_THE THEORY OF THE AVERAGE AS A MEANS OF APPROXIMATION TO THE TRUTH._
1. In the last chapter we were occupied with the Average
mainly under its qualitative rather than its quantitative
aspect. That is, we discussed its general nature, its principal
varieties, and the main uses to which it could be put in
ordinary life or in reasoning processes which did not claim to
be very exact. It is now time to enter more minutely into
the specific question of the employment of the average in
the way peculiarly appropriate to Probability. That is, we
must be supposed to have a certain number of measurements,--in
the widest sense of that term,--placed before us,
and to be prepared to answer such questions as; Why do we
take their average? With what degree of confidence?
Must we in all cases take the average, and, if so, one always
of the same kind?
The subject upon which we are thus entering is one
which, under its most general theoretic treatment, has perhaps
given rise to more profound investigation, to a greater
variety of opinion, and in consequence to a more extensive
history and literature, than any other single problem within
the range of mathematics.[1] But, in spite of this, the main
logical principles underlying the methods and processes in
question are not, I apprehend, particularly difficult to grasp:
though, owing to the extremely technical style of treatment
adopted even in comparatively elementary discussions of the
subject, it is far from easy for those who have but a moderate
command of mathematical resources to disentangle these
principles from the symbols in which they are clothed. The
present chapter contains an attempt to remove these difficulties,
so far as a general comprehension of the subject is concerned.
As the treatment thus adopted involves a considerable
number of subdivisions, the reader will probably
find it convenient to refer back occasionally to the table of
contents at the commencement of this volume.
2. The subject, in the form in which we shall discuss
it, will be narrowed to the consideration of the _average_, on
account of the comparative simplicity and very wide prevalence
of this aspect of the problem. The problem is however
very commonly referred to, even in non-mathematical treatises,
as the Rule or Method of Least Squares; the fact being
that, in such cases as we shall be concerned with, the Rule
of Least Squares resolves itself into the simpler and more
familiar process of taking the arithmetical average. A very
simple example,--one given by Herschel,--will explain the
general nature of the task under a slightly wider treatment,
and will serve to justify the familiar designation.
Suppose that a man had been firing for some time with a
pistol at a small mark, say a wafer on a wall. We may take
it for granted that the shot-marks would tend to group
themselves about the wafer as a centre, with a density varying
in some way inversely with the distance from the centre.
But now suppose that the wafer which marked the centre
was removed, so that we could see nothing but the surface of
the wall spotted with the shot-marks; and that we were
asked to guess the position of the wafer. Had there been
only _one_ shot, common sense would suggest our assuming
(of course very precariously) that this marked the real centre.
Had there been two, common sense would suggest our taking
the mid-point between them. But if three or more were
involved, common sense would be at a loss. It would feel
that some intermediate point ought to be selected, but
would not see its way to a more precise determination, because
its familiar reliance,--the arithmetical average,--does
not seem at hand here. The rule in question tells us how to
proceed. It directs us to select that point which will render
the sum of the squares of all the distances of the various
shot-marks from it the least possible.[2]
This is merely by way of illustration, and to justify the
familiar designation of the rule. The sort of cases with
which we shall be exclusively occupied are those comparatively
simple ones in which only linear magnitude, or some
quality which can be adequately represented by linear magnitude,
is the object under consideration. In respect of these
the Rule of Least Squares reduces itself to the process of
taking the average, in the most familiar sense of that term,
viz. the arithmetical mean; and a single Law of Error, or its
graphical equivalent, a Curve of Facility, will suffice accurately
to indicate the comparative frequency of the different
amounts of the one variable magnitude involved.
3. We may conveniently here again call attention to a
misconception or confusion which has been already noticed
in a former chapter. It is that of confounding the Law of
Error with the Method of Least Squares. These are things
of an entirely distinct kind. The former is of the nature of
a physical fact, and its production is one which in many
cases is entirely beyond our control. The latter,--or any
simplified application of it, such as the arithmetical average,--is
no law whatever in the physical sense. It is rather a
precept or rule for our guidance. The Law states, in any
given case, how the errors tend to occur in respect of their
magnitude and frequency. The Method directs us how to
treat these errors when any number of them are presented
to us. No doubt there is a relation between the two, as will
be pointed out in the course of the following pages; but
there is nothing really to prevent us from using the same
method for different laws of error, or different methods for
the same law. In so doing, the question of distinct right
and wrong would seldom be involved, but rather one of more
or less propriety.
4. The reader must understand,--as was implied in
the illustration about the pistol shots,--that the ultimate
problem before us is an _inverse_ one. That is, we are supposed
to have a moderate number of 'errors' before us and
we are to undertake to say whereabouts is the centre from
which they diverge. This resembles the determination of a
cause from the observation of an effect. But, as mostly
happens in inverse problems, we must commence with the
consideration of the direct problem. In other words, so far
as concerns the case before us, we shall have to begin by
supposing that the ultimate object of our aim,--that is, the
true centre of our curve of frequency,--is already known to
us: in which case all that remains to be done is to study the
consequences of taking averages of the magnitudes which
constitute the errors.
5. We shall, for the present, confine our remarks to
what must be regarded as the typical case where considerations
of Probability are concerned; viz. that in which
the law of arrangement or development is of the Binomial
kind. The nature of this law was explained in Chap. II.,
where it was shown that the frequency of the respective
numbers of occurrences was regulated in accordance with
the magnitude of the successive terms of the expansion of
the binomial (1 + 1)^{n}. It was also pointed out that when n becomes
very great, that is, when the number of influencing
circumstances is very large, and their relative individual
influence correspondingly small, the form assumed by a
curve drawn through the summits of ordinates representing
these successive terms of the binomial tends towards that
assigned by the equation y = Ae^{-h^{2}x^{2}}.
