fififémsi Emma Company
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`1
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`

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`THE LIBRARY
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`OF
`.YHSRmNU
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`OF CALIFORNIA
`
`LOS ANGELES
`
`GIFT or
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`PUBLISHER
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`2
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`...
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`1......
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`3
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`Wwferecélsscrie 92519“'0'
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`5
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`
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`This Handbook, as indicated in the “Foreword,” was prepared to
`assist Western Electric people in performing Western Electric Com-
`pany work. Others elsewhere may also think it useful as a guide
`for applying statistical quality control. The Company has printed
`a limited number of copies beyond its own requirements to meet
`such interest. These copies are available at $7.50, postage paid.
`Inquiries may be addressed to Superintendent, Quality Assurance,
`Western Electric Company, Inc., 195 Broadway, New York 7, N. Y.
`The Handbook is not for sale through ordinary commercial channels.
`
`Western Electric Company, Inc.
`
`© 1956
`BY WESTERN ELECTRIC Co., INC.
`
`First Edition 1956
`
`Second Edition 1958
`
`PRINTED IN THE UNITED STATES OF AMERICA
`
`BY THE MACK PRINTING COMPANY, EASTON, PENNSYLVANIA
`
`
`
`"L321!_...-.*c--on¢~..:....
`
`
`
`
`
`6
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`

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`
`
`Foreword
`
`This book is a Working Handbook prepared by Western Electric people for
`Western Electric use.
`Its primary purpose is to provide a guide for applying
`statistical quality control principles to the company’s manufacturing operations.
`
`The book Was prepared under the guidance of a Handbook Committee ap-
`pointed by the Manufacturing DiVision’s Engineering Staff. The following
`persons have served, during all or part of the time, as members of the formal
`Committee:
`
`D. W. THOMAS, Chairman
`C. S. BARRETT
`. C. W. LESAGE
`E. E. BLANKENSTEIN
`A. S. ORDECKI
`A. T. CHAPMAN
`R. SCHIN
`C. C. COLE
`MISS BONNIE B. SMALL
`F. H. DRUMMOND
`F. STONEHILL
`
`The chairman of the Writing Committee was Miss Bonnie B. Small.
`
`Many engineers and supervisors have assisted in the Writing, assembling and
`checking of this material. The names are so numerous that it is not possible to
`list them all. The Committee wishes to express its appreciation to all
`these
`individuals for contributing so much of their time, effort and material, and also
`for the excellent spirit of cooperation in which the Work was done.
`
`It is with deep regret that it must be recorded here that
`two very active members of this group, Mr. Claude E.
`Adair and Mr. Fred H. Drummond, of the Indian-
`apolis Works, lost their lives in an airplane accident on
`April 1, 1956 while on their way to a Handbook meet-
`ing. Both had made important contributions to the
`writing and planning. For this, as Well as personal
`reasons, their loss has been keenly felt.
`
`A preliminary edition of this book was issued in September 1956. Many
`members of the Western Electric Company, and also of the Bell Telephone Labora-
`tories, Were kind enough to read it. The Committee Wishes to express its appreci-
`ation for the many helpful comments, suggestions and criticisms. The contents
`of this Volume, however, are entirely the responsibility of the Committee.
`
`. j
`
`7
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`7
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`

`
`8
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`

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`Preface by the Writing Committee
`
`It does not
`This book is not a textbook on Statistical Quality Control.
`attempt to discuss quality control theory, nor is it a book of examples of quality
`control applications.
`It is intended primarily to be a description of those pro-
`cedures which, if followed, will tend to preserve the essential features of the quality
`control programs at Western Electric. To a certain extent it may be looked upon
`as a compendium of the techniques and methods which have been found to con-
`tribute most to making these programs successful. Much of the material is based
`on training courses which have been given during the past six or seven years to
`Engineers, key people in the Shop and people at all levels of Management.
