Table Of ContentELEMENTARY STATISTICS
A Workbook
BY
DR. K. HOPE, M.A.,B.A.
Honorary Lecturer, Department of Psychology, Edinburgh
University. Member of Scientific Staff, Medical Research
Council Unit for Research on the Epidem ology of Psychiatric Illness
P E R G A M ON PRESS
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Copyright © 1967 Pergamon Press Ltd.
First edition 1967
Printed in Great Britain by Bell and Bain Ltd., Glasgow
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EDITOR'S INTRODUCTION
"Analysing data," says Dr. Hope, "is like playing the piano;
it is better to do it first and think about it afterwards." This
very sensible approach is likely to be completely misunderstood,
and requires some comment.
We are faced every year, in universities and colleges, with
thousands of able and eager first-year students in the social
and biological sciences who not only have no clue as to what
statistics is all about but who have to overcome a severe emotional
block before they can begin to get to grips with it. And this is
not confined to those who are weak in mathematics anyhow.
It is not only an uncertainty about their computational competence
which fills students with apprehension. The arithmetic of money,
weights and measures, the algebra of equations and series, the
geometry of theorems and constructions, the trigonometry of
surveying and the calculus of physical applications, nowhere
touch, except inadvertently, the insistent daily life of guesswork,
of error, of variability, of probability, of chance, of estimation,
of incomplete information, of human judgement, or even of the
sheer mmerosity of most of our social and biological problems.
Statistics is not empirical science and it is not mathematics:
it is the bridge between the two, bringing precision to observation
and meaning to measurement. There is a two-way traflSc across
this bridge. By an exercise of trained judgement the statistician
has to make a whole series of decisions to translate an array
of qualitative factual data into a structure of mathematical
symbols. Conversely his grasp of the pragmatic significance of
the symbols enables him to travel in the other direction, i.e. to
interpret the practical significance of the mathematical statements.
Learning to use statistics means learning to walk both ways
across this bridge,
vii
Viii EDITOR'S INTRODUCTION
Now walking means both knowing where to go and knowing
how to get there. But for the toddler learning to walk the act of
locomotion itself is a sufficient achievement and satisfaction
without needing to be justified in terms of a target. With five-
finger exercises on the piano the production of regular jingles
of sound is sufficient to keep the fingers practising. Dr. Hope's
provisional title for this book was *'A First Crawl in Statistics"
and this does express the object of the book, though the
theology might not commend itself to the reader.
Traditional educational methods leaned heavily on repeti
tive drill in order to secure proficiency. The drill tended to become
an end in itself. The achievement of "proficiency without
purpose" made for easy teaching, easy examining and easy
administration. But for the pupils it is a matter of endurance
rather than enjoyment. It is therefore a method which will
not work without a large measure of compulsion. It has never
failed to encounter the strictures of educationl reformers. And its
popularity has suffered a sharp decline in modern times.
The emphasis today, rightly enough, is on making education
meaningful. A book of computational drills cannot, without
supporting argument, be expected to stand on its own feet in
an anti-drill cHmate of opinion. But no single text should be
expected to stand alone. The simple logical distinction between the
'*necessary" and the **sufficient" is continually being ignored in
educational theory. Proficiency without purpose is pointless but
it is equally true that purpose without proficiency is useless.
Because drill alone is not enough we do not have to rush to the
other extreme and cry "out with all drill". The learner himself is a
system of inherited rhythms and acquired routines. Repetition is
the method of life as well as the mother of learning. Drill is not
intrinsically the dreary drudgery which sadly sadistic pedagogues
have made of it in the past. The toddler drills himself because
it is just wonderful to walk. And the arithmetician drills himself
because it is just wonderful to be able to calculate
Skill is based on drill but it requires more than drill. We
EDITOR'S INTRODUCTON ix
tend to think of our skills merely as things we can do. But skills,
once acquired, become instruments of thinking. The solution
of any problem demands the exercise of some pattern of skills
and the actual possession of those skills brings an awareness
of what moves are possible. In analysing a problem the man
who cannot spell out the necessary steps towards a solution
is at a severe disadvantage. He may hit on a solution but it is not
likely to be the best solution, economically, strategically or
even intellectually.
We may regard Dr. Hope's book as a useful step towards
a treaty of co-existence between literacy and numeracy; bringing
numeracy to the literate. A complementary step is being envis
aged in the other direction; bringing literacy to the numerate.
PATRICK MEREDITH
FOREWORD
THE author's feeling in embarking on the composition of this
book was that statistics could not possibly be as difficult as it
seemed when he was learning it. He therefore set out to write a
workbook which would serve as an introduction to the standard
textbooks. The order of presentation in this book is neither
historical nor logical. It is determined entirely by educational
considerations.
Examples, said Kant, are no more than the go-cart of judge
ment. But a go-cart is just what the new-bom statistician needs.
Analysing data is like playing the piano; it is better to do it
first and think about it afterwards. But it must be clearly and
emphatically stated that this is not to say that we should do our
experiment first and think about the statistical analysis afterwards.
