exemplification of a number of important but as yet non-standard statistical techniques, and to discussion of various other special topics. An index for the material in all five sections is placed at the end of Section 5. Section 5 contains all the mathematical tables needed for application of the procedures given in Sections 1 through 4.

An understanding of a few basic statistical concepts, as given in Chapter 1, is necessary; otherwise each of the first four sections is largely independent of the others. Each procedure, test, and technique described is illustrated by means of a worked example. A list of authoritative references is included, where appropriate, at the end of each chapter. Step-by-step instructions are given for attaining a stated goal, and the conditions under which a particular procedure is strictly valid are stated explicitly. An attempt is made to indicate the extent to which results obtained by a given procedure are valid to a good approximation when these conditions are not fully met. Alternative procedures are given for handling cases where the more standard procedures cannot be trusted to yield reliable results.

The Handbook is intended for the user with an engineering background who, although he has an occasional need for statistical techniques, does not have the time or inclination to become an expert on statistical theory and methodology.

The Handbook has been written with three types of users in mind. The first is the person who has had a course or two in statistics, and who may even have had some practical experience in applying statistical methods in the past, but who does not have statistical ideas and techniques at his fingertips. For him, the Handbook will provide a ready reference source of once familiar ideas and techniques. The second is the person who feels, or has been advised, that some particular problem can be solved by means of fairly simple statistical techniques, and is in need of a book that will enable him to obtain the solution to his problem with a minimum of outside assistance. The Handbook should enable such a person to become familiar with the statistical ideas, and reasonably adept at the techniques, that are most fruitful in his particular line of research and development work. Finally, there is the individual who, as the head of, or as a member of a service group, has responsibility for analyzing and interpreting experimental and test data brought in by scientists and engineers engaged in ordnance research and development work. This individual needs a ready source of model work sheets and worked examples corresponding to the more common applications of statistics, to free him from the need of translating textbook discussions into step-by-step procedures that can be followed by individuals having little or no previous experience with statistical methods.

It is with this last need in mind that some of the procedures included in the Handbook have been explained and illustrated in detail twice : once for the case where the important question is whether the performance of a new material, product, or process exceeds an established standard; and again for the case where the important question is whether its performance is not up to the specified standards. Small but serious errors are often made in changing greater than procedures into less than procedures.

AUTHORSHIP AND ACKNOWLEDGMENTS

The Handbook on Experimental Statistics was prepared in the Statistical Engineering Laboratory, National Bureau of Standards, under a contract with the Office of Ordnance Research. The project was under the general guidance of Churchill Eisenhart, Chief, Statistical Engineering Laboratory.

Most of the present text is by Mary G. Natrella, who had overall responsibility for the completion of the final version of the Handbook. The original plans for coverage, a first draft of the text, and some original tables were prepared by Paul N. Somerville. Chapter 6 is by Joseph M. Cameron; most of Chapter 1 and all of Chapters 20 and 23 are by Churchill Eisenhart; and Chapter 10 is based on a nearly-final draft by Mary L. Epling.

Other members of the staff of the Statistical Engineering Laboratory have aided in various ways through the years, and the assistance of all who helped is gratefully acknowledged. Particular mention should be made of Norman C. Severo, for assistance with Section 2, and of Shirley Young Lehman for help in the collection and computation of examples.

Editorial assistance, art preparation, and the index were provided by John I. Thompson & Company, Washington, D. C.

Appreciation is expressed for the generous cooperation of publishers and authors in granting permission for the use of their source material. References for tables and other material, taken wholly or in part, from published works, are given on the respective first pages.

June 15, 1962

TABLE OF CONTENTS

CHAPTER 1

SOME BASIC STATISTICAL CONCEPTS AND PRELIMINARY CONSIDERATIONS

1-1  INTRODUCTION

1-2  POPULATIONS, SAMPLES, AND DISTRIBUTIONS

1-3  STATISTICAL INFERENCES AND SAMPLING

1-3.1  Statistical Inferences

1-3.2  Random Sampling

1-4  SELECTION OF A RANDOM SAMPLE

1-5  SOME PROPERTIES OF DISTRIBUTIONS

1-6  ESTIMATION OF m AND σ

1-7  CONFIDENCE INTERVALS

1-8  STATISTICAL TOLERANCE LIMITS

1-9  USING STATISTICS TO MAKE DECISIONS

1-9.1  Approach to a Decision Problem

1-9.2  Choice of Null and Alternative Hypotheses

1-9.3  Two Kinds of Errors

1-9.4  Significance Level and Operating Characteristic (OC) Curve of a Statistical Test

1-9.5  Choice of the Significance Level

1-9.6  A Word of Caution

CHAPTER 2

CHARACTERIZING THE MEASURED PERFORMANCE OF A MATERIAL, PRODUCT, OR PROCESS

2-1  ESTIMATING AVERAGE PERFORMANCE FROM A SAMPLE

2-1.1  General

2-1.2  Best Single Estimate

2-1.3  Some Remarks on Confidence Interval Estimates

2-1.4  Confidence Intervals for the Population Mean When Knowledge of the Variability Cannot Be Assumed

2-1.4.1  Two-sided Confidence Interval

2-1.4.2  One-sided Confidence Interval

2-1.5  Confidence Interval Estimates When We Have Previous Knowledge of the Variability