For all practical purposes therefore we may talk indifferently
of the Binomial or Exponential law; if only on
the ground that the arrangement of the actual phenomena
on one or other of these two schemes would soon become
indistinguishable when the numbers involved are large.
But there is another ground than this. Even when the
phenomena themselves represent a continuous magnitude,
our measurements of them,--which are all with which we
can deal,--are discontinuous. Suppose we had before us the
accurate heights of a million adult men. For all practical
purposes these would represent the variations of a continuous
magnitude, for the differences between two successive
magnitudes, especially near the mean, would be
inappreciably small. But our tables will probably represent
them only to the nearest inch. We have so many assigned
as 69 inches; so many as 70; and so on. The tabular
statement in fact is of much the same character as if we
were assigning the number of 'heads' in a toss of a handful
of pence; that is, as if we were dealing with discontinuous
numbers on the binomial, rather than with a continuous
magnitude on the exponential arrangement.
6. Confining ourselves then, for the present, to this
general head, of the binomial or exponential law, we must
distinguish two separate cases in respect of the knowledge
we may possess as to the generating circumstances of the
variable magnitudes.
(1) There is, first, the case in which the conditions of
the problem are determinable _à priori_: that is, where we are
able to say, prior to specific experience, how frequently each
combination will occur in the long run. In this case the
main or ultimate object for which we are supposing that the
average is employed,--i.e. that of discovering the true mean
value,--is superseded. We are able to say what the mean
or central value in the long run will be; and therefore there
is no occasion to set about determining it, with some trouble
and uncertainty, from a small number of observations. Still
it is necessary to discuss this case carefully, because its
assumption is a necessary link in the reasoning in other
cases.
This comparatively _à priori_ knowledge may present itself
in two different degrees as respects its completeness. In the
first place it may, so far as the circumstances in question
are concerned, be absolutely complete. Consider the results
when a handful of ten pence is repeatedly tossed up. We
know precisely what the mean value is here, viz. equal
division of heads and tails: we know also the chance of six
heads and four tails, and so on. That is, if we had to plot
out a diagram showing the relative frequency of each combination,
we could do so without appealing to experience.
We could draw the appropriate binomial curve from the
generating conditions given in the statement of the problem.
But now consider the results of firing at a target consisting
of a long and narrow strip, of which one point is
marked as the centre of aim.[3] Here (assuming that there
are no causes at work to produce permanent bias) we know
that this centre will correspond to the mean value. And we
know also, in a general way, that the dispersion on each side
of this will follow _a_ binomial law. But if we attempted to
plot out the proportions, as in the preceding case, by erecting
ordinates which should represent each degree of frequency
as we receded further from the mean, we should find that
we could not do so. Fresh data must be given or inferred.
A good marksman and a bad marksman will both distribute
their shot according to the same general law; but the
rapidity with which the shots thin off as we recede from the
centre will be different in the two cases. Another 'constant'
is demanded before the curve of frequency could be correctly
traced out.
7. (2) The second division, to be next considered,
corresponds for all logical purposes to the first. It comprises
the cases in which though we have no à priori knowledge
as to the situation about which the values will tend to
cluster in the long run, yet we have sufficient experience at
hand to assign it with practical certainty. Consider for
instance the tables of human stature. These are often very
extensive, including tens or hundreds of thousands. In such
cases the mean or central value is determinable with just as
great certainty as by any _à priori_ rule. That is, if we took
another hundred thousand measurements from the same
class of population, we should feel secure that the average
would not be altered by any magnitude which our measuring
instruments could practically appreciate.
8. But the mere assignment of the mean or central
value does not here, any more than in the preceding case,
give us all that we want to know. It _might_ so happen that
the mean height of two populations was the same, but that
the law of dispersion about that mean was very different:
so that a man who in one series was an exceptional giant or
dwarf should, in the other, be in no wise remarkable.
To explain the process of thus determining the actual magnitude of the
dispersion would demand too much mathematical detail; but some
indication may be given. What we have to do is to determine the
constant h in the equation[4] y = (h/sqrt{π}) e^{-h^{2} x^{2}}. In
technical language, what we have to do is to determine the _modulus_
of this equation. The quantity 1/h in the above expression is called
the modulus. It measures the degree of contraction or dispersion about
the mean indicated by this equation. When it is large the dispersion
is considerable; that is the magnitudes are not closely crowded up
towards the centre, when it is small they are thus crowded up. The
smaller the modulus in the curve representing the thickness with which
the shot-marks clustered about the centre of the target, the better
the marksman.
9. There are several ways of determining the modulus. In the first of
the cases discussed above, where our theoretical knowledge is
complete, we are able to calculate it _à priori_ from our knowledge of
the chances. We should naturally adopt this plan if we were tossing up
a large handful of pence.
The usual _à posteriori_ plan, when we have the measurements
of the magnitudes or observations before us, is this:--Take
the mean square of the errors, and double this; the result
gives the square of the modulus. Suppose, for instance,
that we had the five magnitudes, 4, 5, 6, 7, 8. The mean of
these is 6: the 'errors' are respectively 2, 1, 0, 1, 2. Therefore
the 'modulus squared' is equal to 10/5; i.e. the modulus is sqrt{2}.
Had the magnitudes been 2, 4, 6, 8, 10; representing
the same mean (6) as before, but displaying a greater dispersion
about it, the modulus would have been larger, viz. sqrt{8}
instead of sqrt{2}.
Mr Galton's method is more of a graphical nature. It
is described in a paper on Statistics by Intercomparison
(_Phil. Mag._ 1875), and elsewhere. It may be indicated as
follows. Suppose that we were dealing with a large number
of measurements of human stature, and conceive that all the
persons in question were marshalled in the order of their
height. Select the average height, as marked by the central
man of the row. Suppose him to be 69 inches. Then raise
(or depress) the scale from this point until it stands at such
a height as just to include one half of the men above (or
below) the mean. (In practice this would be found to require
about 1.71 inches: that is, one quarter of any large
group of such men will fall between 69 and 70.71 inches.)
