Essentials of Statistics for Scientists and Technologists

179 Pages·1967·5.474 MB·English

by  C. Mack (auth.)

Essentials of Statistics
for
Scientists and Technologists
Essentials of Statistics
for
Scientists and Technologists
c.
MACK
Reader in Applied Mathematics
Institute of Technology, Bradford
PLENUM PRESS
(T) NEW YORK
::r:
1967
u.s.
edition published by
Plenum Press. a division of
Plenum Publishing Corporation
227 West 17th Street. New York. N.Y. 10011
ISBN 978-1-4684-7967-6 ISBN 978-1-4684-7965-2 (eBook)
DOl 10.1007/978-1-4684-7965-2
© c. Mack 1966
First published 1967
library of Congress Catalog
Card Number 67-17769
Preface
Statistics is of ever-increasing importance in Science and Technology
and this book presents the essentials of the subject in a form suitable
either as the basis of a course of lectures or to be read and/or used
on its own. It assumes very little in the way of mathematical
knowledge-just the ability to substitute numerically in a few simple
formulae. However, some mathematical proofs are outlined or
given in full to illustrate the derivation of the subject; these can
be omitted without loss of understanding. The book does aim at
making clear the scope and nature of those essential tests and
methods that a scientist or technologist is likely to need; to this
end each chapter has been divided into sections with their own
subheadings and some effort has been made to make the text
unambiguous (if any reader finds a misleading point anywhere I
hope he will write to me about it).
Also with this aim in view, the equality of probability to proportion
of population is stated early, then the normal distribution and the
taking of samples is discussed. This occupies the first five chapters.
With the principles of these chapters understood, the student can
immediately learn the significance tests of Chapter 6 and, if he needs
it, the analysis of variance of Chapter 7. For some scientists this
will be most of what they need. Howcver, they will be in a position
to read and/or use the remaining chapters without undue difficulty.
Chapters 8 to 13 contain material which is of value to almost
everyone at some time, but the remaining chapters are more special
ized in content. The laws (or rules) of probability and the binomial
theorem are dealt with in some detail in Chapter 9, for those inter
ested in the fundamentals or in fraction defective testing. (The
author and some of his colleagues have found the above order of
material to give very good results with scientists and technologists,
who have, as a result, grasped the essentials without encountering
the difficulties that the formal laws of probability followed by the
binomial distribution often give at the start.)
A word to the student
Learn the technical terms, they are usually introduced in inverted
commas. You must do all the examples (all who have learnt statistics
have found this essential). These are largely straightforward applica
tions of the text; they are, by the way, in each section and not
collected at the end of each chapter; this is to enable student or
teacher to identify quickly the nature of each example, and also to
v
vi PREFACE
enable an occasional user to tryout his understanding of a test on
a numerical example (answers are given at the end of the book,
p. 158). In addition Chapter 16 consists largely of a collection of
problems of practical origin which will acquaint the reader with
actual situations requiring statistics (and which are also useful as
examination questions). The Appendix contains enough tables for
most practical purposes of a not too specialized nature.
The pertinent latest theoretical developments and improvements
have been included, among them Welch's test for the significant
difference between two sample means (which is independent of the
true values of the population variances), the so-called 'distribution
free or non-parametric' tests, and my own method of calculating
confidence intervals when finding the fraction defective of articles
whose parts are tested separately (in other words the product of
several binomial parameters).
Acknowledgments
Thanks are due to Messrs Oliver and Boyd, Edinburgh, for
permission to quote data from Statistical Methods in Research and
Production by o. L. Davies. Other quotation acknowledgments are
made in the Appendix.
The original results of section 9.9 are quoted with the editor's
permission from The New Journal of Statistics and Operational
Research, published at the Institute of Technology, Bradford.
Finally, I must acknowledge my deep personal indebtedness to
those who introduced me to statistics, especially to L. H. C. Tippett
and Professor M. S. Bartlett, and to my present colleagues M. L.
Chambers and M. Gent, on whose knowledge and advice I have
leaned heavily in writing this book, as well as to my scientist col
leagues E. Dyson, J. M. Rooum, and H. V. Wyatt who contributed
many of the practical examples of Chapter 16.