2-2  ESTIMATING VARIABILITY OF PERFORMANCE FROM A SAMPLE

2-2.1  General

2-2.2  Single Estimates

2-2.2.1  s² and s

2-2.2.2  The Sample Range as an Estimate of the Standard Deviation

2-2.3  Confidence Interval Estimates

2-2.3.1  Two-sided Confidence Interval Estimates

2-2.3.2  One-sided Confidence Interval Estimates

2-2.4  Estimating the Standard Deviation When No Sample Data are Available

2-3  NUMBER OF MEASUREMENTS REQUIRED TO ESTABLISH THE MEAN WITH PRESCRIBED ACCURACY

2-3.1  General

2-3.2  Estimation of the Mean of a Population Using a Single Sample

2-3.3  Estimation Using a Sample Which is Taken In Two Stages

2-4  NUMBER OF MEASUREMENTS REQUIRED TO ESTABLISH THE VARIABILITY WITH STATED PRECISION

2-5  STATISTICAL TOLERANCE LIMITS

2-5.1  General

2-5.2  Two-sided Tolerance Limits for a Normal Distribution

2-5.3  One-sided Tolerance Limits for a Normal Distribution

2-5.4  Tolerance Limits Which are Independent of the Form of the Distribution

2-5.4.1  Two-sided Tolerance Limits (Distribution-Free)

2-5.4.2  One-sided Tolerance Limits (Distribution-Free)

CHAPTER 3

COMPARING MATERIALS OR PRODUCTS WITH RESPECT TO AVERAGE PERFORMANCE

3-1  GENERAL REMARKS ON STATISTICAL TESTS

3-2  COMPARING THE AVERAGE OF A NEW PRODUCT WITH THAT OF A STANDARD

3-2.1  To Determine Whether the Average of a New Product Differs From the Standard

3-2.1.1  Does the Average of the New Product Differ From the Standard (σ Unknown)?

3-2.1.2  Does the Average of the New Product Differ From the Standard (σ Known)?

3-2.2  To Determine Whether the Average of a New Product Exceeds the Standard

3-2.2.1  Does the Average of the New Product Exceed the Standard (σ Unknown)?

3-2.2.2  Does the Average of the New Product Exceed the Standard (σ Known)?

3-2.3  To Determine Whether the Average of a New Product is Less Than the Standard

3-2.3.1  Is the Average of the New Product Less Than the Standard (σ Unknown)?

3-2.3.2  Is the Average of the New Product Less Than That of the Standard (σ Known)?

3-3  COMPARING THE AVERAGES OF TWO MATERIALS, PRODUCTS, OR PROCESSES

3-3.1  Do Products A and B Differ In Average Performance?

3-3.1.1  (Case 1) — Variability of A and B Is Unknown, But Can Be Assumed to be Equal

3-3.1.2  (Case 2) — Variability of A and B is Unknown, Cannot Be Assumed Equal

3-3.1.3  (Case 3) — Variability in Performance of Each of A and B is Known from Previous Experience, and the Standard Deviations are σA and σB, Respectively

3-3.1.4  (Case 4) — The Observations are Paired

3-3.2  Does the Average of Product A Exceed the Average of Product B?

3-3.2.1  (Case 1) — Variability of A and B is Unknown, But Can Be Assumed to be Equal

3-3.2.2  (Case 2) — Variability of A and B is Unknown, Cannot Be Assumed Equal

3-3.2.3  (Case 3) — Variability in Performance of Each of A and B is Known from Previous Experience, and the Standard Deviations are σA and σB, Respectively

3-3.2.4  (Case 4) — The Observations are Paired

3-4  COMPARING THE AVERAGES OF SEVERAL PRODUCTS

CHAPTER 4

COMPARING MATERIALS OR PRODUCTS WITH RESPECT TO VARIABILITY OF PERFORMANCE

4-1  COMPARING A NEW MATERIAL OR PRODUCT WITH A STANDARD WITH RESPECT TO VARIABILITY OF PERFORMANCE

4-1.1  Does the Variability of the New Product Differ From That of the Standard?

4-1.2  Does the Variability of the New Product Exceed That of the Standard?

4-1.3  Is the Variability of the New Product Less Than That of the Standard?

4-2  COMPARING TWO MATERIALS OR PRODUCTS WITH RESPECT TO VARIABILITY OF PERFORMANCE

4-2.1  Does the Variability of Product A Differ From That of Product B?

4-2.2  Does the Variability of Product A Exceed That of Product B?

CHAPTER 5

CHARACTERIZING LINEAR RELATIONSHIPS BETWEEN TWO VARIABLES

5-1  INTRODUCTION

5-2  PLOTTING THE DATA

5-3  TWO IMPORTANT SYSTEMS OF LINEAR RELATIONSHIPS

5-3.1  Functional Relationships

5-3.2  Statistical Relationships

5-4  PROBLEMS AND PROCEDURES FOR FUNCTIONAL RELATIONSHIPS

5-4.1  FI Relationships (General Case)

5-4.1.1  What Is the Best Line To Be Used for Estimating y From Given Values of x?

5-4.1.2  What Are the Confidence Interval Estimates for: the Line as a Whole; a Point on the Line; a Future Value of Y Corresponding to a Given Value of x?

5-4.1.3  What Is the Confidence Interval Estimate for β1, the Slope of the True Line y = β0 + β1x?

5-4.1.4  If We Observe n′ New Values of Y (With Average ), How Can We Use the Fitted Regression Line to Obtain an Interval Estimate of the Value of x that Produced These Values of Y?