Divide this number by 0.4769 and we have the modulus. In
the case in question it would be equal to about 3.6 inches.
Under the assumption with which we start, viz. that the law of error
displays itself in the familiar binomial form, or in some form
approximating to this, the three methods indicated above will coincide
in their result. Where there is any doubt on this head, or where we do
not feel able to calculate beforehand what will be the rate of
dispersion, we must adopt the second plan of determining the modulus.
This is the only universally applicable mode of calculation: in fact
that it should yield the modulus is a truth of definition; for in
determining the error of mean square we are really doing nothing else
than determining the modulus, as was pointed out in the last chapter.
10. The position then which we have now reached is this. Taking it for
granted that the Law of Error will fall into the symbolic form
expressed by the equation y = (h/sqrt{π}) e^{-h^{2} x^{2}}, we have
rules at hand by which h may be determined. We therefore, for the
purposes in question, know all about the curve of frequency: we can
trace it out on paper: given one value,--say the central one,--we can
determine any other value at any distance from this. That is, knowing
how many men in a million, say, are 69 inches high, we can determine
without direct observation how many will be 67, 68, 70, 71, and so on.
We can now adequately discuss the principal question of logical
interest before us; viz. _why_ do we take averages or means? What is
the exact nature and amount of the advantage gained by so doing? The
advanced student would of course prefer to work out the answers to
these questions by appealing at once to the Law of Error in its
ultimate or exponential form. But I feel convinced that the best
method for those who wish to gain a clear conception of the logical
nature of the process involved, is to begin by treating it as a
question of combinations such as we are familiar with in elementary
algebra; in other words to take a finite number of errors and to see
what comes of averaging these. We can then proceed to work out
arithmetically the results of combining two or more of the errors
together so as to get a new series, not contenting ourselves with the
general character merely of the new law of error, but actually
calculating what it is in the given case. For the sake of simplicity
we will not take a series with a very large number of terms in it, but
it will be well to have enough of them to secure that our law of error
shall roughly approximate in its form to the standard or exponential
law.
For this purpose the law of error or divergence given by supposing our
effort to be affected by ten causes, each of which produces an equal
error, but which error is equally likely to be positive and negative
(or, as it might perhaps be expressed, 'ten equal and indifferently
additive and subtractive causes') will suffice. This is the lowest
number formed according to the Binomial law, which will furnish to the
eye a fair indication of the limiting or Exponential law.[5] The whole
number of possible cases here is 2^{10} or 1024; that is, this is the
number required to exhibit not only all the cases which can occur (for
there are but eleven really distinct cases), but also the relative
frequency with which each of these cases occurs in the long run. Of
this total, 252 will be situated at the mean, representing the 'true'
result, or that given when five of the causes of disturbance just
neutralize the other five. Again, 210 will be at what we will call one
unit's distance from the mean, or that given by six causes combining
against four; and so on; until at the extreme distance of five places
from the mean we get but one result, since in only one case out of the
1024 will all the causes combine together in the same direction. The
set of 1024 efforts is therefore a fair representation of the
distribution of an infinite number of such efforts. A graphical
representation of the arrangement is given here.
[Figure: Binomial distribution for the tenth power.]
11. This representing a complete set of single observations or
efforts, what will be the number and arrangement in the corresponding
set of combined or reduced observations, say of two together? With
regard to the number we must bear in mind that this is not a case of
the combinations of things which cannot be repeated; for any given
error, say the extreme one at F, can obviously be repeated twice
running. Such a repetition would be a piece of very bad luck no doubt,
but being possible it must have its place in the set. Now the possible
number of ways of combining 1024 things two together, where the same
thing may be repeated twice running, is 1024 × 1024 or 1048576. This
then is the number in a complete cycle of the results taken two and
two together.
12. So much for their number; now for their arrangement or
distribution. What we have to ascertain is, firstly, how many times
each possible pair of observations will present itself; and, secondly,
where the new results, obtained from the combination of each pair, are
to be placed. With regard to the first of these enquiries;--it will be
readily seen that on one occasion we shall have F repeated twice; on
20 occasions we shall have F combined with E (for F coming first we
may have it followed by any one of the 10 at E, or any one of these
may be followed by F); E can be repeated in 10 × 10, or 100 ways, and
so on.
Now for the position of each of these reduced observations, the
relative frequency of whose component elements has thus been pointed
out. This is easy to determine, for when we take two errors there is
(as was seen) scarcely any other mode of treatment than that of
selecting the mid-point between them; this mid-point of course
becoming identical with each of them when the two happen to
coincide. It will be seen therefore that F will recur once on the new
arrangement, viz. by its being repeated twice on the old one. G midway
between E and F, will be given 20 times. E, on our new arrangement,
can be got at in two ways, viz. by its being repeated twice (which
will happen 100 times), and by its being obtained as the mid-point
between D and F (which will happen 90 times). Hence E will occur 190
times altogether.
The reader who chooses to take the trouble may work out the frequency
of all possible occurrences in this way, and if the object were simply
to illustrate the principle in accordance with which they occur, this
might be the best way of proceeding. But as he may soon be able to
observe, and as the mathematician would at once be able to prove, the
new 'law of facility of error' can be got at more quickly deductively,
viz. by taking the successive terms of the expansion of (1 + 1)^{20}.
They are given, below the line, in the figure on p. 476.
