_THE NATURE AND USE OF AN AVERAGE, AND ON THE DIFFERENT KINDS OF
AVERAGE._[1]
1. There is much need of some good account, accessible to the ordinary
English reader, of the nature and properties of the principal kinds
of Mean. The common text-books of Algebra suggest that there are
only three such, viz. the arithmetical, the geometrical and the
harmonical:--thus including two with which the statistician has
little or nothing to do, and excluding two or more with which he
should have a great deal to do. The best three references I can give
the reader are the following. (1) The article _Moyenne_ in the
_Dictionnaire des Sciences Médicales_, by Dr Bertillon. This is
written somewhat from the Quetelet point of view. (2) A paper by
Fechner in the _Abhandlungen d. Math. phys. Classe
d. Kön. Sächs. Gesellschaft d. Wiss_. 1878; pp. 1-76. This contains
a very interesting discussion, especially for the statistician, of a
number of different kinds of mean. His account of the median is
remarkably full and valuable. But little mathematical knowledge is
demanded. (3) A paper by Mr F. Y. Edgeworth in the
_Camb. Phil. Trans._ for 1885, entitled _Observations and
Statistics_. This demands some mathematical knowledge. Instead of
dealing, as such investigations generally do, with only one Law of
Error and with only one kind of mean, it covers a wide field of
investigation.
1. We have had such frequent occasion to refer to
_averages_, and to the kind of uniformity which they are apt
to display in contrast with individual objects or events, that
it will now be convenient to discuss somewhat more minutely
what are the different kinds of available average, and what
exactly are the functions they perform.
The first vague notion of an average, as we now understand
it, seems to me to involve little more than that of a
something _intermediate_ to a number of objects. The objects
must of course resemble each other in certain respects, otherwise
we should not think of classing them together; and
they must also differ in certain respects, otherwise we should
not distinguish between them. What the average does for
us, under this primitive form, is to enable us conveniently to
retain the group together as a whole. That is, it furnishes a
sort of representative value of the quantitative aspect of the
things in question, which will serve for certain purposes to
take the place of any single member of the group.
It would seem then that the first dawn of the conception
which science reduces to accuracy under the designation of
an average or mean, and then proceeds to subdivide into
various distinct species of means, presents itself as performing
some of the functions of a general name. For what
is the main use of a general name? It is to reduce a plurality
of objects to unity; to group a number of things
together by reference to some qualities which they possess
in common. The ordinary general name rests upon a considerable
variety of attributes, mostly of a qualitative
character, whereas the average, in so far as it serves the
same sort of purpose, rests rather upon a single quantitative
attribute. It directs attention to a certain kind and degree
of magnitude. When the grazier says of his sheep that 'one
with another they will fetch about 50 shillings,' or the
farmer buys a lot of poles which 'run to about 10 feet,' it is
true that they are not strictly using the equivalent of either
a general or a collective name. But they are coming very
near to such use, in picking out a sort of type or specimen of
the magnitude to which attention is to be directed, and in
classing the whole group by its resemblance to this type.
The grazier is thinking of his sheep: not in a merely general
sense, as sheep, and therefore under that name or conception,
but as sheep of a certain approximate money value.
Some will be more, some less, but they are all near enough
to the assigned value to be conveniently classed together as
if by a name. Many of our rough quantitative designations
seem to be of this kind, as when we speak of 'eight-day
clocks' or 'twelve-stone men,' &c.; unless of course we intend
(as we sometimes do in these cases) to assign a maximum
or minimum value. It is not indeed easy to see how else we
could readily convey a merely general notion of the quantitative
aspect of things, except by selecting a type as above,
or by assigning certain limits within which the things are
supposed to lie.
2. So far there is not necessarily any idea introduced
of comparison,--of comparison, that is, of one group with
another,--by aid of such an average. As soon as we begin
to think of this we have to be more precise in saying what
we mean by an average. We can easily see that the number
of possible kinds of average, in the sense of intermediate
values, is very great; is, in fact, indefinitely great. Out of
the general conception of an intermediate value, obtained by
some treatment of the original magnitudes, we can elicit as
many subdivisions as we please, by various modes of treatment.
There are however only three or four which for our
purposes need be taken into account.
(1) In the first place there is the arithmetical average
or mean. The rule for obtaining this is very simple: add
all the magnitudes together, and divide the sum by their
number. This is the only kind of average with which the
unscientific mind is thoroughly familiar. But we must not
let this simplicity and familiarity blind us to the fact that
there are definite reasons for the employment of this average,
and that it is therefore appropriate only in definite circumstances.
The reason why it affords a safe and accurate
intermediate value for the actual divergent values, is that
for many of the ordinary purposes of life, such as purchase
and sale, we come to exactly the same result, whether we
take account of those existent divergences, or suppose all
the objects equated to their average. What the grazier
must be understood to mean, if he wishes to be accurate, by
saying that the average price of his sheep is 50 shillings, is,
that so far as that flock is concerned (and so far as he is
concerned), it comes to exactly the same thing, whether they
are each sold at different prices, or are all sold at the 'average'
price. Accordingly, when he compares his sales of one
year with those of another; when he says that last year the
sheep averaged 48 shillings against the 50 of this year; the
employment of this representative or average value is a great
simplification, and is perfectly accurate for the purpose in
question.