This practice, which is only too common among research workers,
has results which are comparable to the achievement of those
pianists who venture upon the Hammerklavier without practising
their scales. This workbook contains the scales. Scales alone
never made a pianist and a workbook will never make a research
worker, but unless the research worker submits himself to a
discipline such as the discipline of these pages he will never
advance far in the design and analysis of experiments.
The statistical textbooks, excellent as they are as works of
reference, are, pedagogically, at the stage of the eighteenth-
century grammar school. In the interests of purism they refuse
to separate the mechanical from the interpretative, and their
examples employ arithmetic so complicated that a calculating
machine is necessary if the reader is to follow them. By the time
the learner has calculated a value to four decimal places he has
quite forgotten what to do with it. There is no difficult arithmetic
xi
XU FOREWORD
in this book. Anyone in the author's acquaintance who claimed
an "arithmetic block" was browbeaten into working through the
examples in order to eliminate difficult long divisions and im
possible square roots. The purist will object that the exclusive
use of simple whole numbers like 2 and 4 renders the distributions
non-normal. This is rather Hke complaining that scales are not
music; it is true but irrelevant. The methods taught here are all
parametric, that is they make certain restrictive assumptions (such
as normality of distribution) about the nature of the data to be
analysed. From time to time the reader is exhorted to ensure
that the data of his experiment conform to the assumptions and
he is shown how to test whether they do in fact conform. It is
of no concern to him whether the artificial numbers on which he
practises his statistical techniques are normally distributed or not.
Non-parametric methods such as chi-square, the Fisher-
Yates test and Kendall's ranking methods are not expounded.
The most important reason for their exclusion is that, in the
author's experience, the readability and effectiveness of a book
are in inverse relation to its size. A second consideration is that
the parametric methods seem to cause the learner most difficulty.
A third reason is that some at least of the non-parametric methods
are adequately and clearly taught in several books. Anyone who
has mastered the parametric methods should have little difficulty
in learning chi-square and the binomial test.
The author wishes to express his thanks to the numerous
research workers and others who have been good enough to
read the manuscript and comment upon it. He is particularly
grateful to those who, in the course of teaching themselves
statistics from the manuscript, pointed out its obscurities and
infelicities. The responsibihty for any errors remains with the
author.
The author wishes to acknowledge the courtesy of Biometrika
in permitting him to reproduce part of Table 18 of Biometrika
Tables for Statisticians, vol. I.
CHAPTER 1
PRELIMINARIES
Σ ("capital sigma", or just "sigma") means sum.
EXAMPLES
If JTi = 5 = 2 Z>, = 4 i
= 3 a2 = 5 ¿2 = 2 4
= 4 3
2
then ΣΧ =12 Σαό = 8 + 10 = 18 Σι^ = 16+9+4 = 29.
Calculate the following (1) [i.e. the answers will be found
under heading (1) in Appendix 3].
b X Y
5-0 2 7
70 1 2
2-2
ΣΧ= ΣΓ=
Zb= ΣΧΥ=
Im VW
4 -1 2 3
2 -2 3 1
3 2
Σν^= Σ>ν2 =
Σ/= Σm=
Σ/,«=
Note the difference between ΣΧ^ and (ΣΛΤ)^. If there are
brackets, do the calculations inside the brackets and then do those
outside.
1
ELEMENTARY 8ΤΑΤΙ8Ή€8
EXAMPLE
Β
3
4
7
ΣΒ^ 9 + 16 = 25
{ΣΒΫ 7^ =49
Calculate (2):
t Ρ
3 1
2 Ο
1 -2
Σί ΣΡ
Σ/2 ΣΡ2
(ΣΟ' (ΣΡ)^
If we have a number of scores or measurements and we want
to summarize them briefly, we calculate their mean (or average)
and their variance. In order to calculate the variance we express
each score as a deviation from the mean.
Example of the calculation of mean and variance.
Scores on a test {X)
Smith 0
Jones 2
Brown 4
Hughes 6
ΣΧ 12
η 4
— 3 This is the mean and it is symbolized
η
X(XhdiT).
Now we express each score as a deviation from this mean and we
symboUze the deviation scores by a small letter x. For example,
Smith's Λ: is 0-3 = -3.
PRELIMINARIES 3
Deviation scores (x)
Smith -3
Jones -1
Brown +1
Hughes +3
Σχ 0
Deviation scores always sum to nought and this provides us with
a check on our arithmetic. Now square each deviation score and
sum the squares.
x^
9
1
1
9
20
5 This is the variance.
η
Calculate the mean and the variance of the following samples
(3):
Xl X2 X3 X4 X S Xe ^7
0 0 2 0 4 2 3
15 4 0 12 4 4 2
10 7 3 4 0 -1
20 9 3 4 -1 4
5 7 4 3
4 3 2
6 10
2 3 0
0 1
6
The mean (or average) is a measure of central tendency. The
variance is a measure of scatter (example Xs above has no
scatter and so its variance is zero). Mean and variance are the
two most frequently used statistics, and their usefulness is that they
are a way of summarizing the nature of a distribution of values.