13. There are two apparent obstacles to any direct comparison between
the distribution of the old set of simple observations, and the new
set of combined or reduced ones. In the first place, the number of
the latter is much greater. This, however, is readily met by reducing
them both to the same scale, that is by making the same total number
of each. In the second place, half of the new positions have no
representatives amongst the old, viz. those which occur midway between
F and E, E and D, and so on. This can be met by the usual plan of
interpolation, viz. by filling in such gaps by estimating what would
have been the number at the missing points, on the same scale, had
they been occupied. Draw a curve through the vertices of the
ordinates at A, B, C, &c., and the lengths of the ordinates at the
intermediate points will very fairly represent the corresponding
frequency of the errors of those magnitudes respectively. When the
gaps are thus filled up, and the numbers thus reduced to the same
scale, we have a perfectly fair basis of comparison. (See figure on
next page.)
Similarly we might proceed to group or 'reduce' three observations, or
any greater number. The number of possible groupings naturally becomes
very much larger, being (1024)^{3} when they are taken three
together. As soon as we get to three or more observations, we have (as
already pointed out) a variety of possible modes of treatment or
reduction, of which that of taking the arithmetical mean is but one.
14. The following figure is intended to illustrate the nature of the
advantage secured by thus taking the arithmetical mean of several
observations.
The curve ABCD represents the arrangement of a given number of
'errors' supposed to be disposed according to the binomial law already
mentioned, when the angles have been smoothed off by drawing a curve
through them. A'CD' represents the similar arrangement of the same
number when given not as simple errors, but as averages of pairs of
errors. A"BD", again, represents the similar arrangement obtained as
averages of errors taken three together. They are drawn as carefully
to scale as the small size of the figure permits.
[Figure: Gaussian distributions for two or three observations.]
15. A glance at the above figure will explain to the reader, better
than any verbal description, the full significance of the statement
that the result of combining two or more measurements or observations
together and taking the average of them, instead of stopping short at
the single elements, is to make large errors comparatively more
scarce. The advantage is of the same general description as that of
fishing in a lake where, of the same number of fish, there are more
big and fewer little ones than in another water: of dipping in a bag
where of the same number of coins there are more sovereigns and fewer
shillings; and so on. The extreme importance, however, of obtaining a
perfectly clear conception of the subject may render it desirable to
work this out a little more fully in detail.
For one thing, then, it must be clearly understood that the result of
a set of 'averages' of errors is nothing else than another set of
'errors,' No device can make the attainment of the true result
certain,--to suppose the contrary would be to misconceive the very
foundations of Probability,--no device even can obviate the
possibility of being actually worse off as the result of our
labour. The average of two, three, or any larger number of single
results, may give a worse result, i.e. one further from the ultimate
average, than was given by the first observation we made. We must
simply fall back upon the justification that big deviations are
rendered scarcer in the long run.
Again; it may be pointed out that though, in the above investigation,
we have spoken only of the arithmetical average as commonly understood
and employed, the same general results would be obtained by resorting
to almost any symmetrical and regular mode of combining our
observations or errors. The two main features of the regularity
displayed by the Binomial Law of facility were (1) ultimate symmetry
about the central or true result, and (2) increasing relative
frequency as this centre was approached. A very little consideration
will show that it is no peculiar prerogative of the arithmetical mean
to retain the former of these and to increase the latter. In saying
this, however, a distinction must be attended to for which it will be
convenient to refer to a figure.
16. Suppose that O, in the line D'OD, was the point aimed at by any
series of measurements; or, what comes to the same thing for our
present purpose, was the ultimate average of all the measurements
made. What we mean by a symmetrical arrangement of the values in
regard to O, is that for every error OB, there shall be in the long
run a precisely corresponding opposite one OB'; so that when we erect
the ordinate BQ, indicating the frequency with which B is yielded, we
must erect an equal one B'Q'. Accordingly the two halves of the curve
on each side of P, viz. PQ and PQ' are precisely alike.
[Figure: Symmetry of a distribution.]
It then readily follows that the secondary curve, viz. that marking
the law of frequency of the averages of two or more simple errors,
will also be symmetrical. Consider any three points B, C, D: to these
correspond another three B', C', D'. It is obvious therefore that any
regular and symmetrical mode of dealing with all the groups, of which
BCD is a sample, will result in symmetrical arrangement about the
centre O. The ordinary familiar arithmetical average is but one out of
many such modes. One way of describing it is by saying that the
average of B, C, D, is assigned by choosing a point such that the sum
of the squares of its distances from B, C, D, is a minimum. But we
might have selected a point such that the cubes, or the fourth powers,
or any higher powers should be a minimum. These would all yield curves
resembling in a general way the dotted line in our figure. Of course
there would be insuperable practical objections to any such courses as
these; for the labour of calculation would be enormous, and the
results so far from being better would be worse than those afforded by
the employment of the ordinary average. But so far as concerns the
general principle of dealing with discordant and erroneous results, it
must be remembered that the familiar average is but one out of
innumerable possible resources, all of which would yield the same sort
of help.
17. Once more. We saw that a resort to the average had the effect of
'humping up' our curve more towards the centre, expressive of the fact
that the errors of averages are of a better, i.e. smaller kind. But it
must be noticed that exactly the same characteristics will follow, as
a general rule, from any other such mode of dealing with the
individual errors. No strict proof of this fact can be given here, but
a reference to one of the familiar results of taking combinations of
things will show whence this tendency arises. Extreme results, as
yielded by an average of any kind, can only be got in one way, viz. by
repetitions of extremes in the individuals from which the averages
were obtained. But intermediate results can be got at in two ways,
viz. either by intermediate individuals, or by combinations of
individuals in opposite directions. In the case of the Binomial Law of
Error this tendency to thicken towards the centre was already strongly
predominant in the individual values before we took them in hand for
our average; but owing to this characteristic of combinations we may
lay it down (broadly speaking) that any sort of average applied to any
sort of law of distribution will give a result which bears the same
general relation to the individual values that the dotted lines above
bear to the black line.[6]
18. This being so, the speculative advantages of one method of
combining, or averaging, or reducing, our observations, over another
method,--irrespective, that is, of the practical conveniences in
carrying them out,--will consist solely in the degree of rapidity with
which it tends thus to cluster the result about the centre. We shall
have to subject this merit to a somewhat further analysis, but for the
present purpose it will suffice to say that if one kind of average
gave the higher dotted line in the figure on p. 479 and another gave
the lower dotted line, we should say that the former was the better
one. The advantage is of the same general kind as that which is
furnished in algebraical calculation, by a series which converges
rapidly towards the true value as compared with one which converges
slowly. We can do the work sooner or later by the aid of either; but
we get nearer the truth by the same amount of labour, or get as near
by a less amount of labour, on one plan than on the other.