3. (2) Now consider this case. A certain population is
found to have doubled itself in 100 years: can we talk of an
'average' increase here of 1 per cent. annually? The circumstances
are not quite the same as in the former case, but
the analogy is sufficiently close for our purpose. The answer
is decidedly, No. If 100 articles of any kind are sold for £100,
we say that the average price is £1. By this we mean that
the total amount is the same whether the entire lot are sold
for £100, or whether we split the lot up into individuals
and sell each of these for £1. The average price here is a
convenient fictitious substitute, which can be applied for
each individual without altering the aggregate total. If
therefore the question be, Will a supposed increase of 1 p. c.
in each of the 100 years be equivalent to a total increase to
double the original amount? we are proposing a closely
analogous question. And the answer, as just remarked, must
be in the negative. An annual increase of 1 p. c. continued
for 100 years will more than double the total; it will multiply
it by about 2.7. The true annual increment required is measured
by sqrt[100]{2}; that is, the population may be said to have
increased 'on the average' 0.7 p. c. annually.
We are thus directed to the second kind of average discussed
in the ordinary text-books of algebra, viz. the geometrical.
When only two quantities are concerned, with a single
intermediate value between them, the geometrical mean constituting
this last is best described as the mean proportional
between the two former. Thus, since 3 : sqrt{15} :: sqrt{15} : 5,
sqrt{15} is the geometrical mean between 3 and 5. When a
number of geometrical means have to be interposed between
two quantities, they are to be so chosen that every term in
the entire succession shall bear the same constant ratio to
its predecessor. Thus, in the example in the last paragraph,
99 intermediate steps were to be interposed between 1 and 2,
with the condition that the 100 ratios thus produced were to
be all equal.
It would seem therefore that wherever accurate quantitative
results are concerned, the selection of the appropriate
kind of average must depend upon the answer to the question,
What particular intermediate value may be safely
substituted for the actual variety of values, so far as the
precise object in view is concerned? This is an aspect of
the subject which will have to be more fully considered in
the next chapter. But it may safely be laid down that for
purposes of general comparison, where accurate numerical
relations are not required, almost any kind of intermediate
value will answer our purpose, provided we adhere to the
same throughout. Thus, if we want to compare the statures
of the inhabitants of different counties or districts in England,
or of Englishmen generally with those of Frenchmen,
or to ascertain whether the stature of some particular class
or district is increasing or diminishing, it really does not
seem to matter what sort of average we select provided, of
course, that we adhere to the same throughout our investigations.
A very large amount of the work performed by
averages is of this merely comparative or non-quantitative
description; or, at any rate, nothing more than this is really
required. This being so, we should naturally resort to the
arithmetical average; partly because, having been long in
the field, it is universally understood and appealed to, and
partly because it happens to be remarkably simple and easy
to calculate.
4. The arithmetical mean is for most ordinary purposes
the simplest and best. Indeed, when we are dealing
with a small number of somewhat artificially selected magnitudes,
it is the only mean which any one would think of
employing. We should not, for instance, apply any other
method to the results of a few dozen measurements of lengths
or estimates of prices.
When, however, we come to consider the results of a very
large number of measurements of the kind which can be
grouped together into some sort of 'probability curve' we
begin to find that there is more than one alternative before
us. Begin by recurring to the familiar curve represented
on p. 29; or, better still, to the initial form of it represented
in the next chapter (p. 476). We see that there are three
different ways in which we may describe the vertex of the
curve. We may call it the position of the _maximum_ ordinate;
or that of the _centre_ of the curve; or (as will be seen
hereafter) the point to which the arithmetical average of
all the different values of the variable magnitude directs us.
These three are all distinct ways of describing a position;
but when we are dealing with a symmetrical curve at all
resembling the binomial or exponential form they all three
coincide in giving the same result: as they obviously do in
the case in question.
As soon, however, as we come to consider the case of
asymmetrical, or lop-sided curves, the indications given by
these three methods will be as a rule quite distinct; and
therefore the two former of these deserve brief notice as
representing different kinds of means from the arithmetical
or ordinary one. We shall see that there is something about
each of them which recommends it to common sense as being
in some way natural and appropriate.
5. (3) The first of these selects from amongst the
various different magnitudes that particular one which is
most frequently represented. It has not acquired any technical
designation,[1] except in so far as it is referred to, by
its graphical representation, as the "maximum ordinate"
method. But I suspect that some appeal to such a mean
or standard is really far from uncommon, and that if we
could draw out into clearness the conceptions latent in the
judgments of the comparatively uncultivated, we should find
that there were various classes of cases in which this mean
was naturally employed. Suppose, for instance, that there
was a fishery in which the fish varied very much in size
but in which the commonest size was somewhat near the
largest or the smallest. If the men were in the habit of
selling their fish by _weight_, it is probable that they would
before long begin to acquire some kind of notion of what
is meant by the arithmetical mean or average, and would
perceive that this was the most appropriate test. But if the
fish were sorted into sizes, and sold by numbers in each of
these sizes, I suspect that this appeal to a maximum ordinate
would begin to take the place of the other. That is,
the most numerous class would come to be selected as a
sort of type by which to compare the same fishery at one
time and another, or one fishery with others. There is also,
as we shall see in the next chapter, some scientific ground
for the preference of this kind of mean in peculiar cases;
viz. where the quantities with which we deal are true
'errors,' in the estimate of some magnitude, and where also
it is of much more importance to be exactly right, or very
nearly right, than to have merely a low average of error.
6. (4) The remaining kind of mean is that which is
now coming to be called the "median." It is one with
which the writings of Mr Galton have done so much to
familiarize statisticians, and is best described as follows.
Conceive all the objects in question to be marshalled in the
order of their magnitude; or, what comes to the same thing,
conceive them sorted into a number of equally numerous
classes; then the middle one of the row, or the middle one
in the middle class, will be the _median_. I do not think
that this kind of mean is at all generally recognized at
present, but if Mr Galton's scheme of natural measurement
by what he calls "per-centiles" should come to be generally
adopted, such a test would become an important one.