`
`The book is written in non-technical language, and no attempt has been made
`to Write for the professional statistician or the mathematician. The techniques
`described are essentially those which have been used in all types of industry since
`their development during the 1920's by Dr. Shewhart. Perhaps the most dis-
`tinctive features of the Western Electric program are (a) the emphasis on Engi-
`neering and Operating applications rather than Inspection, and (b) emphasis on
`the control chart, and particularly the process capability study, as the foundation
`of the entire program.
`
`The book also stresses the importance of the Quality Control Team as a means
`of putting the quality control methods to direct practical use.
`
`On certain subjects, such as designed experiments and correlation, it has not
`been possible to give more than a brief discussion of principles and an explanation
`of some of the terminology.
`It was thought best to include this, however briefly,
`rather than to omit these subjects entirely. The material on acceptance sampling
`has purposely been kept brief, partly because of the emphasis on Operating and
`Engineering, and partly because the subject has been covered adequately elsewhere.
`
`Much of this book has been written in the imperative form. This is to facilitate
`its use as a practical working Handbook. The fact that the book states that
`“samples should be taken this Way” or that “patterns should be marked this way”
`should not be taken to imply that this is the only way to do it.
`
`It should also be kept in mind that this book does not attempt, in any sense,
`to cover the entire field of Statistical Quality Control.
`It does describe certain
`procedures and methods which the Western Electric Company has found it desirable
`to emphasize in order to secure the wanted results from its quality control programs.
`
`BONNIE B. SMALL
`Chairman
`
`Wfiting Committee
`
`vii
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`9
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`10
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`Table of Contents
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`Foreword .
`Preface
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`vii
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`Section I
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`Fundamental Principles
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`Part A.
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`Introduction to Statistical Quality Control
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`Part B.
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`A-1 Meaning of the Term “Statistical Quality Control” .
`A-2 Meaning of “Process” .
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`A-3 Essential Techniques in Statistical Quality Control
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`Introduction to Control Charts
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`B-1 Statistical Phenomena in the World Around Us .
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`B-2 Principal Kinds of Control Charts .
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`B-3 X and R Charts .
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`B-4
`p—Charts and Other Attributes Charts .
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`B-5 Charts for Individual Measurements with Control Limits
`Based on the Moving Range .
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`B-6 Tests for Unnatural Patterns .
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`B-7 Tests to Be Used When the Control Limits Are Not Sym-
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`B-8 Other Unnatural Patterns .
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`B-9 Simple Interpretation of Control Charts .
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`Part C. Essential Elements in a Quality Control Program
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`C-1
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`Process Capability Studies to Obtain Information and Solve
`Problems .
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`C-2 Process Control Charts to Secure Tangible Results in the
`Shop .
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`C-3 Statistical Sampling Plans to Reduce the Cost of Inspection.
`C-4 Quality Control Meetings to Make Quality Control Work .
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`3
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`Section II
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`Engineering Applications
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`Part A. Process Capability Studies
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`A-1 The Scientific Foundation of a Process Capability Study .
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`A-2 Obtaining the Data .
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`A-3 Analyzing the Data .
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`A-4 Making an Estimate of the Process Capability .
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`A-5 Using the Information from a Process Capability Study .
`A-6 Translating a Process Capability Study into a Shop Control
`Chart
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`A-7 Simple Examples of Process Capability Studies .
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`A-8 Performance Studies .
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`61
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`11
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`Part B.
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`Designed Experiments
`B-1 Place of Designed Experiments in a Process Capability Study
`B-2 Experiment I (Comparison of Two Methods) .
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`B-3 Experiment II (Error of Measurement)
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`B-4 Experiment III (Four Factor Experiment) .
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`B-5 Explanation of
`the Four Factor Analysis (With Special
`Reference to the Control Chart Method) .
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`B-6 Directions for Plotting .
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`B-7 Drawing Conclusions from Experimental Control Charts .