As we are here considering the case in which the individual
observations are supposed to be grouped in accordance with the
Binomial Law, it will suffice to say that in this case there is no
doubt that the arithmetical average is not only the simplest and
easiest to deal with, but is also the best in the above sense of the
term. And since this Binomial Law, or something approximating to it,
is of very wide prevalence, a strong _primâ facie_ case is made out
for the general employment of the familiar average.
19. The analysis of a few pages back carried the results of the
averaging process as far as could be conveniently done by the help of
mere arithmetic. To go further we must appeal to higher mathematics,
but the following indication of the sort of results obtained will
suffice for our present purpose. After all, the successive steps,
though demanding intricate reasoning for their proof, are nothing more
than generalizations of processes which could be established by simple
arithmetic.[7] Briefly, what we do is this:--
(1) We first extend the proof from the binomial form, with its finite
number of elements, to the limiting or exponential form. Instead of
confining ourselves to a small number of discrete errors, we then
recognize the possibility of any number of errors of any magnitude
whatever.
(2) In the next place, instead of confining ourselves to the
consideration of an average of two or three only,--already, as we have
seen, a tedious piece of arithmetic,--we calculate the result of an
average of any number, n. The actual result is extremely simple. If
the modulus of the single errors is c, that of the average of n of
these will be c ÷ sqrt{n}.
(3) Finally we draw similar conclusions in reference to the _sum or
difference_ of two averages of any numbers. Suppose, for instance,
that m errors were first taken and averaged, and then n similarly
taken and averaged. These averages will be nearly, but not quite,
equal. Their sum or difference,--these, of course, are
indistinguishable in the end, since positive and negative errors are
supposed to be equal and opposite,--will itself be an 'error', every
magnitude of which will have a certain assignable probability or
facility of occurrence. What we do is to assign the modulus of _these_
errors. The actual result again is simple. If c had been the modulus
of the single errors, that of the sum or difference of the averages of
m and n of them will be c sqrt{1/m + 1/n}.
20. So far, the problem under investigation has been of a direct
kind. We have supposed that the ultimate mean value or central
position has been given to us; either _à priori_ (as in many games of
chance), or from more immediate physical considerations (as in aiming
at a mark), or from extensive statistics (as in tables of human
stature). In all such cases therefore the main desideratum is already
taken for granted, and it may reasonably be asked what remains to be
done. The answers are various. For one thing we may want to estimate
the value of an average of many when compared with an average of a
few. Suppose that one man has collected statistics including 1000
instances, and another has collected 4000 similar instances. Common
sense can recognize that the latter are better than the former; but it
has no idea _how much_ better they are. Here, as elsewhere,
quantitative precision is the privilege of science. The answer we
receive from this quarter is that, in the long run, the modulus,--and
with this the probable error, the mean error, and the error of mean
square, which all vary in proportion,--diminishes inversely as the
square root of the number of measurements or observations. (This
follows from the second of the above formulæ.) Accordingly the
probable error of the more extensive statistics here is one half that
of the less extensive. Take another instance. Observation shows that
"the mean height of 2,315 criminals differs from the mean height of
8,585 members of the general adult population by about two inches"
(v. Edgeworth, Methods of Statistics: _Stat. Soc. Journ._ 1885). As
before, common sense would feel little doubt that such a difference
was significant, but it could give no numerical estimate of the
significance. Appealing to science, we see that this is an
illustration of the third of the above formulæ. What we really want to
know is the odds against the averages of two large batches differing
by an assigned amount: in this case by an amount equalling twenty-five
times the modulus of the variable quantity. The odds against this are
many billions to one.
21. The number of direct problems which will thus admit of solution is
very great, but we must confine ourselves here to the main inverse
problem to which the foregoing discussion is a preliminary. It is
this. Given _a few only_ of one of these groups of measurements or
observations; what can we do with these, in the way of determining
that mean about which they would ultimately be found to cluster?
Given a large number of them, they would betray the position of their
ultimate centre with constantly increasing certainty: but we are now
supposing that there are only a few of them at hand, say half a dozen,
and that we have no power at present to add to the number.
In other words,--expressing ourselves by the aid of graphical
illustration, which is perhaps the best method for the novice and for
the logical student,--in the direct problem we merely have to draw the
curve of frequency from a knowledge of its determining elements;
viz. the position of the centre, and the numerical value of the
modulus. In the inverse problem, on the other hand, we have three
elements at least, to determine. For not only must we, (1), as before,
determine whereabouts the centre may be assumed to lie; and (2), as
before, determine the value of the modulus or degree of dispersion
about this centre. This does not complete our knowledge. Since neither
of these two elements is assigned with certainty, we want what is
always required in the Theory of Chances, viz. some estimate of their
probable truth. That is, after making the best assignment we can as to
the value of these elements, we want also to assign numerically the
'probable error' committed in such assignment. Nothing more than this
can be attained in Probability, but nothing less than this should be
set before us.