There are some conspicuous advantages about this kind of
mean. For one thing, in most statistical enquiries, it is
far the simplest to calculate; and, what is more, the process
of determining it serves also to assign another important
element to be presently noticed, viz. the 'probable error.'
Then again, as Fechner notes, whereas in the arithmetical
mean a few exceptional and extreme values will often cause
perplexity by their comparative preponderance, in the case
of the median (where their number only and not their extreme
magnitude is taken into account) the importance of
such disturbance is diminished.
7. A simple illustration will serve to indicate how these
three kinds of mean coalesce into one when we are dealing
with symmetrical Laws of Error, but become quite distinct
as soon as we come to consider those which are unsymmetrical.
[Figure: Various definitions of the mean.]
Suppose that, in measuring a magnitude along OBDC,
where the extreme limits are OB and OC, the law of error
is represented by the triangle BAC: the length OD will
be at once the arithmetical mean, the median, and the most
frequent length: its frequency being represented by the
maximum ordinate AD. But now suppose, on the other
hand, that the extreme lengths are OD and OC, and that
the triangle ADC represents the law of error. The most
frequent length will be the same as before, OD, marked by
the maximum ordinate AD. But the mean value will now
be OX, where DX = 1/3 DC; and the median will be OY,
where DY = (1 - 1/sqrt{2}) DC.
Another example, taken from natural phenomena, may
be found in the heights of the barometer as taken at the
same hour on successive days. So far as 4857 of these may
be regarded as furnishing a sufficiently stable basis of experience,
it certainly seems that the resulting curve of frequency
is asymmetrical. The mean height here was found
to be 29.98: the median was 30.01: the most frequent
height was 30.05. The close approximation amongst these
is an indication that the asymmetry is slight.[2]
8. It must be clearly understood that the average, of
whatever kind it may be, from the mere fact of its being a
single substitute for an actual plurality of observed values,
must let slip a considerable amount of information. In fact
it is only introduced for economy. It may entail no loss
when used for some one assigned purpose, as in our example
about the sheep; but for purposes in general it
cannot possibly take the place of the original diversity, by
yielding all the information which they contained. If all
this is to be retained we must resort to some other method.
Practically we generally do one of two things: either (1) we
put all the figures down in statistical tables, or (2) we appeal
to a diagram. This last plan is convenient when the
data are very numerous, or when we wish to display or to
discover the nature of the law of facility under which they
range.
The mere assignment of an average lets drop nearly all
of this, confining itself to the indication of an intermediate
value. It gives a "middle point" of some kind, but says
nothing whatever as to how the original magnitudes were
grouped about this point. For instance, whether two magnitudes
had been respectively 25 and 27, or 15 and 37, they
would yield the same arithmetical average of 26.
9. To break off at this stage would clearly be to leave
the problem in a very imperfect condition. We therefore
naturally seek for some simple test which shall indicate
how closely the separate results were grouped about their
average, so as to recover some part of the information which
had been let slip.
If any one were approaching this problem entirely anew,--that
is, if he had no knowledge of the mathematical exigencies
which attend the theory of "Least Squares,"--I apprehend
that there is but one way in which he would set
about the business. He would say, The average which we
have already obtained gave us a rough indication, by assigning
an intermediate point amongst the original magnitudes.
If we want to supplement this by a rough indication
as to how near together these magnitudes lie, the
best way will be to treat their departures from the mean
(what are technically called the "errors") in precisely the
same way, viz. by assigning _their_ average. Suppose there
are 13 men whose heights vary by equal differences from
5 feet to 6 feet, we should say that their average height
was 66 inches, and their average departure from this average
was 3-3/13 inches.
Looked at from this point of view we should then proceed
to try how each of the above-named averages would
answer the purpose. Two of them,--viz. the arithmetical
mean and the median,--will answer perfectly; and, as we
shall immediately see, are frequently used for the purpose.
So too we could, if we pleased, employ the geometrical
mean, though such employment would be tedious, owing
to the difficulty of calculation. The 'maximum ordinate'
clearly would not answer, since it would generally (v. the
diagram on p. 443) refer us back again to the average
already obtained, and therefore give no information.
The only point here about which any doubt could arise
concerns what is called in algebra the _sign_ of the errors.
Two equal and opposite errors, added algebraically, would
cancel each other. But when, as here, we are regarding
the errors as substantive quantities, to be considered on
their own account, we attend only to their real magnitude,
and then these equal and opposite errors are to be put upon
exactly the same footing.
10. Of the various means already discussed, two, as
just remarked, are in common use. One of these is familiarly
known, in astronomical and other calculations, as
the 'Mean Error,' and is so absolutely an application of the
same principle of the arithmetical mean to the errors, that
has been already applied to the original magnitudes, that it
needs no further explanation. Thus in the example in the
last section the mean of the heights was 66 inches, the
mean of the errors was 3-3/13 inches.
The other is the Median, though here it is always known
under another name, i.e. as the 'Probable Error';--a technical
and decidedly misleading term. It is briefly defined
as that error which we are as likely to exceed as to fall
short of: otherwise phrased, if we were to arrange all the
errors in the order of their magnitude, it corresponds to that
one of them which just bisects the row. It is therefore the
'median' error: or, if we arrange all the magnitudes in successive
order, and divide them into four equally numerous
classes,--what Mr Galton calls 'quartiles,'--the first and
third of the consequent divisions will mark the limits of
the 'probable error' on each side, whilst the middle one will
mark the 'median.' This median, as was remarked, coincides,
in symmetrical curves, with the arithmetical mean.