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`B-8 Some Suggestions on Planning the Experiment
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`Part C.
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`Specifications
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`C-1
`Specifications in General .
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`C-2 Relationship Between Process and Specification.
`C-3 Specification Conflicts and What Can Be Done to Avoid
`Them .
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`C-4 Statistical Addition of Tolerances .
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`C-5 Clearance and Fits
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`75
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`126
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`Distributions
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`D-1 Characteristics of Frequency Distributions .
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`D-2 Distributions Derived from Samples.
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`D-3 Methods of Plotting a Frequency Distribution .
`D-4 Practical Uses of Frequency Distributions .
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`Correlation
`E-1 Graphical Methods of Studying Correlation.
`E-2 Regression Lines
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`E-3 Formal Correlation Analysis .
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`E—4 Other Information on Correlation .
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`129
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`Part D.
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`Part E.
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`Part F.
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`Control Chart Patterns
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`F-1 Control Chart Theory .
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`F-2
`Interpretation of X’ Charts .
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`F-3
`Interpretation of R Charts .
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`F-4
`Joint Interpretation of X’ Charts and R Charts .
`F-5
`Interpretation of p-Charts and Other Attributes Charts .
`F-6
`Interpretation of a Chart for Individual Measurements
`F—7 Analysis of Patterns.
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`F-8 Cycles .
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`F-9 Freaks .
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`F-10 Gradual Change in Level .
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`F-11 “Grouping” or “Bunching”.
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`F-12 Instability .
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`F-13 Interaction.
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`F-14 Mixtures.
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`F-15 Natural Pattern .
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`F-16 Stable Forms of Mixture.
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`F-17 Stratification .
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`F-18 Sudden Shift in Level
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`F-19 Systematic Variables
`F-20 Tendency of One Chart to Follow Another .
`F-21 Trends .
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`F-22 Unstable Forms of Mixture .
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`F-23 Calculation of Tests for Unnatural Patterns
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`Section III
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`Shop Applications
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`Part A.
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`Process Control Charts
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`Planning the Control Charts .
`A-1
`A-2 Detailed Procedures in Setting Up the Charts .
`A-3 Other Methods of Charting .
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`A-4 Making Changes in Shop Control Charts .
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`Part B.
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`Introduction of Charts in the Shop
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`B-1 Explaining the Charts to the People.
`B-2
`Simple Examples of the Advantages of Control Charts.
`B-3 General Instruction for Process Control
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`B-4
`Instructions for Process Checkers .
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`Part C.
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`Action on Control charts
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`Importance of Promptness in Acting on Shop Charts
`C-1
`C-2 First Type of Action: To Be Taken by the Process Checker
`C-3
`Second Type of Action: To Be Taken by the Operator, Ma-
`chine Setter, Layout Operator or Other Responsible Person.
`C-4 Third Type of Action: To Be Taken by the Supervisor
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`C-5 Fourth Type of Action: To Be Taken by the Quality Con-
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`C-6 Using Shop Charts to Experiment with the Process .
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`C-7 Meaning of an “Economical State of Control” .
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`C-8 Summary Control Charts
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`Quality Control Teams
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`D-1 Regular Meetings of the Team .
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`D-2 Quality Control Coverage .
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`D-3 Reports on Progress .
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`D-4 Cost Reduction .
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`D-5 Control Chart Audits .
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`D-6 Routine Duties in Connection with Process Control Charts.
`D—7 Manual for Statistical Clerks .
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`Part D.
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`Section IV
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`Inspection Procedures
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`Part A.
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`Principles of Inspection
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`Place of Inspection in the Quality Control Program .
`A-1
`A-2 Why Inspection Can Be Reduced But Never Completely
`Eliminated .
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`13
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`A-3 Uses of Sorting and Sampling Inspection.
`A-4
`Inspection Planning .
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`Part B.
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`Acceptance Sampling
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`B-1 Elementary Concepts .