5-4.1.5  Using the Fitted Regression Line, How Can We Choose a Value (x′) of x Which We May Expect with Confidence (1 − α) Will Produce a Value of Y Not Less Than Some Specified Value Q?

5-4.1.6  Is the Assumption of Linear Regression Justified?

5-4.2  FI Relationships When the Intercept Is Known To Be Equal to Zero (Lines Through the Origin)

5-4.2.1  Line Through Origin, Variance of Y’s Independent of x

5-4.2.2  Line Through Origin, Variance Proportional to

5-4.2.3  Line Through Origin, Standard Deviation Proportional to x (σY·x = xσ)

5-4.2.4  Line Through Origin, Errors of Y’s Cumulative (Cumulative Data)

5-4.3  FII Relationships

5-4.3.1  A Simple Method of Fitting the Line In the General Case

5-4.3.2  An Important Exceptional Case

5-4.4  Some Linearizing Transformations

5-5  PROBLEMS AND PROCEDURES FOR STATISTICAL RELATIONSHIPS

5-5.1  SI Relationships

5-5.1.1  What Is the Best Line To Be Used for Estimating for Given Values of X?

5-5.1.2  What Are the Confidence Interval Estimates for: the Line as a Whole; a Point on the Line; a Single Y Corresponding to a New Value of X?

5-5.1.3  Give a Confidence Interval Estimate for β1, the Slope of the True Regression Line, ?

5-5.1.4  What Is the Best Line for Predicting From Given Values of Y?

5-5.1.5  What Is the Degree of Relationship of the Two Variables X and Y as Measured by ρ, the Correlation Coefficient?

5-5.2  SII Relationships

5-5.2.1  What Is the Best Line To Be Used for Estimating From Given Values of X?

5-5.2.2  What Are the Confidence Interval Estimates for: the Line as a Whole; a Point on the Line; a Single Y Corresponding to a New Value of X?

5-5.2.3  What Is the Confidence Interval Estimate for β1, the Slope of the True Line ?

CHAPTER 6

POLYNOMIAL AND MULTIVARIABLE RELATIONSHIPS ANALYSIS BY THE METHOD OF LEAST SQUARES

6-1  INTRODUCTION

6-2  LEAST SQUARES THEOREM

6-3  MULTIVARIABLE FUNCTIONAL RELATIONSHIPS

6-3.1  Use and Assumptions

6-3.2  Discussion of Procedures and Examples

6-3.3  Procedures and Examples

6-4  MULTIPLE MEASUREMENTS AT ONE OR MORE POINTS

6-5  POLYNOMIAL FITTING

6-6  INEQUALITY OF VARIANCE

6-6.1  Discussion of Procedures and Examples

6-6.2  Procedures and Examples

6-7  CORRELATED MEASUREMENT ERRORS

6-7.1  Discussion of Procedures and Examples

6-7.2  Procedures and Examples

6-8  USE OF ORTHOGONAL POLYNOMIALS WITH EQUALLY SPACED x VALUES

6-8.1  Discussion of Procedures and Examples

6-8.2  Procedures and Examples

6-9  MATRIX METHODS

6-9.1  Formulas Using Triangular Factorization of Normal Equations

6-9.2  Triangularization of Matrices

6-9.3  Remarks

CHAPTER 7

CHARACTERIZING THE QUALITATIVE PERFORMANCE OF A MATERIAL, PRODUCT, OR PROCESS

7-1  GENERAL

7-2  BEST SINGLE ESTIMATE OF THE TRUE PROPORTION P

7-3  CONFIDENCE INTERVAL ESTIMATES OF THE TRUE PROPORTION P

7-3.1  Two-Sided Confidence Intervals

7-3.1.1  Exact Limits for n ≤ 30

7-3.1.2  Exact Limits for n > 30

7-3.1.3  Approximate Limits for n > 30

7-3.2  One-Sided Confidence Intervals

7-3.2.1  Exact Limits for n ≤ 30

7-3.2.2  Exact Limits for n > 30

7-3.2.3  Approximate Limits for n > 30

7-4  SAMPLE SIZE REQUIRED TO ESTIMATE THE TRUE PROPORTION

7-4.1  Determining the Sample Size Required to Estimate the True Proportion With a Specified Limit Of Error In Both Directions (i.e., When It Is Required To Estimate P Within ±δ)

7-4.1.1  Graphical Method

7-4.1.2  Numerical Method

7-4.2  Determining the Sample Size Required To Estimate the True Proportion With a Specified Limit Of Error In Only One Direction (i.e., When It Is Required To Estimate P Within +δ; or, To Estimate P Within −δ)

CHAPTER 8

COMPARING MATERIALS OR PRODUCTS WITH RESPECT TO A TWO-FOLD CLASSIFICATION OF PERFORMANCE (COMPARING TWO PERCENTAGES)

8-1  COMPARING AN OBSERVED PROPORTION WITH A STANDARD PROPORTION

8-1.1  Does the New Product Differ From the Standard With Regard To the Proportion of Items Which Show the Characteristic of Interest? (Does P Differ From P0?)

8-1.1.1  Procedure for n ≤ 30

8-1.1.2  Procedure for n > 30

8-1.2  Does the Characteristic Proportion for the New Product Exceed That For the Standard? (Is P > P0?)

8-1.2.1  Procedure for n ≤ 30

8-1.2.2  Procedure for n > 30

8-1.3  Is the Characteristic Proportion for the New Product Less Than That for the Standard? (Is P < P0?)