It is best to stand by accepted nomenclature, but the
reader must understand that such an error is not in any
strict sense 'probable.' It is indeed highly improbable that
in any particular instance we should happen to get just this
error: in fact, if we chose to be precise and to regard it as
one exact magnitude out of an infinite number, it would be
infinitely unlikely that we should hit upon it. Nor can it be
said to be probable that we shall be _within_ this limit of the
truth, for, by definition, we are just as likely to exceed as to
fall short. As already remarked (see note on p. 441), the
'maximum ordinate' would have the best right to be regarded
as indicating the really most probable value.
11. (5) _The error of mean square._ As previously suggested, the plan
which would naturally be adopted by any one who had no concern with
the higher mathematics of the subject, would be to take the 'mean
error' for the purpose of the indication in view. But a very different
kind of average is generally adopted in practice to serve as a test of
the amount of divergence or dispersion. Suppose that we have the
magnitudes x_{1}, x_{2}, ... x_{n}; their ordinary average is
1/n (x_{1} + x_{2} + ... + x_{n}), and their 'errors' are the
differences between this and x_{1}, x_{2}, ... x_{n}. Call these errors
e_{1}, e_{2}, ... e_{n}, then the arithmetical mean of these errors
(irrespective of sign) is 1/n (e_{1} + e_{2} + ... + e_{n}). The Error
of Mean Square,[3] on the other hand, is the square root of
1/n (e_{1}^{2} + e_{2}^{2} + ... + e_{n}^{2}).
The reasons for employing this latter kind of average in
preference to any of the others will be indicated in the following
chapter. At present we are concerned only with the
general logical nature of an average, and it is therefore
sufficient to point out that any such intermediate value will
answer the purpose of giving a rough and summary indication
of the degree of closeness of approximation which our
various measures display to each other and to their common
average. If we were to speak respectively of the 'first' and
the 'second average,' we might say that the former of these
assigns a rough single substitute for the plurality of original
values, whilst the latter gives a similar rough estimate of the
degree of their departure from the former.
12. So far we have only been considering the general
nature of an average, and the principal kinds of average
practically in use. We must now enquire more particularly
what are the principal purposes for which averages are employed.
In this respect the first thing we have to do is to raise
doubts in the reader's mind on a subject on which he
perhaps has not hitherto felt the slightest doubt. Every
one is more or less familiar with the practice of appealing to
an average in order to secure accuracy. But distinctly what
we begin by doing is to sacrifice accuracy; for in place of
the plurality of actual results we get a single result which
very possibly does not agree with any one of them. If I find
the temperature in different parts of a room to be different,
but say that the average temperature is 61°, there may perhaps
be but few parts of the room where this exact temperature
is realized. And if I say that the average stature of
a certain small group of men is 68 inches, it is probable that
no one of them will present precisely this height.
The principal way in which accuracy can be thus secured
is when what we are really aiming at is not the magnitudes
before us but something else of which they are an indication.
If they are themselves 'inaccurate,'--we shall see presently
that this needs some explanation,--then the single average,
which in itself agrees perhaps with none of them, may be
much more nearly what we are actually in want of. We shall
find it convenient to subdivide this view of the subject into
two parts; by considering first those cases in which quantitative
considerations enter but slightly, and in which no determination
of the particular Law of Error involved is demanded,
and secondly those in which such determination cannot be
avoided. The latter are only noticed in passing here, as a
separate chapter is reserved for their fuller consideration.
13. The process, as a practical one, is familiar enough
to almost everybody who has to work with measures of any
kind. Suppose, for instance, that I am measuring any object
with a brass rod which, as we know, expands and contracts
according to the temperature. The results will vary slightly,
being sometimes a little too great and sometimes a little too
small. All these variations are physical facts, and if what
we were concerned with was the properties of brass they
would be the one important fact for us. But when we are
concerned with the length of the object measured, these facts
become superfluous and misleading. What we want to do is
to escape their influence, and this we are enabled to effect by
taking their (arithmetical) average, provided only they are
as often in excess as in defect.[4] For this purpose all that is
necessary is that equal excesses and defects should be
equally prevalent. It is not necessary to know what is the
law of variation, or even to be assured that it is of one particular
kind. Provided only that it is in the language of
the diagram on p. 29, symmetrical, then the arithmetical
average of a suitable and suitably varied number of measurements
will be free from this source of disturbance. And
what holds good of this cause of variation will hold good of
all others which obey the same general conditions. In fact
the equal prevalence of equal and opposite errors seems to
be the sole and sufficient justification of the familiar process
of taking the average in order to secure accuracy.
14. We must now make the distinction to which attention
requires so often to be drawn in these subjects
between the cases in which there respectively is, and is not,
some objective magnitude aimed at: a distinction which the
common use of the same word "errors" is so apt to obscure.
When we talked, in the case of the brass rod, of excesses
and defects being equal, we meant exactly what we said, viz.
that for every case in which the 'true' length (i.e. that determined
by the authorized standard) is exceeded by a given
fraction of an inch, there will be a corresponding case in
which there is an equal defect.
On the other hand, when there is no such fixed objective
standard of reference, it would appear that all that we mean
by equal excesses and defects is permanent symmetry of
arrangement. In the case of the measuring rod we were
able to start with something which existed, so to say, before
its variations; but in many cases any starting point which
we can find is solely determined by the average.
Suppose, for instance, we take a great number of observations
of the height of the barometer at a certain place,
at all times and seasons and in all weathers, we should
generally consider that the average of all these showed the
'true' height for that place. What we really mean is that
the height at any moment is determined partly (and principally)
by the height of the column of air above it, but partly
also by a number of other agencies such as local temperature,
moisture, wind, &c. These are sometimes more and sometimes
less effective, but their range being tolerably constant,
and their distribution through this range being
tolerably symmetrical, the average of one large batch of
observations will be almost exactly the same as that of any
other. This constancy of the average _is_ its truth. I am
quite aware that we find it difficult not to suppose that
there must be something more than this constancy, but we
are probably apt to be misled by the analogy of the other
class of cases, viz. those in which we are really aiming at
some sort of mark.