Contents
Introduction or 'What is statistics?' 1
2 The presentation of data 3
3 Probability, its meaning, real and theoretical popula-
tions 15
4 Basic properties of the normal distribution 23
5 Some properties of sampling distributions 30
6 Applications of normal sampling theory; significance
tests 38
7 Normal sampling theory: test for difference between
several sample means, analysis of variance, design of
experiments 52
8 Normal sampling theory: estimation of 'parameters'
by confidence intervals, by maximum likelihood 63
9 The binomial distribution: laws of probability,
applications of the binomial distribution, the multi-
nomial distribution 72
10 The Poisson, negative exponential, and rectangular
distributions 89
II The X2 test for 'goodness of fit': test for 'association' 95
12 Fitting lines and curves to data, least squares method 106
13 Regression curves and lines, correlation coefficient,
normal bivariate distribution 116
14 Some distribution-independent (or 'distribution-free'
or 'non-parametric') tests 126
15 Note on sampling techniques and quality control 132
16 Some problems of practical origin 135
Appendix 151
Answers 158
Index 171
vii
I
Introduction or 'What is statistics?'
I do believe, Statist though I am none nor like to be,
That this will prove a war-Cymbeline Act II Scene I
Often the best way to understand a subject is to trace its historical
development and, pursuing this policy, we note that the word
'statist' first appeared in the English language while Shakespeare
was a young man (c. 1584). It meant 'a person knowledgeable in
state affairs' as might be inferred from the quotation above.
Almost exactly two hundred years later the word 'statistics' first
appeared denoting the science of collecting, presenting, classifying
and/or analysing data of importance in state or public affairs (note
that the ending 'ics' usually connotes a science, e.g. physics). Within
a further fifty years the word 'statistics' began to be applied to the
data itself and, later, particular types of data acquired particular
adjectives. Thus 'vital statistics' came to be used for the figures of
births and deaths (from vita the Latin for life). Later still, any
collection of data was called 'statistics' and, so, today the word
means either 'data' or the 'science of collecting and analysing data',
and whichever of the two meanings applies is determined from the
context. Though, perhaps, the very latest use of the term 'vital
statistics' to denote the most important measurements of bathing
beauties should be mentioned (together with the Duke of Edinburgh's
witty remark that 'Nowadays, statistics appears to mean the use of
three figures to define one figure').
In this book we shall concentrate more on the analysis of data
than on its collection or presentation and this analysis derives largely
from the study of quite a different type of heavenly body, i.e.
astronomy. In astronomy very great accuracy is essential and the
need arose to make the maximum use of all the observations on a
star to deduce its 'best' position. The methods developed by Gauss
for this purpose (which were based on the theory of dice-throwing
developed in 1654 by Pascal and Fermat, the very first theoretical
work of all) were later applied to state and biological data, and were
greatly extended and developed by Karl Pearson and others from
1900 onwards to this end. Later, agricultural experiments (e.g. the
measurements of wheat yields with different strains, with different
fertilizers, etc.) led to further important analytical developments
largely due to Fisher. From 1920 onwards an ever-increasing flow
2 ESSENTIALS OF STATISTICS
of both problems and solutions has arisen in every branch of science
and technology, from industrial administration, from economics, etc.
Indeed, it would be fair to say that some knowledge of statistics is
essential to any education today. However, the reader need not be
alarmed at the apparent need to acquire a mass of knowledge, for
the subject matter has been reduced to a few clear-cut principles which,
once grasped, apply to most problems. The aim of this book is to
make those principles clear and also to give a clear idea of their use.
We conclude this introduction with some problems which illustrate
the need for, and use of, statistics. Now, shoe manufacturers are
faced with the problem of what proportion of size 5, size 6, etc.