`B-2 Methods of Calculating the Probability of Acceptance .
`B-3 Economic Importance of OC Curves.
`Classification of Sampling Plans According to AQL, LTPD
`B-4
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`B-5
`Sampling Plans for Continuous Processes .
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`Sampling Plans for Lot-by-Lot Inspection .
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`B-7
`Special Types of Sampling Plans
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`B-8
`Proper Grouping of Inspection Items
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`B-9
`How to Select a Quality Level for Sampling .
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`B-10 Factors Determining the Choice of a Particular Plan.
`.
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`Part C. General Instruction for Inspection .
`
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`Part D.
`
`Inspection Levels
`
`Acknowledgments .
`References .
`.
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`Index
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`
`xii
`
`14
`
`

`
`Section I
`
`Fundamental
`
`Principles
`
`15
`
`

`
`16
`
`

`
`PART A
`
`Introduction to Statistical Quality Control
`
`Statistical Quality Control is a scientific method of analyzing data and using the analysis to solve
`practical problems.
`It can be applied to anything that it is possible to express in the form of
`numbers.
`In a manufacturing plant it can be applied to
`
`0 Engineering problems
`- Operating problems
`
`0 Inspection problems
`- Management problems
`0 Accounting and clerical problems
`or virtually any other field of activity. Few of us can think of anything, in connection with our
`own jobs, that is not associated in some way with numbers.
`
`
`
`A-1 MEANING OF THE TERM
`“STATISTICAL QUALITY
`
`CONTROL”
`
`The word “Statistical” means “having to do
`with numbers,” or more particularly with
`“drawing conclusions from numbers.”
`The word “Quality” means much more than
`the goodness or badness of a product.
`It
`refers to the qualities or characteristics of the
`thing or process being studied.
`The word “Control” means “to keep some-
`thing within boundaries,”
`or
`“to make
`something behave the way we want
`it
`to
`behave.”
`Taken together, and applied to a manufac-
`turing operation, the words Statistical Quality
`Control mean this:
`
`STATISTICAL With the help of numbers, or
`data,
`
`way we want it to behave.
`
`QUALITY
`
`We study the characteristics
`of our process
`
`CONTROL
`
`In order to make it behave the
`
`The term “process,” as used in the previous
`statement, 1S capable of assuming many dif-
`ferent meanings.
`
`A-2 MEANING OF “PROCESS”
`
`A “process” is any set of conditions, or set
`of causes, which work together to produce a
`given result.
`In a manufacturing plant we
`usually think first of a series of fabricating and
`assembling operations which result, for exam-
`ple, in the production of cable, electron tubes,
`relays, switchboards and other apparatus or
`equipment.
`However, since the word “process” means
`merely a system of causes, a process may be far
`simpler than the ones mentioned above, or it
`may be far more complex.
`In statistical qual-
`ity control the “process” we choose to study
`may be:
`0 a single machine, or a single fixture or
`element of a machine.
`0 a single human being, or a single mo-
`tion performed by a human being.
`a piece of test equipment.
`a method of measurement or gaging.
`a method of assembly.
`the act of typing (or performing any
`clerical operation).
`0 a group of many machines turning out
`different or similar pieces of product.
`0 a group of many human beings (for
`example, a pay group or a Shop).
`0 a
`combination
`of human beings,
`machines, materials, methods, pieces
`of equipment, etc. For example, the
`procedures required to manufacture
`3
`
`17
`
`

`
`-u.... _ .____.p._
`
`switches, rubber or wire.
`such as
`0 a method of processing,
`chemical treatment or plating.
`- a mental activity, such as visual check-
`ing or making calculations.
`0 intangible human elements, such as
`attitudes, motives and skills.
`0 the whole mass of causes which result
`in a Guided Missile System (for ex-
`ample), including thousands of resis-
`tors, capacitors, tubes and other com-
`ponents furnished by hundreds or
`thousands of different suppliers.