15. As regards the practical methods available for
determining the various kinds of average there is very little
to be said; as the arithmetical rules are simple and definite,
and involve nothing more than the inevitable drudgery
attendant upon dealing with long rows of figures. Perhaps
the most important contribution to this part of the subject is
furnished by Mr Galton's suggestion to substitute the median
for the mean, and thus to elicit the average with sufficient
accuracy by the mere act of grouping a number of objects
together. Thus he has given an ingenious suggestion for
obtaining the average height of a number of men without
the trouble and risk of measuring them all. "A barbarian
chief might often be induced to marshall his men in the
order of their heights, or in that of the popular estimate of
their skill in any capacity; but it would require some apparatus
and a great deal of time to measure each man
separately, even supposing it possible to overcome the usually
strong repugnance of uncivilized people to any such proceeding"
(_Phil. Mag._ Jan. 1875). That is, it being known
from wide experience that the heights of any tolerably
homogeneous set of men are apt to group themselves symmetrically,--the
condition for the coincidence of the three
principal kinds of mean,--the middle man of a row thus
arranged in order will represent the mean or average man,
and him we may subject to measurement. Moreover, since
the intermediate heights are much more thickly represented
than the extreme ones, a moderate error in the selection of
the central man of a long row will only entail a very small
error in the selection of the corresponding height.
16. We can now conveniently recur to a subject which
has been already noticed in a former chapter, viz. the attempt
which is sometimes made to establish a distinction
between an average and a mean. It has been proposed to
confine the former term to the cases in which we are dealing
with a fictitious result of our own construction, that is, with
a mere arithmetical deduction from the observed magnitudes,
and to apply the latter to cases in which there is
supposed to be some objective magnitude peculiarly representative
of the average.
Recur to the three principal classes, of things appropriate
to Probability, which were sketched out in Ch. II. §4. The
first of these comprised the results of games of chance. Toss
a die ten times: the total number of pips on the upper
side may vary from ten up to sixty. Suppose it to be
thirty. We then say that the average of this batch of
ten is three. Take another set of ten throws, and we may
get another average, say four. There is clearly nothing
objective peculiarly corresponding in any way to these
averages. No doubt if we go on long enough we shall
find that the averages tend to centre about 3.5: we then
call this _the_ average, or the 'probable' number of points;
and this ultimate average might have been pretty constantly
asserted beforehand from our knowledge of the constitution
of a die. It has however no other truth or reality
about it of the nature of a type: it is simply the limit
towards which the averages tend.
The next class is that occupied by the members of most
natural groups of objects, especially as regards the characteristics
of natural species. Somewhat similar remarks may
be repeated here. There is very frequently a 'limit' towards
which the averages of increasing numbers of individuals tend
to approach; and there is certainly some temptation to regard
this limit as being a sort of type which all had been
intended to resemble as closely as possible. But when we
looked closer, we found that this view could scarcely be
justified; all which could be safely asserted was that this
type represented, for the time being, the most numerous
specimens, or those which under existing conditions could
most easily be produced.
The remaining class stands on a somewhat different
ground. When we make a succession of more or less successful
attempts of any kind, we get a corresponding series
of deviations from the mark at which we aimed. These we
may treat arithmetically, and obtain their averages, just as
in the former cases. These averages are fictions, that is to
say, they are artificial deductions of our own which need
not necessarily have anything objective corresponding to
them. In fact, if they be averages of a _few_ only they
most probably will not have anything thus corresponding
to them. Anything answering to a type can only be sought
in the 'limit' towards which they ultimately tend, for this
limit coincides with the fixed point or object aimed at.
17. Fully admitting the great value and interest of
Quetelet's work in this direction,--he was certainly the first
to direct public attention to the fact that so many classes of
natural objects display the same characteristic property,--it
nevertheless does not seem desirable to attempt to mark
such a distinction by any special use of these technical
terms. The objections are principally the two following.
In the first place, a single antithesis, like this between
an average and a mean, appears to suggest a very much
simpler state of things than is actually found to exist in
nature. A reference to the three classes of things just
mentioned, and a consideration of the wide range and diversity
included in each of them, will serve to remind us
not only of the very gradual and insensible advance from
what is thus regarded as 'fictitious' to what is claimed as
'real;' but also of the important fact that whereas the 'real
type' may be of a fluctuating and evanescent character, the
'fiction' may (as in games of chance) be apparently fixed
for ever. Provided only that the conditions of production
remain stable, averages of large numbers will always practically
present much the same general characteristics. The
far more important distinction lies between the average of
a few, with its fluctuating values and very imperfect and
occasional attainment of its ultimate goal, and the average
of many and its gradually close approximation to its ultimate
value: i.e. to its objective point of aim if there happen
to be such.
Then, again, the considerations adduced in this chapter
will show that within the field of the average itself there is
far more variety than Quetelet seems to have recognized.
He did not indeed quite ignore this variety, but he practically
confined himself almost entirely to those symmetrical
arrangements in which three of the principal means coalesce
into one. We should find it difficult to carry out his distinction
in less simple cases. For instance, when there is
some degree of asymmetry, it is the 'maximum ordinate'
which would have to be considered as a 'mean' to the
exclusion of the others; for no appeal to an arithmetical
average would guide us to this point, which however is to
be regarded, if any can be so regarded, as marking out the
position of the ultimate type.