shoes they should make and with what range of widths, etc. Since
it is impossible to measure everyone in the country a small fraction
or 'sample' of the population has to be measured. These questions
then arise: how many people should be measured; how should they
be selected, what reliability can be placed on the answer, etc.? The
modern theory of statistics supplies answers to these problems. (The
Shoe and Allied Trade Research Association has carried out a
number of such surveys since its inception in 1948 and it may be a
coincidence but I can now buy a good-fitting pair of shoes 'off the
peg' and have lost the feeling of deformity that remarks like 'Rather
a low instep, sir' had given me-my thanks are due to the manu
facturers and to SATRA.) Again no manufacturer can guarantee
that every article he makes is perfect, indeed, it is quite uneconomic
to attempt to do so, but he likes to be sure that 95 %, say, of the
articles reach a given standard. Again, it is usually uneconomic to
test every article (it may even be impossible, e.g. testing the reliability
of striking of matches). The problems of how many should be
tested and what proportion of these can be safely allowed to be
below the given standard are answered by statistical theory. Simi
larly, in scientific work, no experiment exactly repeats itself. Thus,
suppose we measure the velocity of light several times by a particular
method. Each time we shall get a slightly different answer and we
have to decide what the 'best' answer is. Again, suppose we measure
the velocity by a different method; then we shall get a different 'best'
answer and we have to compare this with the first to determine if
they are consistent.
These and other problems will be dealt with in this book, though
we shall not deal with the theory in mathematical detail, but will
outline sufficient to enable a scientist or technologist to appreciate
the scope of each method or test. With this aim in mind we start
with one of the early problems in the development of statistics, one
which is essential to a clear understanding of the subject, namely
the presentation of data.
2
The presentation of data
2.1 Graphical presentation (frequency diagram, histogram)
One answer to the problem of presenting a mass of data in a clear
manner is to present it graphically. One very useful form of graphical
presentation is the 'frequency diagram'. In Fig. 2.1 the contents of
Table 2.1, which gives the wages 'statistics' of a small factory, are
shown as a frequency diagram.
50
40
>-
.g, 30
.:0:,> '
It 20
10
o~~~~~~~~~~~~~~~~
69 72 75 78 81 84 87 90
Wage-rote (Variate]
FIGURE 2.1. Frequency diagram showing the wage-rate
data of Table 2.1
Table 2.1 WAGE-RATES IN PENCE PER HOUR OF THE MEN IN A
FACTORY
Rate in pence per hour 69 72 75 78 81 84 87 90
Number of men 10 17 23 55 47 28 18 13
Before describing the nature of a frequency diagram in detail, we
define a few basic terms which the reader should know. Data in
statistics usually consist of measurements or readings of some
3
4 ESSENTIALS OF STATISTICS
property of each person or object in a specified group. Thus, the
wage-rates of Table 2.1 are measurements of the earning power of
that gr6up consisting of the men in the factory. We call the property
measured the 'variate', and the number of persons (or objects) who
have a given value of the variate we call the 'frequency' of that
variate-value. In Table 2.1 the 'variate' is the wage-rate and the
number of men with a given wage-rate is the frequency (e.g. the
frequency of a variate-value of 81 pence per hour is 47). The
essential feature of a frequency diagram is that the heights of the
ordinates are proportional to the frequencies.
Example
2.1(i). Draw frequency diagrams for the following two sets of
data:
(a) Lengths of words selected at random from a dictionary
Number of letters in the
word 3 4 5 6 7 8 9 10 11 12 13
Frequency. 26 81 100 96 89 61 68 42 31 13 11
(b) Observations ofp roportion of sky covered by cloud at
I Greenwich
Proportion of sky covered 0'0 0'1 0·2 0·3 0'4 0'5 0·6 0'7 0·8 0·9 1·0
Frequency . . . 320 129 74 68 45 45 55 65 90 48 676
The frequency diagram gives a good pictorial description of the
data if there are not too many different values of the variate (in
Table 2.1 there are 8 different variate-values). When there are 15 or
more variate-values it is usually better to group some of the variate
values together and to present the results in the form of a 'histogram',
as it is called. Fig. 2.2 presents the data of Table 2.2 (this gives the
observed heights of 8,585 men) in 'histogram' form. In a histogram
the area of each box is proportional to the frequency (which, in this
case, is the number of men with heights within a given box-width).
There are one or two details of presentation by histogram worth
remarking on. (1) Where there are only a few observations as with
heights less than 60 in. the observations are usually grouped into
one box; similarly with heights greater than 74 in. (2) The width
of the boxes is usually the same except for grouped frequencies.
(3) By the height 62-in. is meant heights of 62 in. or more up to but
not including 63 in.; similarly 63-in. means 63 in. or more up to
64 and so on. (4) (This point can be omitted on a first reading.) The
accuracy of the measurements must be carefully noted-in this case
each height was measured to the nearest tin.; thus, any height