8-1.3.1  Procedure for n ≤ 30

8-1.3.2  Procedure for n > 30

8-1.4  Sample Size Required To Detect a Difference Of Prescribed Magnitude From a Standard Proportion When the Sign of the Difference IS NOT Important

8-1.5  Sample Size Required To Detect a Difference Of Prescribed Magnitude From a Standard Proportion When the Sign of the Difference IS Important

8-2  COMPARING TWO OBSERVED PROPORTIONS

8-2.1  Comparing Two Proportions When the Sample Sizes Are Equal

8-2.1.1  Does the Characteristic Proportion for Product A Differ From That for Product B? (Does PA Differ From PB?)

8-2.1.2  Does the Characteristic Proportion for Product A Exceed That for Product B? (Is PA Larger Than PB?)

8-2.2  Comparing Two Proportions When the Sample Sizes Are Unequal and Small (nA ≠ nB; Both No Greater Than 20)

8-2.2.1  Does the Characteristic Proportion for Product A Differ From That for Product B?

8-2.2.2  Does the Characteristic Proportion for Product A Exceed That for Product B? (Is PA Larger than PB?)

8-2.3  Comparing Two Proportions When the Sample Sizes Are Large

8-2.3.1  Does the Characteristic Proportion for Product A Differ From That for Product B? (Does PA Differ From PB?)

8-2.3.2  Is the Characteristic Proportion for Product A Larger Than That for Product B? (Is PA Larger Than PB?)

8-2.4  Sample Size Required to Detect a Difference Between Two Proportions

8-2.4.1  Sample Size Required to Detect a Difference of Prescribed Magnitude Between Two Proportions When the Sign of the Difference IS NOT Important

8-2.4.2  Sample Size Required to Detect a Difference of Prescribed Magnitude Between Two Proportions When the Sign of the Difference IS Important

CHAPTER 9

COMPARING MATERIALS OR PRODUCTS WITH RESPECT TO SEVERAL CATEGORIES OF PERFORMANCE (CHI-SQUARE TESTS)

9-1  COMPARING A MATERIAL OR PRODUCT WITH A STANDARD

9-1.1  When the Comparison Is With a Standard Material or Product

9-1.2  When the Comparison Is With a Theoretical Standard

9-2  COMPARING TWO OR MORE MATERIALS OR PRODUCTS

9-3  A TEST OF ASSOCIATION BETWEEN TWO METHODS OF CLASSIFICATION

CHAPTER 10

SENSITIVITY TESTING

10-1  EXPERIMENTAL SITUATION

10-2  KÄRBER METHOD OF ANALYSIS

10-2.1  General Solution For the Kärber Method

10-2.1.1  Procedure

10-2.1.2  Example

10-2.2  Simplified Solution (Kärber Method) For the Special Case When Test Levels Are Equally Spaced and Equal Numbers of Items Are Tested at Each Level

10-2.2.1  Procedure

10-2.2.2  Example

10-3  PROBIT METHOD OF ANALYSIS

10-3.1  Graphical Probit Solution

10-3.1.1  Procedure

10-3.1.2  Example

10-3.2  Exact Probit Solution

10-3.2.1  Procedure

10-3.2.2  Example

10-3.3  Testing Whether the Line Is An Adequate Representation of the Data

10-3.3.1  Procedure

10-3.3.2  Example

10-3.4  Using the Probit Regression Line For Prediction

10-3.4.1  Level of Stimulus x′ At Which a Specified Proportion P′ of the Individuals Would Be Expected To Respond

10-3.4.2  Level of Stimulus x′ At Which 50% of the Individuals Would Be Expected To Respond

10-3.4.3  Proportion of Individuals Which Would Be Expected To Respond At a Specified Level of Stimulus

10-4  THE UP-AND-DOWN DESIGN

10-5  SENSITIVITY TESTS WHEN THE STIMULUS LEVELS CANNOT BE CONTROLLED

CHAPTER 11

GENERAL CONSIDERATIONS IN PLANNING EXPERIMENTS

11-1  THE NATURE OF EXPERIMENTATION

11-2  EXPERIMENTAL PATTERN

11-3  PLANNED GROUPING

11-4  RANDOMIZATION

11-5  REPLICATION

11-6  THE LANGUAGE OF EXPERIMENTAL DESIGN

CHAPTER 12

FACTORIAL EXPERIMENTS

12-1  INTRODUCTION

12-1.1  Some General Remarks and Terminology

12-1.2  Estimates of Experimental Error for Factorial-Type Designs

12-1.2.1  Internal Estimates of Error

12-1.2.2  Estimates of Error from Past Experience

12-2  FACTORIAL EXPERIMENTS (EACH FACTOR AT TWO LEVELS)

12-2.1  Symbols

12-2.2  Analysis

12-2.2.1  Estimation of Main Effects and Interactions

12-2.2.2  Testing for Significance of Main Effects and Interactions

12-3  FACTORIAL EXPERIMENTS WHEN UNIFORM CONDITIONS CANNOT BE MAINTAINED THROUGHOUT THE EXPERIMENT (EACH FACTOR AT TWO LEVELS)

12-3.1  Some Experimental Arrangements

12-3.2  Analysis of Blocked Factorial Experiments When Each Factor Is at Two Levels

12-3.2.1  Estimation of Main Effects and Interactions

12-3.2.2  Testing for Significance of Main Effects and Interactions

12-4  FRACTIONAL FACTORIAL EXPERIMENTS (EACH FACTOR AT TWO LEVELS)