`0 anything else which results in a series
`of numbers and is connected with
`unsolved problems.
`
`In its narrowest sense the term “process”
`refers to the operation of a single cause.
`In
`its broadest sense it may refer to the operation
`of a very complicated “cause system.” This
`is why it is possible to make “process capa-
`bility studies” in connection with practically
`any type of Engineering, Operating, Inspection
`or Management problem;
`including such
`problems as overall merchandise losses,
`the
`overall cost of maintenance or plant-wide in-
`spection ratios.
`
`A-3 ESSENTIAL TECHNIQUES IN
`
`STATISTICAL QUALITY
`
`CONTROL
`
`The following techniques are essential in the
`quality control programs to be described in this
`Handbook:
`
`(1) Process Capability Studies (the basic
`use of control charts).
`
`(2) Process Control Charts (the use of con-
`trol charts in the shop).
`
`(3) Statistical Sampling Inspection (which
`may or may not make use of control
`charts).
`
`(4) Statistical Design of Experiment (a de-
`vice for comparing variables and deter-
`mining their significance. This also is
`frequently handled with control charts).
`
`The principal concept presented in this Hand-
`book is that of the control chart, which con-
`sists of a plotted series of observations or
`“samples.” The unique feature of the control
`chart is its ability to form data into patterns
`which, when tested statistically, can lead to in-
`formation about the process.
`The Process Capability Study is used by
`engineers, Operating supervisors and manage-
`ment to obtain information about how a proc-
`ess behaves.
`Process Control Charts are used in the shop
`to increase yields, cut down scrap and repair,
`and help the people to do better work.
`Statistical Sampling Inspection is used by in-
`spectors as a scientific basis for accepting or
`rejecting product.
`Design of Experiment is used in research
`and development problems, to study the ef-
`fect of many variables at once, to throw light
`on mysterious or puzzling causes and efiects,
`and to explore the unknown.
`These four techniques are described more
`fully in later parts of the Handbook.
`
`
`
`18
`
`

`
`
`
`PART B
`
`Introduction to Control Charts
`
`B-1 STATISTICAL PHENOMENA
`
`IN THE WORLD AROUND
`
`US
`
`B-1.1 Fluctuating patterns
`
`If data are collected which have a bearing
`on any problem, any series of events or any
`manufacturing situation,
`these data are al-
`ways found to exhibit variation.
`Instead of
`being exactly the same from point to point or
`from time to time,
`the numbers vary.
`If
`plotted on a piece of graph paper, so the varia-
`tions can be studied, the numbers always form
`a fluctuating, zig-zag pattern. Some typical
`examples of fluctuating patterns are shown in
`Figures 1-4.
`
`In the first case an operator was assembling
`spoolheads onto a core. The supervisor kept
`a record of the percentage of loose spoolhead
`assemblies made by this operator day after
`day. He found that the percentages varied.
`The percentage of loose spoolheads was not
`the same every day.
`Figure 2 is a record obtained from an auto-
`matic welding operation. This is an entirely
`different type of operation from Figure 1.
`It
`involves different materials, different facilities,
`a different operator, different problems. When
`the supervisor kept a record of the daily per-
`centage of oficenter welds, he found that the
`percentages varied in much the same manner
`as Figure 1. The percentage of offcenter welds
`was not the same every day.
`
`3%
`
`4%
`
`2%
`
`0%
`
`Fig. 1. Daily percentage of loose spoolheads.
`
`
`
`Fig. 2. Daily percentage of offcenter welds.
`
`
`
`
`Fig. 3. Successive parts coming from a screw machine
`(diameter).
`
`Fig. 4. Electrical measurement on a series of assemblies
`(noise level).