18. We have several times pointed out that it is a
characteristic of the things with which Probability is concerned
to present, in the long run, a continually intensifying
uniformity. And this has been frequently described as what
happens 'on the average.' Now an objection may very
possibly be raised against regarding an arrangement of
things by virtue of which order thus emerges out of disorder
as deserving any special notice, on the ground that from the
nature of the arithmetical average it could not possibly be
otherwise. The process by which an average is obtained, it
may be urged, insures this tendency to equalization amongst
the magnitudes with which it deals. For instance, let there
be a party of ten men, of whom four are tall and four are
short, and take the average of any five of them. Since this
number cannot be made up of tall men only, or of short men
only, it stands to reason that the averages cannot differ so
much amongst themselves as the single measures can. Is
not then the equalizing process, it may be asked, which is
observable on increasing the range of our observations,
one which can be shown to follow from necessary laws of
arithmetic, and one therefore which might be asserted _à priori_?
Whatever force there may be in the above objection arises
principally from the limitations of the example selected, in
which the number chosen was so large a proportion of the
total as to exclude the bare possibility of only extreme cases
being contained within it. As much confusion is often felt
here between what is necessary and what is matter of experience,
it will be well to look at an example somewhat
more closely, in order to determine exactly what are the
really necessary consequences of the averaging process.
19. Suppose then that we take ten digits at random
from a table (say) of logarithms. Unless in the highly unlikely
case of our having happened upon the same digit ten
times running, the average of the ten _must_ be intermediate
between the possible extremes. Every conception of an
average of any sort not merely involves, but actually means,
the taking of something intermediate between the extremes.
The average therefore of the ten must lie closer to 4.5 (the
average of the extremes) than did some of the single digits.
Now suppose we take 1000 such digits instead of 10. We
can say nothing more about the larger number, with demonstrative
certainty, than we could before about the smaller.
If they were unequal to begin with (i.e. if they were not all
the same) then the average _must_ be intermediate, but more
than this cannot be proved arithmetically. By comparison with
such purely arithmetical considerations there is what may be
called a physical fact underlying our confidence in the growing
stability of the average of the larger number. It is that
the constituent elements from which the average is deduced
will themselves betray a growing uniformity:--that the proportions
in which the different digits come out will become
more and more nearly equal as we take larger numbers of
them. If the proportions in which the 1000 digits were
distributed were the same as those of the 10 the averages
would be the same. It is obvious therefore that the arithmetical
process of obtaining an average goes a very little
way towards securing the striking kind of uniformity which
we find to be actually presented.
20. There is another way in which the same thing
may be put. It is sometimes said that whatever may have
been the arrangement of the original elements the process of
continual averaging will necessarily produce the peculiar
binomial or exponential law of arrangement. This statement
is perfectly true (with certain safeguards) but it is not
in any way opposed to what has been said above. Let us
take for consideration the example above referred to. The
arrangement of the individual digits in the long run is the
simplest possible. It would be represented, in a diagram,
not by a curve but by a finite straight line, for each digit
occurs about as often as any other, and this exhausts all the
'arrangement' that can be detected. Now, when we consider
the results of taking averages of ten such digits, we see
at once that there is an opening for a more extensive arrangement.
The totals may range from 0 up to 100, and therefore
the average will have 100 values from 0 to 9; and what
we find is that the frequency of these numbers is determined
according to the Binomial[5] or Exponential Law. The most
frequent result is the true mean, viz. 4.5, and from this they
diminish in each direction towards 0 and 10, which will each
occur but once (on the average) in 10^{10} occasions.
The explanation here is of the same kind as in the former
case. The resultant arrangement, so far as the averages are
concerned, is only 'necessary' in the sense that it is a necessary
result of certain physical assumptions or experiences.
If all the digits tend to occur with equal frequency, and if
they are 'independent' (i.e. if each is associated indifferently
with every other), then it is an arithmetical consequence
that the averages when arranged in respect of their magnitude
and prevalence will display the Law of Facility above
indicated. Experience, so far as it can be appealed to, shows
that the true randomness of the selection of the digits,--i.e.
their equally frequent recurrence, and the impartiality of
their combination,--is very fairly secured in practice. Accordingly
the theoretic deduction that whatever may have
been the original Law of Facility of the individual results
we shall always find the familiar Exponential Law asserting
itself as the law of the averages, is fairly justified by experience
in such a case.
The further discussion of certain corrections and refinements
is reserved to the following chapter.
21. In regard to the three kinds of average employed
to test the amount of dispersion,--i.e. the mean error, the
probable error, and the error of mean square,--two important
considerations must be borne in mind. They will
both recur for fuller discussion and justification in the course
of the next chapter, when we come to touch upon the Method
of Least Squares, but their significance for logical purposes
is so great that they ought not to be entirely passed by at
present.
(1) In the first place, then, it must be remarked that in
order to know what in any case is the real value of an error
we ought in strictness to know what is the position of the
limit or ultimate average, for the amount of an error is
always theoretically measured from this point. But this is
information which we do not always possess. Recurring
once more to the three principal classes of events with which
we are concerned, we can readily see that in the case of
games of chance we mostly do possess this knowledge. Instead
of appealing to experience to ascertain the limit, we
practically deduce it by simple mechanical or arithmetical
considerations, and then the 'error' in any individual case or
group of cases is obviously found by comparing the results
thus obtained with that which theory informs us would ultimately
be obtained in the long run. In the case of deliberate
efforts at an aim (the third class) we may or may
not know accurately the value or position of this aim. In
astronomical observations we do not know it, and the method
of Least Squares is a method for helping us to ascertain it as
well as we can; in such experimental results as firing at a
mark we do know it, and may thus test the nature and
amount of our failure by direct experience. In the remaining
case, namely that of what we have termed natural kinds
or groups of things, not only do we not know the ultimate
limit, but its existence is always at least doubtful, and in
many cases may be confidently denied. Where it does exist,
that is, where the type seems for all practical purposes permanently
fixed, we can only ascertain it by a laborious resort
to statistics. Having done this, we may then test by it the
results of observations on a small scale. For instance, if we
find that the ultimate proportion of male to female births is
about 106 to 100, we may then compare the statistics of
some particular district or town and speak of the consequent
'error,' viz. the departure, in that particular and special
district, from the general average.