12-4.1  The Fractional Factorial Designs

12-4.2  Analysis

12-4.2.1  Estimates of Main Effects and Interactions

12-4.2.2  Testing for Significance of Main Effects and Interactions

CHAPTER 13

RANDOMIZED BLOCKS, LATIN SQUARES, AND OTHER SPECIAL-PURPOSE DESIGNS

13-1  INTRODUCTION

13-2  COMPLETELY-RANDOMIZED PLANS

13-2.1  Planning

13-2.2  Analysis

13-3  RANDOMIZED BLOCK PLANS

13-3.1  Planning

13-3.2  Analysis

13-3.2.1  Estimation of the Treatment Effects

13-3.2.2  Testing and Estimating Differences in Treatment Effects

13-3.2.3  Estimation of Block Effects

13-3.2.4  Testing and Estimating Differences in Block Effects

13-4  INCOMPLETE BLOCK PLANS

13-4.1  General

13-4.2  Balanced Incomplete Block Plans

13-4.2.1  Planning

13-4.2.2  Analysis

13-4.2.2.1  Estimating Treatment Effects

13-4.2.2.2  Testing and Estimating Differences in Treatment Effects

13-4.2.2.3  Estimating Block Effects

13-4.2.2.4  Testing and Estimating Differences in Block Effects

13-4.3  Chain Block Plans

13-4.3.1  Planning

13-4.3.2  Analysis

13-4.3.2.1  Estimating Treatment and Block Effects

13-4.3.2.2  Testing and Estimating Differences in Treatment Effects

13-5  LATIN SQUARE PLANS

13-5.1  Planning

13-5.2  Analysis

13-5.2.1  Estimation of Treatment Effects

13-5.2.2  Testing and Estimating Differences in Treatment Effects

13-5.2.3  Estimation of Row (or Column) Effects

13-5.2.4  Testing and Estimating Differences in Row (or Column) Effects

13-6  YOUDEN SQUARE PLANS

13-6.1  Planning

13-6.2  Analysis

13-6.2.1  Estimation of Treatment Effects

13-6.2.2  Testing and Estimating Differences in Treatment Effects

13-6.2.3  Estimation of Column Effects

13-6.2.4  Testing and Estimating Differences in Column Effects

13-6.2.5  Estimation of Row Effects

13-6.2.6  Testing and Estimating Differences in Row Effects

CHAPTER 14

EXPERIMENTS TO DETERMINE OPTIMUM CONDITIONS OR LEVELS

14-1  INTRODUCTION

14-2  THE RESPONSE FUNCTION

14-3  EXPERIMENTAL DESIGNS

14-4  FINDING THE OPTIMUM

14-5  RECOMMENDED SOURCES FOR FURTHER STUDY

CHAPTER 15

SOME SHORTCUT TESTS FOR SMALL SAMPLES FROM NORMAL POPULATIONS

15-1  GENERAL

15-2  COMPARING THE AVERAGE OF A NEW PRODUCT WITH THAT OF A STANDARD

15-2.1  Does the Average of the New Product Differ From the Standard?

15-2.2  Does the Average of the New Product Exceed the Standard?

15-2.3  Is the Average of the New Product Less Than the Standard?

15-3  COMPARING THE AVERAGES OF TWO PRODUCTS

15-3.1  Do the Products A and B Differ In Average Performance?

15-3.2  Does the Average of Product A Exceed the Average of Product B?

15-4  COMPARING THE AVERAGES OF SEVERAL PRODUCTS, DO THE AVERAGES OF t PRODUCTS DIFFER?.

15-5  COMPARING TWO PRODUCTS WITH RESPECT TO VARIABILITY OF PERFORMANCE

15-5.1  Does the Variability of Product A Differ From that of Product B?

15-5.2  Does the Variability of Product A Exceed that of Product B?

CHAPTER 16

SOME TESTS WHICH ARE INDEPENDENT OF THE FORM OF THE DISTRIBUTION

16-1  GENERAL

16-2  DOES THE AVERAGE OF A NEW PRODUCT DIFFER FROM A STANDARD?

16-2.1  Does the Average of a New Product Differ From a Standard? The Sign Test

16-2.2  Does the Average of a New Product Differ From a Standard? The Wilcoxon Signed-Ranks Test

16-3  DOES THE AVERAGE OF A NEW PRODUCT EXCEED THAT OF A STANDARD?

16-3.1  Does the Average of a New Product Exceed that of a Standard? The Sign Test

16-3.2  Does the Average of a New Product Exceed that of a Standard? The Wilcoxon Signed-Ranks Test

16-4  IS THE AVERAGE OF A NEW PRODUCT LESS THAN THAT OF A STANDARD?

16-4.1  Is the Average of a New Product Less Than that of a Standard? The Sign Test

16-4.2  Is the Average of a New Product Less Than that of a Standard? The Wilcoxon Signed-Ranks Test

16-5  DO PRODUCTS A AND B DIFFER IN AVERAGE PERFORMANCE?

16-5.1  Do Products A and B Differ in Average Performance? The Sign Test For Paired Observations

16-5.2  Do Products A and B Differ in Average Performance? The Wilcoxon-Mann-Whitney Test For Two Independent Samples

16-6  DOES THE AVERAGE OF PRODUCT A EXCEED THAT OF PRODUCT B?