`
`5
`
`19
`
`

`
`The same thing was found on plotting the
`dimensions of successive parts coming from a
`screw machine (Figure 3), and also the elec-
`trical characteristics of a series of assembled
`units (Figure 4). Any series of numbers
`from a process,
`if plotted in sequence, will
`form a
`fiuctuating pattern. Even repeat
`measurements made on the same object will
`not be exactly the same, time after time or day
`after day.
`Similar variation is found in accounting
`figures, production figures, records of attend-
`ance, temperatures, pressures, medical reports
`or any other set of numbers from an industrial
`process. We do not know of any type of
`manufacture in which variation is not present.
`
`B-1.2 What causes the fluctuations in a
`fluctuating pattern?
`Fluctuations in the data are caused by a
`large number of minute variations or differ-
`ences: differences in materials, equipment, the
`surrounding atmospheric conditions, the physi-
`cal and mental reactions of people. Most of
`these difierences are extremely small. They
`cause the pattern to fluctuate in what is known
`as a “natural” or “normal” manner. Occasion-
`ally, however, there will be a large or unusual
`difference, much more important than all the
`other differences put together. For example,
`material is taken from a different batch;
`the
`machine setter makes a new setting; an in-
`experienced operator takes the place of an
`experienced operator. These large causes make
`the pattern fluctuate in an “unnatural” or
`“abnormal” manner.
`there are definite
`Experience shows that
`detectable differences between the “natural”
`and “unnatura ” patterns.
`It is possible to
`discover and study these differences by means
`of simple calculations based on well-known
`statistical laws.
`Once we know that a pattern is unnatural,
`it is possible to go further and find the cause
`of the unnaturalness. This makes it possible to
`detect,
`identify and study the behavior of
`causes.
`
`B-1.3 Distributions
`the only statistical
`Fluctuations are not
`phenomena which are observable in nature.
`6
`
`It has long been known that if we take large
`numbers of observations on some physical
`quantity (such as the charge on an electron),
`or large numbers of measurements on an indus-
`trial product (such as the diameter of a wire
`or shaft),
`these measurements will
`tend to
`group themselves around some central value
`with a. certain amount of variation or “scatter”
`on either side. The pattern or shape formed
`by the grouped measurements is called a “fre-
`quency distribution.” We observe that if the
`causes which produce the observations or
`measurements remain essentially unchanged,
`the distribution tends to have certain dis-
`tinguishable and stable characteristics. These
`characteristics become more definite as the
`number of observations or measurements in-
`creases. We conclude from this that,
`if the
`cause system is constant, the observed distribu-
`tion tends to approach, as a statistical limit,
`some distribution function or “law.”
`This tendency to form a distribution is ob-
`served throughout nature.
`It is one of the
`most fundamental of all natural laws.
`Experience tells us that
`the two sets _ of
`statistical phenomena——distributions and fluc-
`tuations—are not separate and unrelated. A
`distribution can be thought of as a composite
`mass of fluctuations, and the fluctuations can
`be thought of as con ned within the limits of a
`distribution.
`It is therefore possible to make
`use of statistical limits, derived from distribu-
`tions, to predict the behavior of a fluctuating
`pattern when there are no abnormal causes.
`
`This can be stated formally as follows:
`
`Whenever we have a series of events pro-
`ceeding from a given system of causes,
`those events will not in general be identical
`with each other.
`Instead, they will fluc-
`tuate or vary in a manner described as
`“random.” Nevertheless,
`if
`the cause
`system remains unchanged, the frequencies
`with which the events occur will tend to
`approach an objective probability, or set
`of probabilities, as the number of events
`increases indefinitely.
`
`Translated into industrial terms, this can be
`stated as follows:
`
`Whenever we have a series of observations
`or measurements, obtained from a given
`
`20
`
`

`
`those measurements will not in
`process,
`general be identical with each other.
`In-
`stead they will vary in such a way as to
`form a fluctuating pattern. Nevertheless,
`if nothing disturbs the process,
`these
`fluctuating measurements will be held
`within definite mathematical
`limits.