What we have therefore to do in the vast majority of
practical cases is to take the average of a finite number of
measurements or observations,--of all those, in fact, which
we have in hand,--and take _this_ as our starting point in
order to measure the errors. The errors in fact are not
known for certain but only probably calculated. This however
is not so much of a theoretic defect as it may seem at
first sight; for inasmuch as we seldom have to employ these
methods,--for purposes of calculation, that is, as distinguished
from mere illustration,--except for the purpose of discovering
what the ultimate average is, it would be a sort of
_petitio principii_ to assume that we had already secured it.
But it is worth while considering whether it is desirable to
employ one and the same term for 'errors' known to be
such, and whose amount can be assigned with certainty, and
for 'errors' which are only probably such and whose amount
can be only probably assigned. In fact it has been proposed[6]
to employ the two terms 'error' and 'residual' respectively
to distinguish between the magnitudes thus determined, that
is, between the (generally unknown) actual error and the observed
error.
22. (2) The other point involves the question to what
extent either of the first two tests (pp. 446,7) of the closeness
with which the various results have grouped themselves
about their average is trustworthy or complete. The answer
is that they are necessarily incomplete. No single estimate
or magnitude can possibly give us an adequate account of a
number of various magnitudes. The point is a very important
one; and is not, I think, sufficiently attended to, the
consequence being, as we shall see hereafter, that it is far
too summarily assumed that a method which yields the
result with the least 'error of mean square' must necessarily
be the best result for all purposes. It is not however by any
means clear that a test which answers best for one purpose
must do so for all.
It must be clearly understood that each of these tests is
an 'average,' and that every average necessarily rejects a
mass of varied detail by substituting for it a single result.
We had, say, a lot of statures: so many of 60 inches, so
many of 61, &c. We replace these by an 'average' of 68,
and thereby drop a mass of information. A portion of this
we then seek to recover by reconsidering the 'errors' or
departures of these statures from their average. As before,
however, instead of giving the full details we substitute an
average of the errors. The only difference is that instead of
taking the same kind of average (i.e. the arithmetical) we
often prefer to adopt the one called the 'error of mean
square.'
23. A question may be raised here which is of sufficient
importance to deserve a short consideration. When we have
got a set of measurements before us, why is it generally
held to be sufficient simply to assign: (1) the mean value;
and (2) the mean departure from this mean? The answer
is, of course, partly given by the fact that we are only supposed
to be in want of a rough approximation: but there is
more to be said than this. A further justification is to be
found in the fact that we assume that we need only contemplate
the possibility of a single Law of Error, or at any
rate that the departures from the familiar Law will be but
trifling. In other words, if we recur to the figure on p. 29,
we assume that there are only two unknown quantities or
disposable constants to be assigned; viz. first, the position of
the centre, and, secondly, the degree of eccentricity, if one
may so term it, of the curve. The determination of the
mean value directly and at once assigns the former, and the
determination of the mean error (in either of the ways referred
to already) indirectly assigns the latter by confining us
to one alone of the possible curves indicated in the figure.
Except for the assumption of one such Law of Error the
determination of the mean error would give but a slight
intimation of the sort of outline of our Curve of Facility.
We might then have found it convenient to adopt some plan
of successive approximation, by adding a third or fourth
'mean.' Just as we assign the mean value of the magnitude,
and its mean departure from this mean; so we might
take this mean error (however determined) as a fresh starting
point, and assign the mean departure from it. If the point
were worth further discussion we might easily illustrate by
means of a diagram the sort of successive approximations
which such indications would yield as to the ultimate form
of the Curve of Facility or Law of Error.
* * * * *
As this volume is written mainly for those who take an interest in the
logical questions involved, rather than as an introduction to the actual
processes of calculation, mathematical details have been throughout avoided
as much as possible. For this reason comparatively few references have
been made to the exponential equation of the Law of Error, or to the
corresponding 'Probability integral,' tables of which are given in several
handbooks on the subject. There are two points however in connection
with these particular topics as to which difficulties are, or should be,
felt by so many students that some notice may be taken of them here
(1) In regard to the ordinary algebraical expression for the law of
error, viz. y = (h/sqrt{π}) e^{-h^{2}x^{2}}, it will have been
observed that I have always spoken of y as being _proportional_ to the
number of errors of the particular magnitude x. It would hardly be
correct to say, absolutely, that y _represents_ that number, because
of course the actual number of errors of any precise magnitude, where
continuity of possibility is assumed, must be indefinitely small. If
therefore we want to pass from the continuous to the discrete, by
ascertaining the actual number of errors between two consecutive
divisions of our scale, when, as usual in measurements, all within
certain limits are referred to some one precise point, we must modify
our formula. In accordance with the usual differential notation, we
must say that the number of errors falling into one subdivision (dx)
of our scale _is_ dx (h/sqrt{π}) e^{-h^{2} x^{2}}, where dx is a
(small) unit of length, in which both h^{-1} and x must be measured.