16-6.1  Does the Average of Product A Exceed that of Product B? The Sign Test For Paired Observations

16-6.2  Does the Average of Product A Exceed that of Product B? The Wilcoxon-Mann-Whitney Test For Two Independent Samples

16-7  COMPARING THE AVERAGES OF SEVERAL PRODUCTS, DO THE AVERAGES OF t PRODUCTS DIFFER?

CHAPTER 17

THE TREATMENT OF OUTLIERS

17-1  THE PROBLEM OF REJECTING OBSERVATIONS

17-2  REJECTION OF OBSERVATIONS IN ROUTINE EXPERIMENTAL WORK

17-3  REJECTION OF OBSERVATIONS IN A SINGLE EXPERIMENT

17-3.1  When Extreme Observations In Either Direction are Considered Rejectable

17-3.1.1  Population Mean and Standard Deviation Unknown — Sample in Hand is the Only Source of Information

17-3.1.2  Population Mean and Standard Deviation Unknown — Independent External Estimate of Standard Deviation is Available

17-3.1.3  Population Mean Unknown — Value for Standard Deviation Assumed

17-3.1.4  Population Mean and Standard Deviation Known

17-3.2  When Extreme Observations In Only One Direction are Considered Rejectable

17-3.2.1  Population Mean and Standard Deviation Unknown — Sample in Hand is the Only Source of Information

17-3.2.2  Population Mean and Standard Deviation Unknown — Independent External Estimate of Standard Deviation is Available

17-3.2.3  Population Mean Unknown — Value for Standard Deviation Assumed

17-3.2.4  Population Mean and Standard Deviation Known

CHAPTER 18

THE PLACE OF CONTROL CHARTS IN EXPERIMENTAL WORK

18-1  PRIMARY OBJECTIVE OF CONTROL CHARTS

18-2  INFORMATION PROVIDED BY CONTROL CHARTS

18-3  APPLICATIONS OF CONTROL CHARTS

CHAPTER 19

STATISTICAL TECHNIQUES FOR ANALYZING EXTREME-VALUE DATA

19-1  EXTREME-VALUE DISTRIBUTIONS

19-2  USE OF EXTREME-VALUE TECHNIQUES

19-2.1  Largest Values

19-2.2  Smallest Values

19-2.3  Missing Observations

CHAPTER 20

THE USE OF TRANSFORMATIONS

20-1  GENERAL REMARKS ON THE NEED FOR TRANSFORMATIONS

20-2  NORMALITY AND NORMALIZING TRANSFORMATIONS

20-2.1  Importance of Normality

20-2.2  Normalization By Averaging

20-2.3  Normalizing Transformations

20-3  INEQUALITY OF VARIANCES, AND VARIANCE-STABILIZING TRANSFORMATIONS

20-3.1  Importance of Equality of Variances

20-3.2  Types of Variance Inhomogeneity

20-3.3  Variance-Stabilizing Transformations

20-4  LINEARITY, ADDITIVITY, AND ASSOCIATED TRANSFORMATIONS

20-4.1  Definition and Importance of Linearity and Additivity

20-4.2  Transformation of Data To Achieve Linearity and Additivity

20-5  CONCLUDING REMARKS

CHAPTER 21

THE RELATION BETWEEN CONFIDENCE INTERVALS AND TESTS OF SIGNIFICANCE

21-1  INTRODUCTION

21-2  A PROBLEM IN COMPARING AVERAGES

21-3  TWO WAYS OF PRESENTING THE RESULTS

21-4  ADVANTAGES OF THE CONFIDENCE-INTERVAL APPROACH

21-5  DEDUCTIONS FROM THE OPERATING CHARACTERISTIC (OC) CURVE

21-6  RELATION TO THE PROBLEM OF DETERMINING SAMPLE SIZE

21-7  CONCLUSION

CHAPTER 22

NOTES ON STATISTICAL COMPUTATIONS

22-1  CODING IN STATISTICAL COMPUTATIONS

22-2  ROUNDING IN STATISTICAL COMPUTATIONS

22-2.1  Rounding of Numbers

22-2.2  Rounding the Results of Single Arithmetic Operations

22-2.3  Rounding the Results of a Series of Arithmetic Operations

CHAPTER 23

EXPRESSION OF THE UNCERTAINTIES OF FINAL RESULTS

23-1  INTRODUCTION

23-2  SYSTEMATIC ERROR AND IMPRECISION BOTH NEGLIGIBLE (CASE 1)

23-3  SYSTEMATIC ERROR NOT NEGLIGIBLE, IMPRECISION NEGLIGIBLE (CASE 2)

23-4  NEITHER SYSTEMATIC ERROR NOR IMPRECISION NEGLIGIBLE (CASE 3)

23-5  SYSTEMATIC ERROR NEGLIGIBLE, IMPRECISION NOT NEGLIGIBLE (CASE 4)

LIST OF ILLUSTRATIONS

1-1  Histogram representing the distribution of 5,000 Rockwell hardness readings

1-2  Normal curve fitted to the distribution of 5,000 Rockwell hardness readings

1-3  Frequency distributions of various shapes

1-4  Three different normal distributions

1-5  Percentage of the population in various intervals of a normal distribution

1-6  Sampling distribution of for random samples of size n from a normal population with mean m

1-7  Sampling distribution of s² for samples of size n from a normal population with σ = 1

1-8  Computed confidence intervals for 100 samples of size 4 drawn at random from a normal population with m = 50,000 psi, σ = 5,000 psi. Case A shows 50% confidence intervals; Case B shows 90% confidence intervals