`In
`the aggregate, a large number of these
`measurements will tend to form a pre-
`dictable distribution.
`
`Translated into everyday language,
`I
`statements above mean this:
`
`the
`
`a. Everything varies.
`b. Individual things are unpredictable.
`c. Groups of things from a constant system
`of causes tend to be predictable.
`
`Check your understanding of these funda-
`mental concepts by studying the following
`simple examples:
`
`E:vample 1
`
`a. People live to different ages.
`b. No one knows how long he himself will live.
`c. Insurance companies can tell with great
`accuracy what percentage of people will live
`to be 60, 65, 70 etc.
`
`B-1.4 Statistical limits for fluctuating
`patterns
`
`By making use of certain equations, derived
`from statistical laws, it is possible to calculate
`“limits” for any given pattern.
`If a pattern
`is natural, its fluctuations will fit within these
`limits.
`If a pattern is unnatural, its fluctua-
`tions will not fit these limits. The following
`are examples of the calculation of statistical
`limits.
`
`(1) Statistical limits for the chart on the spool-
`head operation (Figure 1).
`To calculate limits for this chart, proceed
`as follows:
`loose
`take the percentage of
`a. First
`spoolheads turned out by the operator,
`on the average, over a period of time.
`In this case it was 4%.
`The average percentage is called 13
`(pronounced p-bar). The “p” means
`percentage or proportion, and the bar
`above it means average.
`b. Then take the average number of
`spoolheads assembled by this operator
`' during the day.
`In the example used,
`this number was 400.
`The number assembled is called it
`(meaning number).
`the calculation of
`c. The equation for
`statistical limits is as follows:
`
`Example 2
`
`Limits of fluctuation
`
`a. You cannot write the letter “a” twice in
`exactly the same way.
`b. You have no way of knowing how your next
`“a” will differ from the last one.
`c. Nevertheless there is something about your
`“a’s” that makes them recognizably different
`from my “a’s.”
`
`Example 3
`
`a. All patterns fluctuate.
`b. The individual points are unpredictable.
`c. A group or series of points from a constant
`process will tend to follow a pattern that obeys
`a fixed law.
`
`These concepts are carefully developed by
`Shewhart in Reference No. 37.
`
`i3‘/1_3<1 — 23’)
`
`'n
`
`*3‘)
`
`:04. (.96)
`40
`
`= :3 x .0098
`
`= $0294
`
`The limits of fluctuation are therefore 2.94%
`on either side of the average, or 6.94% and
`1.06%. The pattern for the spoolhead process
`should stay inside of these limits.
`
`(2) Statistical limits for the chart on the weld-
`ing operation (Figure 2).
`To calculate limits for this chart, proceed
`as follows:
`a. Use the same equation that was used
`in (1) above.
`b. The average percentage of oiicenter
`welds (13) was again 4%.
`
`7
`
`21
`
`

`
`c. The average number of welds made by
`the operator per day (n) was in this
`case 1000.
`
`d. Limits of fluctuation = :3 ‘/‘fl;—’3)
`
`*3 ‘l
`
`Hi (-_95_)
`1000
`
`= :!:3 = .0062
`
`= =|:.0186
`
`The limits of fluctuation are therefore 1.86%
`on either side of the average, or 5.86% and
`2.14%. The pattern for the welding process
`should stay inside of these limits.
`
`limits for the charts on (a)
`(3) Statistical
`screw machine and (b) electrical charac-
`teristic.
`
`Limits of fluctuation around the_average of the
`data = i2.66 MR,
`
`where the symbol “MR” refers to the
`average difference between successive
`pairs of measurements. This is explained
`on page 21.
`
`The limits of fluctuation for the screw ma-
`chine chart turn out to be .3102 and .3086.
`The limits of fluctuation for the electrical chart
`are 1.95 and 0.75. The patterns for these two
`processes should stay inside of these limits.