22. (1) As regards the first of these questions, the answer is very
simple. Whether the number of measurements or observations be few or
many, we must make the assumption that their average _is_ the point we
want; that is, that the average of the few will coincide with the
ultimate average. This is the best, in fact the only assumption we can
make. We should adopt this plan, of course, in the extreme case of
there being only _one_ value before us, by just taking that one; and
our confidence increases slowly with the number of values before
us. The only difference therefore here between knowledge resting upon
such data, and knowledge resting upon complete data, lies not in the
result obtained but in the confidence with which we entertain it.
23. (2) As regards the second question, viz. the determination of the
modulus or degree of dispersion about the mean, much the same may be
said. That is, we adopt the same rule for the determination of the
E.M.S. (error of mean square) by which the modulus is assigned, as we
should adopt if we possessed full Information. Or rather we are
confined to _one_ of the rules given on p. 473, viz. the second, for
by supposition we have neither the _à priori_ knowledge which would be
able to supply the first, nor a sufficient number of observations to
justify the third. That is, we reckon the errors, measured from the
average, and calculate their mean square: twice this is equal to the
square of the modulus of the probable curve of facility.[8]
24. (3) The third question demands for its solution somewhat advanced
mathematics; but the results can be indicated without much
difficulty. A popular way of stating our requirement would be to say
that we want to know how likely it is that the mean of the few, which
we have thus accepted, shall coincide with the true mean. But this
would be to speak loosely, for the chances are of course indefinitely
great against such precise coincidence. What we really do is to assign
the 'probable error'; that is, to assign a limit which it is as likely
as not that the discrepancy between the inferred mean and the true
mean should exceed.[9] To take a numerical example: suppose we had
made several measurements of a wall with a tape, and that the average
of these was 150 feet. The scrupulous surveyor would give us this
result, with some such correction as this added,--'probable error 3
inches'. All that this means is that we may assume that the true value
is 150 feet, with a confidence that in half the cases (of this
description) in which we did so, we should really be within three
inches of the truth.
The expression for this probable error is a simple multiple of the
modulus: it is the modulus multiplied by 0.4769.... That it should be
some function of the modulus, or E.M.S., seems plausible enough; for
the greater the errors,--in other words the wider the observed
discrepancy amongst our measurements,--the less must be the confidence
we can feel in the accuracy of our determination of the mean. But, of
course, without mathematics we should be quite unable to attempt any
numerical assignment.
25. The general conclusion therefore is that the determination of the
curve of facility,--and therefore ultimately of every conclusion which
rests upon a knowledge of this curve,--where only a few observations
are available, is of just the same kind as where an infinity are
available. The rules for obtaining it are the same, but the confidence
with which it can be accepted is less.
The knowledge, therefore, obtainable by an average of a small number
of measurements of any kind, hardly differs except in degree from that
which would be attainable by an indefinitely extensive series of
them. We know the same sort of facts, only we are less certain about
them. But, on the other hand, the knowledge yielded by an average even
of a small number differs in kind from that which is yielded by a
single measurement. Revert to our marksman, whose bullseye is supposed
to have been afterwards removed. If he had fired only a single shot,
not only should we be less certain of the point he had aimed at, but
we should have no means whatever of guessing at the quality of his
shooting, or of inferring in consequence anything about the probable
remoteness of the next shot from that which had gone before. But
directly we have a plurality of shots before us, we not merely feel
more confident as to whereabouts the centre of aim was, but we also
gain some knowledge as to how the future shots will cluster about the
spot thus indicated. The quality of his shooting begins at once to be
betrayed by the results.
26. Thus far we have been supposing the Law of Facility to be of the
Binomial type. There are several reasons for discussing this at such
comparative length. For one thing it is the only type which,--or
something approximately resembling which,--is actually prevalent over
a wide range of phenomena. Then again, in spite of its apparent
intricacy, it is really one of the simplest to deal with; owing to the
fact that every curve of facility derived from it by taking averages
simply repeats the same type again. The curve of the average only
differs from that of the single elements in having a smaller modulus;
and its modulus is smaller in a ratio which is exceedingly easy to
give. If that of the one is c, that of the other (derived by averaging
n single elements) is c/sqrt{n}.
But for understanding the theory of averages we must consider other
cases as well. Take then one which is intrinsically as simple as it
possibly can be, viz. that in which all values within certain assigned
limits are equally probable. This is a case familiar enough in
abstract Probability, though, as just remarked, not so common in
natural phenomena. It is the state of things when we act at random
directly upon the objects of choice;[10] as when, for instance, we
choose digits at random out of a table of logarithms.
The reader who likes to do so can without much labour work out the
result of taking an average of two or three results by proceeding in
exactly the same way which we adopted on p. 476. The 'curve of
facility' with which we have to start in this case has become of
course simply a finite straight line. Treating the question as one of
simple combinations, we may divide the line into a number of equal
parts, by equidistant points; and then proceed to take these two and
two together in every possible way, as we did in the case discussed
some pages back.
If we did so, what we should find would be this. When an average of
_two_ is taken, the 'curve of facility' of the average becomes a
triangle with the initial straight line for base; so that the ultimate
mean or central point becomes the likeliest result even with this
commencement of the averaging process. If we were to take averages of
three, four, and so on, what we should find would be that the Binomial
law begins to display itself here. The familiar bell shape of the
exponential curve would be more and more closely approximated to,
until we obtained something quite indistinguishable from it.
27. The conclusion therefore is that when we are dealing with averages
involving a considerable number it is not necessary, in general, to
presuppose the binomial law of distribution in our original data. The
law of arrangement of what we may call the derived curve, viz. that
corresponding to the averages, will not be appreciably affected
thereby. Accordingly we seem to be justified in bringing to bear all
the same apparatus of calculation as in the former case. We take the
initial average as the probable position of the true centre or
ultimate average: we estimate the probability that we are within an
assignable distance of the truth in so doing by calculating the 'error
of mean square'; and we appeal to this same element to determine the
modulus, i.e. the amount of contraction or dispersion, of our derived
curve of facility.