1-9  Computed 50% confidence intervals for the population mean m from 100 samples of 4, 40 samples of 100, and 4 samples of 1000

1-10  Computed statistical tolerance limits for 99.7% of the population from 100 samples of 4, 40 samples of 100, and 4 samples of 1000

2-1  The standard deviation of some simple distributions

2-2  Number of degrees of freedom required to estimate the standard deviation within P% of its true value with confidence coefficient γ

3-1  OC curves for the two-sided t-test (α = .05)

3-2  OC curves for the two-sided t-test (α = .01)

3-3  OC curves for the two-sided normal test (α = .05)

3-4  OC curves for the two-sided normal test (α = .01)

3-5  OC curves for the one-sided t-test (α = .05)

3-6  OC curves for the one-sided t-test (α = .01)

3-7  OC curves for the one-sided normal test (α = .05)

3-8  OC curves for the one-sided normal test (α = .01)

3-9  Probability of rejection of hypothesis mA = mB when true, plotted against θ

4-1  Operating characteristics of the one-sided χ²-test to determine whether the standard deviation σ1 of a new product exceeds the standard deviation σ0 of a standard. (α = .05)

4-2  Operating characteristics of the one-sided χ²-test to determine whether the standard deviation σ1 of a new product is less than the standard deviation σ0 of a standard. (α = .05)

4-3  Operating characteristics of the one-sided F-test to determine whether the standard deviation σA of product A exceeds the standard deviation σB of product B. (α = .05; nA = nB)

4-4  Operating characteristics of the one-sided F-test to determine whether the standard deviation σA of product A exceeds the standard deviation σB of product B.

(α = .05; nA = nB, 3nA = 2nB, 2nA = nB)

4-5  Operating characteristics of the one-sided F-test to determine whether the standard deviation σA of product A exceeds the standard deviation σB of product B.

(α = .05; nA = nB, 2nA = 3nB, nA = 2nB)

5-1  Time required for a drop of dye to travel between distance markers

5-2  Linear functional relationship of Type FI (only Y affected by measurement errors)

5-3  Linear functional relationship of Type FII (Both X and Y affected by measurement errors)

5-4  A normal bivariate frequency surface

5-5  Contour ellipses for normal bivariate distributions having different values of the five parameters, mX, mY, σX, σY, ρX Y

5-6  Diagram showing effect of restrictions of X or Y on the regression of Y on X

5-7  Young’s modulus of sapphire rods as a function of temperature — an FI relationship

5-8  Young’s modulus of sapphire rods as a function of temperature, showing computed regression line and confidence interval for the line

5-9  Relationship between two methods of determining a chemical constituent — an FII relationship

5-10  Relationship between the weight method and the center groove method of estimating tread life — an SI relationship

5-11  Relationship between weight method and center groove method — the line shown with its confidence band is for estimating tread life by center groove method from tread life by weight method

5-12  Relationship between weight method and center groove method — showing the two regression lines

5-13  Relationship between weight method and center groove method when the range of the weight method has been restricted — an SII relationship

10-1  Probit regression line (fitted by eye)

12-1  Examples of response curves showing presence or absence of interaction

12-2  A one-half replicate of a 2⁷ factorial

12-3  A one-quarter replicate of a 2⁷ factorial

12-4  A one-eighth replicate of a 2⁷ factorial

14-1  A response surface

14-2  Yield contours for the surface of Figure 14-1 with 2² factorial design

19-1  Theoretical distribution of largest values

19-2  Annual maxima of atmospheric pressure, Bergen, Norway, 1857-1926

20-1  Normalizing effect of some frequently used transformations

20-2  Variance-stabilizing effect of some frequently used transformations

21-1  Reprint of Figure 3-1. OC curves for the two-sided t-test (α = .05)

21-2  Reprint of Figure 1-8. Computed confidence intervals for 100 samples of size 4 drawn at random from a normal population with m = 50,000 psi, σ = 5,000 psi. Case A shows 50% confidence intervals; Case B shows 90% confidence intervals

LIST OF TABLES

2-1  Table of factors for converting the range of a sample of n to an estimate of σ, the population standard deviation. Estimate of σ = range/dn

3-1  Summary of techniques for comparing the average of a new product with that of a standard

3-2  Summary of techniques for comparing the average performance of two products

5-1  Summary of four cases of linear relationships

5-2  Computational arrangement for Procedure 5-4.1.2.1

5-3  Computational arrangement for test of linearity

5-4  Some linearizing transformations

5-5  Computational arrangement for Procedure 5-5.1.2.1

5-6  Computational arrangement for Procedure 5-5.2.2.1

6-1  Sample table of orthogonal polynomials

8-1  Observed frequencies from two samples in two mutually exclusive categories (a 2 × 2 table)

8-2  Rearrangement of Table 8-1 for convenient use in testing significance with Table A-29

9-1  Computational arrangement for Data Sample 9-1.1

9-2  Computational arrangement for Data Sample 9-1.2

9-3  Table of — computational arrangement for Data Sample 9-2

9-4  Table of — computational arrangement for Data Sample 9-3

10-1  Kärber method of analysis for fuze peak voltage test data

10-2  Simplified solution for the Kärber method of analysis when the test levels (x) are equally spaced and equal numbers of objects (n) are tested at each level

10-3  Graphical probit solution using Data Sample 10-1

10-4  Exact probit solution

10-5  Exact probit solution (second iteration)