The difficulty felt by most students is in applying the formula to
actual statistics, in other words in putting in the correct units. To
take an actual numerical example, suppose that 1460 men have been
measured in regard to their height "true to the nearest inch," and let
it be known that the modulus here is 3.6 inches. Then dx = 1 (inch);
h^{-1} = 3.6 inches. Now sum (h/sqrt{π}) e^{-h^{2}x^{2}} dx = 1; that
is, the sum of all the consecutive possible values is equal to
_unity_. When therefore we want the sum, as here, to be 1460, we must
express the formula thus;--y = 1460/(sqrt{π} × 3.6) e^{-(x/3.6)^{2}},
or y = 228 e^{-(x/3.6)^{2}}.
Here x stands for the number of inches measured from the central or
mean height, and y stands for the number of men referred to that
height in our statistical table. (The values of e^{-t^{2}} for
successive values of t are given in the handbooks.)
For illustration I give the calculated numbers by this formula for values
of x from 0 to 8 inches, with the actual numbers observed in the Cambridge
measurements recently set on foot by Mr Galton.
inches calculated observed
x = 0 y = 228 = 231
x = 1 y = 212 = 218
x = 2 y = 166 = 170
x = 3 y = 111 = 110
x = 4 y = 82 = 66
x = 5 y = 32 = 31
x = 6 y = 11 = 10
x = 7 y = 4 = 6
x = 8 y = 1 = 3
Here the average height was 69 inches: dx, as stated, = 1 inch. By
saying, 'put x = 0,' we mean, calculate the number of men who are assigned
to 69 inches; i.e. who fall between 68.5 and 69.5. By saying, 'put x = 4,'
we mean, calculate the number who are assigned to 65 or to 73; i.e. who lie
between 64.5 and 65.5, or between 72.5 and 73.5. The observed results, it
will be seen, keep pretty close to the calculated: in the case of the former
the _means_ of equal and opposite divergences from the mean have been taken,
the actual results not being always the same in opposite directions.
(2) The other point concerns the interpretation of the familiar
probability integral, (2/sqrt{π}) integral_{0}^{t} e^{-t^{2}} dt.
Every one who has calculated the chance of an event, by the help
of the tables of this integral given in so many handbooks, knows that
if we assign any numerical value to t, the corresponding value of the
above expression assigns the chance that an error taken at random
shall lie within that same limit, viz. t. Thus put t = 1.5, and we
have the result 0.96; that is, only 4 per cent. of the errors will
exceed 'one and a half.' But when we ask, 'one and a half' _what?_ the
answer would not always be very ready. As usual, the main difficulty
of the beginner is not to manipulate the formulæ, but to be quite
clear about his units.
It will be seen at once that this case differs from the preceding in that
we cannot now choose our unit as we please. Where, as here, there is only
one variable (t), if we were allowed to select our own unit, the inch, foot,
or whatever it might be, we might get quite different results. Accordingly
some comparatively natural unit must have been chosen for us in which we
are bound to reckon, just as in the circular measurement of an angle as
distinguished from that by degrees.
The answer is that the unit here is the _modulus_, and that to put 't = 1.5'
is to say, 'suppose the error half as great again as the modulus'; the
modulus itself being an error of a certain assignable magnitude depending
upon the nature of the measurements or observations in question. We shall
see this better if we put the integral in the form
(2/sqrt{π}) integral_{0}^{hx} e^{-h^{2} x^{2}} d(hx);
which is precisely equivalent, since the value of a definite integral is
independent of the particular variable employed. Here hx is the same as
x : 1/h; i.e. it is the ratio of x to 1/h, or x measured in terms of 1/h.
But 1/h is the modulus in the equation
(y = (h/sqrt{π}) e^{-h^{2} x^{2}})
for the law of error. In other words the numerical value of an error in
this formula, is the number of times, whole or fractional, which it
contains the modulus.
1. This kind of mean is called by Fechner and others the "_dichteste
Werth_." The most appropriate appeal to it that I have seen is by
Prof. Lexis (_Massenerscheinungen_, p. 42) where he shows that it
indicates clearly a sort of normal length of human life, of about 70
years; a result which is almost entirely masked when we appeal to
the arithmetical average.
This mean _ought_ to be called the 'probable' value (a name however
in possession of another) on the ground that it indicates the point
of likeliest occurrence; i.e. if we compare all the indefinitely
small and equal units of variation, the one corresponding to this
will tend to be most frequently represented.
2. A diagram illustrative of this number of results was given in
_Nature_ (Sept. 1, 1887). In calculating, as above, the different
means, I may remark that the original results were given to three
decimal places; but, in classing them, only one place was
noted. That is, 29.9 includes all values between 29.900 and 29.999.
Thus the value most frequently entered in my tables was 30.0, but on
the usual principles of interpolation this is reckoned as 30.05.
3. There is some ambiguity in the phraseology in use here. Thus Airy
commonly uses the expression 'Error of Mean Square' to represent, as
here, sqrt{(sum e^{2})/n}. Galloway commonly speaks of the 'Mean
Square of the Errors' to represent (sum e^{2})/n. I shall adhere to
the former usage and represent it briefly by E.M.S. Still more
unfortunate (to my thinking) is the employment, by Mr Merriman and
others, of the expression 'Mean Error,' (widely in use in its more
natural signification,) as the equivalent of this E.M.S.
The technical term 'Fluctuation' is applied by Mr F. Y. Edgeworth to
the expression (2 sum e^{2})/n.
4. Practically, of course, we should allow for the expansion or
contraction. But for purposes of logical explanation we may
conveniently take this variation as a specimen of one of those
disturbances which may be neutralised by resort to an average.
5. More strictly _multinomial_: the relative frequency of the
different numbers being indicated by the coefficients of the powers
of x in the development of
(1 + x + x^{2} + ... + x^{9})^{10}.
6. By Mr Merriman, in his work on _Least Squares_.