The same general considerations will apply to most other kinds of Law
of Facility. Broadly speaking,--we shall come to the examination of
certain exceptions immediately,--whatever may have been the primitive
arrangement (i.e. that of the single results) the arrangement of the
derived results (i.e. that of the averages) will be more crowded up
towards the centre. This follows from the characteristic of
combinations already noticed, viz. that extreme values can only be got
at by a repetition of several extremes, whereas intermediate values
can be got at either by repetition of intermediates or through the
counteraction of opposite extremes. Provided the original
distribution be symmetrical about the centre, and provided the limits
of possible error be finite, or if infinite, that the falling off of
frequency as we recede from the mean be very rapid, then the results
of taking averages will be better than those of trusting to single
results.
28. We will now take notice of an exceptional case. We shall do so,
not because it is one which can often actually occur, but because the
consideration of it will force us to ask ourselves with some
minuteness what we mean in the above instances by calling the results
of the averages 'better' than those of the individual values. A
diagram will bring home to us the point of the difficulty better than
any verbal or symbolic description.
[Figure: Distribution for two samples from a non-Gaussian distribution.]
The black line represents a Law of Error easily stated in words, and
one which, as we shall subsequently see, can be conceived as occurring
in practice. It represents a state of things under which up to a
certain distance from O, on each side, viz. to A and B, the
probability of an error diminishes uniformly with the distance from O;
whilst beyond these points, up to E and F, the probability of error
remains constant. The dotted line represents the resultant Law of
Error obtained by taking the average of the former two and two
together. Now is the latter 'better' than the former? Under it,
certainly, great errors are less frequent and intermediate ones more
frequent; but then on the other hand the _small_ errors are less
frequent: is this state of things on the whole an improvement or not?
This requires us to reconsider the whole question.
29. In all the cases discussed in the previous sections the
superiority of the curve of averages over that of the single results
showed itself at every point. The big errors were scarcer and the
small errors were commoner; it was only just at one intermediate point
that the two were on terms of equality, and this point was not
supposed to possess any particular significance or
importance. Accordingly we had no occasion to analyse the various
cases included under the general relation. It was enough to say that
one was better than the other, and it was sufficient for all purposes
to take the 'modulus' as the measure of this superiority. In fact we
are quite safe in simply saying that the _average_ of those average
results is better than that of the individual ones.
When however we proceed in what Hume calls "the sifting humour," and
enquire _why_ it is sufficient thus to trust to the average; we find,
in addition to the considerations hitherto advanced, that some
postulate was required as to the _consequences_ of the errors we
incur. It involved an estimate of what is sometimes called the
'detriment' of an error. It seemed to take for granted that large and
small errors all stand upon the same general footing of being
mischievous in their consequences, but that their evil effects
increase in a greater ratio than that of their own magnitude.
30. Suppose, for comparison, a case in which the importance of an
error is directly proportional to its magnitude (of course we suppose
positive and negative errors to balance each other in the long run):
it does not appear that any advantage would be gained by taking
averages. Something of this sort may be considered to prevail in cases
of mere purchase and sale. Suppose that any one had to buy a very
large number of yards of cloth at a constant price per yard: that he
had to do this, say, five times a day for many days in succession. And
conceive that the measurement of the cloth was roughly estimated on
each separate occasion, with resultant errors which are as likely to
be in excess as in defect. Would it make the slightest difference to
him whether he paid separately for each piece; or whether the five
estimated lengths were added together, their average taken, and he
were charged with this average price for each piece? In the latter
case the errors which will be made in the estimation of each piece
will of course be less in the long run than they would be in the
former: will this be of any consequence? The answer surely is that it
will not make the slightest difference to either party in the
bargain. In the long run, since the same parties are concerned, it
will not matter whether the intermediate errors have been small or
large.
Of course nothing of this sort can be regarded as the general rule. In
almost every case in which we have to make measurements we shall find
that large errors are much more mischievous than small ones, that is,
mischievous in a greater ratio than that of their mere magnitude. Even
in purchase and sale, where _different_ purchasers are concerned, this
must be so, for the pleasure of him who is overserved will hardly
equal the pain of him who is underserved. And in many cases of
scientific measurement large errors may be simply fatal, in the sense
that if there were no reasonable prospect of avoiding them we should
not care to undertake the measurement at all.
31. If we were only concerned with practical considerations we might
stop at this point; but if we want to realize the full logical import
of average-taking as a means to this particular end, viz. of
estimating some assigned magnitude, we must look more closely into
such an exceptional case as that which was indicated in the figure on
p. 493. What we there assumed was a state of things in reference to
which extremely small errors were very frequent, but that when once we
got beyond a certain small range all other errors, within considerable
limits, were equally likely.
It is not difficult to imagine an example which will aptly illustrate
the case in point: at worst it may seem a little far-fetched.
Conceive then that some firm in England received a hurried order to
supply a portion of a machine, say a steam-engine, to customers at a
distant place; and that it was absolutely essential that the work
should be true to the tenth of an inch for it to be of any use. But
conceive also that two specifications had been sent, resting on
different measurements, in one of which the length of the requisite
piece was described as sixty and in the other sixty-one inches. On the
assumption of any ordinary law of error, whether of the binomial type
or not, there can be no doubt that the firm would make the best of a
very bad job by constructing a piece of 60 inches and a half:
i.e. they would have a better chance of being within the requisite
tenth of an inch by so doing, than by taking either of the two
specifications at random and constructing it accurately to this. But
if the law were of the kind indicated in our diagram,[11] then it
seems equally certain that they would be less likely to be within the
requisite narrow margin by so doing. As a mere question of
probability,--that is, if such estimates were acted upon again and
again,--there would be fewer failures encountered by simply choosing
one of the conflicting measurements at random and working exactly to
this, than by trusting to the average of the two.