10-6  Test of linearity — final probit equation

11-1  Some requisites and tools for sound experimentation

12-1  Results of flame tests of fire-retardant treatments (factorial experiment of Data Sample 12-2)

12-2  Yates’ method of analysis using Data Sample 12-2

12-3  Some blocked factorial plans (for use when factorial experiment must be sub-divided into homogeneous groups)

12-4  Some fractional factorial plans

12-5  Results of flame tests of fire-retardant treatments (fractional factorial experiment of Data Sample 12-4)

12-6  Yates’ method of analysis using Data Sample 12-4

13-1  Schematic presentation of results for completely-randomized plans

13-2  Schematic presentation of results for randomized block plans

13-3  Balanced incomplete block plans (4 ≤ t ≤ 10, r ≤ 10)

13-4  Schematic representation of results for a balanced incomplete block plan

13-5  Schematic representation of a chain block plan

13-6  Schematic representation of the chain block plan described in Data Sample 13-4.3.2

13-7  Spectographic determination of nickel (Data Sample 13-4.3.2)

13-8  Selected Latin squares

13-9  Youden square arrangements (r ≤ 10)

16-1  Work table for Data Sample 16-7

18-1  Tests for locating and identifying specific types of assignable causes

18-2  Factors for computing 3-sigma control limits

20-1  Some frequently used transformations

Note: Tables A-1 through A-37 follow the last chapter of text.

A-1  Cumulative normal distribution—values of P

A-2  Cumulative normal distribution—values of zP

A-3  Percentiles of the χ² distribution

A-4  Percentiles of the t distribution

A-5  Percentiles of the F distribution

A-6  Factors for two-sided tolerance limits for normal distributions

A-7  Factors for one-sided tolerance limits for normal distributions

A-8  Sample sizes required to detect prescribed differences between averages when the sign of the difference is not important

A-9  Sample sizes required to detect prescribed differences between averages when the sign of the difference is important

A-10  Percentiles of the studentized range, q

A-11  Percentiles of

A-12  Percentiles for

A-13  Percentiles for

A-14  Criteria for rejection of outlying observations

A-15  Critical values of L for Link-Wallace Test

A-16  Percentage points of the extreme studentized deviate from sample mean

A-17  Confidence belts for the correlation coefficient

A-18  Weighting coefficients for probit analysis

A-19  Maximum and minimum working probits and range

A-20  Factors for computing two-sided confidence limits for σ

A-21  Factors for computing one-sided confidence limits for σ

A-22  Confidence limits for a proportion (two-sided)

A-23  Confidence limits for a proportion (one-sided)

A-24  Confidence belts for proportions for n > 30

A-25  Sample size required for comparing a proportion with a standard proportion when the sign of the difference is not important

A-26  Sample size required for comparing a proportion with a standard proportion when the sign of the difference is important

A-27  Table of arc sine transformation for proportions

A-28  Minimum contrasts required for significance in 2 × 2 tables with equal samples

A-29  Tables for testing significance in 2 × 2 tables with unequal samples

A-30  Tables for distribution-free tolerance limits (two-sided)

A-31  Tables for distribution-free tolerance limits (one-sided)

A-32  Confidence associated with a tolerance limit statement

A-33  Critical values of r for the sign test

A-34  Critical values of Tα(n) for the Wilcoxon signed-ranks test

A-35  Critical values of smaller rank sum for the Wilcoxon-Mann-Whitney Test

A-36  Short table of random numbers

A-37  Short table of random normal deviates

SECTION 1

BASIC STATISTICAL CONCEPTS

AND

STANDARD TECHNIQUES FOR ANALYSIS AND INTERPRETATION OF MEASUREMENT DATA

DISCUSSION OF TECHNIQUES IN CHAPTERS 2 THROUGH 6

The techniques described in Chapters 2 through 6 apply to the analysis of results of experiments expressed as measurements in some conventional units on a continuous scale. They do not apply to the analysis of data in the form of proportions, percentages, or counts.

It is assumed that the underlying population distributions are normal or nearly normal. Where this assumption is not very important, or where the actual population distribution would show only slight departure from normality, an indication is given of the effect upon the conclusions derived from the use of the techniques. Where the normality assumption is critical, or where the actual population distribution shows substantial departure from normality, or both, suitable warnings are given.

Table A-37 is a table of three-decimal-place random normal deviates that exemplify sampling from a normal distribution with zero mean (m = 0) and unit standard deviation (σ = 1). To construct numbers that will simulate measurements that are normally distributed about a true value of, say, 0.12, with a standard deviation of, say, 0.02, multiply the table entries by 0.02 and then add 0.12. The reader who wishes to get a feel for the statistical behavior of sample data, and to try out and judge the usefulness of particular statistical techniques, is urged to carry out a few dry runs with such simulated measurements of known characteristics.

All A-Tables referenced in these Chapters are contained in ORDP 20-114, Section 5.

CHAPTER 1

SOME BASIC STATISTICAL CONCEPTS AND PRELIMINARY CONSIDERATIONS

1-1  INTRODUCTION

Statistics deals with the collection, analysis, interpretation, and presentation of numerical data. Statistical methods may be divided into two classes—descriptive and inductive. Descriptive statistical methods are those which are used to summarize or describe data. They are the kind we see used everyday in the newspapers and magazines. Inductive statistical methods are used when we wish to generalize from a small body of data to a larger system of similar data. The generalizations usually are