`
`In a similar way, limits can be calculated
`for any other type of data. Detailed instruc-
`tions are given on pages 12-23.
`limits to a
`When we add the statistical
`fluctuating pattern, the result is called a “con-
`trol chart.” The control chart is one of the
`most sensitive devices known for analyzing
`data and obtaining information.
`
`Since these involve a different type of data
`(individual measurements rather
`than per-
`centages),
`it is necessary to use a different
`equation.
`
`B-1.5 Meaning of a control chart
`
`The following are control charts for the four
`operations discussed above.
`
`Fig. 5. Control chart for the percentage of loose
`spoolheads.
`
`
`
`
`Fig. 7. Control chart for screw machine parts
`(diameter).
`
`Fig. 6. Control chart for the percentage of offcenter
`welds.
`
`
`
`
`Fig. 8. Control chart for electrical assemblies
`(noise level).
`
`22
`
`

`
`The statistical limits are drawn in as dotted
`lines and are called “control limits.” The con-
`trol limits used in this Handbook, unless other-
`wise stated, are “3 sigma control limits.” *
`The control
`limits are used to determine
`whether the pattern is “natural” or “unnat-
`ural.” The following procedure is used:
`
`(1) Check the fluctuating pattern to see
`whether it is in conflict with the natural
`statistical limits. The pattern is in con-
`flict
`if it
`(a)
`jumps outside the control
`limits or (b) forms unnatural clusters of
`points inside the control limits.
`Tests for unnatural points or clusters of
`points are given on pages 23-28.
`
`Interpretation of Figures 5-8
`
`the spoolhead operation
`for
`chart
`The
`(Figure 5) is interpreted as follows:
`
`(1) There are no x’s.
`(2) There is no evidence that the process is
`out of control.
`
`(3) It is not being disturbed by any unusual,
`outside causes.
`
`The chart for the welding operation (Figure
`6) is interpreted as follows:
`
`(1) Four out of the 10 points are marked
`with x’s.
`
`(2) There is strong evidence that the proc-
`ess is out of control.
`
`(2) Mark any unnatural points or clusters of
`points with “x’s.”
`
`(3) It is being disturbed by large and un-
`necessary outside causes.
`
`(3) If the pattern is not in conflict with the
`limits (that is, there are no x’s), consider
`it a “natural” pattern.
`In general, the
`longer the series of points without evi-
`dence of unnaturalness, the stronger is the
`evidence that this is a natural pattern.
`An occasional
`“:12”
`(perhaps once in a
`hundred points) may be the result of chance
`alone, and is not considered to make the
`pattern unnatural.
`
`(4) If the pattern is in conflict with the
`limits (that is, there are x’s), consider the
`pattern “unnatural” and the process “out
`of control.” The more numerous the x’s,
`in general, the stronger is the evidence of
`lack of control.
`
`When a pattern is natural, it means that
`there are no abnormal extraneous causes work-
`
`ing in the process. When the pattern is un-
`natural, it means that outside disturbances are
`present and are affecting the process.
`When a pattern is unnatural, those familiar
`with the process should investigate to find
`what the outside disturbances are.
`
`On most control charts, We prefer to have a
`longer series of points than those shown in
`Figures 5-8.
`In the following discussion these
`charts should be considered as typical portions
`of a more complete control chart.
`
`“' “Sigma” (usually written a) is a unit of measure
`which is used to describe the width or spread of a dis-
`tribution or pattern. The fluctuations in a “natural”
`pattern tend to spread about :l:3 sigma.
`
`The chart for the screw machine process
`(Figure 7) is interpreted as follows:
`
`(1) Three out of the 10 points are marked
`with x’s.
`
`(2) There is strong evidence that the process
`is out of control.
`.
`
`(3) The pattern shows a continuous move-
`ment in one direction which, in the pres-
`ence of x’s, indicates a trend.
`
`The chart for