This suggests some further reflections as to the taking of
averages. We will turn now to another exceptional case,
but one involving somewhat different considerations than
those which have been just discussed. As before, it may be
most conveniently introduced by commencing with an example.
32. Suppose then that two scouts were sent to take the calibre of a
gun in a hostile fort,--we may conceive that the fort was to be
occupied next day, and used against the enemy, and that it was
important to have a supply of shot or shell,--and that the result is
that one of them reports the calibre to be 8 inches and the other 9.
Would it be wise to assume that the mean of these two, viz. 8-1/2
inches, was a likelier value than either separately?
The answer seems to be this. If we have reason to suppose that the
possible calibres partake of the nature of a _continuous_
magnitude,--i.e. that all values, with certain limits, are to be
considered as admissible, (an assumption which we always make in our
ordinary inverse step from an observation or magnitude to the thing
observed or measured)--then we should be justified in selecting the
average as the likelier value. But if, on the other hand, we had
reason to suppose that _whole_ inches are always or generally
preferred, as is in fact the case now with heavy guns, we should do
better to take, even at hazard, one of the two estimates set before
us, and trust this alone instead of taking an average of the two.
33. The principle upon which we act here may be stated thus. Just as
in the direct process of calculating or displaying the 'errors',
whether in an algebraic formula or in a diagram, we generally assume
that their possibility is continuous, i.e. that all intermediate
values are possible; so, in the inverse process of determining the
probable position of the original from the known value of two or more
errors, we assume that that position is capable of falling at any
point whatever between certain limits. In such an example as the
above, where we know or suspect a discontinuity of that possibility of
position, the value of the average may be entirely destroyed.
In the above example we were supposed to know that the calibre of the
guns was likely to run in English inches or in some other recognized
units. But if the battery were in China or Japan, and we knew nothing
of the standards of length in use there, we could no longer appeal to
this principle. It is doubtless highly probable that those calibres
are not of the nature of continuously varying magnitudes; but in an
entire ignorance of the standards actually adopted, we are to all
intents and purposes in the same position as if they were of that
continuous nature. When this is so the objections to trusting to the
average would no longer hold good, and if we had only one opportunity,
or a very few opportunities, we should do best to adhere to the
customary practice.
34. When however we are able to collect and compare a large number of
measurements of various objects, this consideration of the probable
discontinuity of the objects we thus measure,--that is, their tendency
to assume some one or other of a finite number of distinct magnitudes,
instead of showing an equal readiness to adapt themselves to all
intermediate values,--again assumes importance. In fact, given a
sufficient number of measurable objects, we can actually deduce with
much probability the standard according to which the things in
question were made.
This is the problem which Mr Flinders Petrie has attacked with so much
acuteness and industry in his work on _Inductive Metrology_, a work
which, merely on the ground of its speculative interest, may well be
commended to the student of Probability. The main principles on which
the reasoning is based are these two:--(1) that all artificers are
prone to construct their works according to round numbers, or simple
fractions, of their units of measurement; and (2) that, aiming to
secure this, they will stray from it in tolerable accordance with the
law of error. The result of these two assumptions is that if we
collect a very large number of measurements of the different parts and
proportions of some ancient building,--say an Egyptian temple,--whilst
no assignable length is likely to be permanently unrepresented, yet we
find a marked tendency for the measurements to cluster about certain
determinate points in our own, or any other standard scale of
measurement. These points mark the length of the standard, or of some
multiple or submultiple of the standard, employed by the old
builders. It need hardly be said that there are a multitude of
practical considerations to be taken into account before this method
can be expected to give trustworthy results, but the leading
principles upon which it rests are comparatively simple.
35. The case just considered is really nothing else than the
recurrence, under a different application, of one which occupied our
attention at a very early stage. We noticed (Chap. II.) the
possibility of a curve of facility which instead of having a single
vertex like that corresponding to the common law of error, should
display two humps or vertices. It can readily be shown that this
problem of the measurements of ancient buildings, is nothing more than
the reopening of the same question, in a slightly more complex form,
in reference to the question of the functions of an average.
Take a simple example. Suppose an instance in which great errors, of a
certain approximate magnitude, are distinctly more likely to be
committed than small ones, so that the curve of facility, instead of
rising into one peak towards the centre, as in that of the familiar
law of error, shows a depression or valley there. Imagine, in fact,
two binomial curves, with a short interval between their centres. Now
if we were to calculate the result of taking averages here we should
find that this at once tends to fill up the valley; and if we went on
long enough, that is, if we kept on taking averages of sufficiently
large numbers, a peak would begin to arise in the centre. In fact the
familiar single binomial curve would begin to make its appearance.
36. The question then at once suggests itself, ought we to do this?
Shall we give the average free play to perform its allotted function
of thus crowding things up towards the centre? To answer this question
we must introduce a distinction. If that peculiar double-peaked curve
had been, as it conceivably might, a true error-curve,--that is, if it
had represented the divergences actually made in aiming at the real
centre,--the result would be just what we should want. It would
furnish an instance of the advantages to be gained by taking averages
even in circumstances which were originally unfavourable. It is not
difficult to suggest an appropriate illustration. Suppose a man firing
at a mark from some sheltered spot, but such that the range crossed a
broad exposed valley up or down which a strong wind was generally
blowing. If the shot-marks were observed we should find them
clustering about two centres to the right and left of the
bullseye. And if the results were plotted out in a curve they would
yield such a double-peaked curve as we have described. But if the
winds were equally strong and prevalent in opposite directions, we
should find that the averaging process redressed the consequent
disturbance.
If however the curve represented, as it is decidedly more likely to
do, some outcome of natural phenomena in which there was, so to say, a
real double aim on the part of nature, it would be otherwise. Take,
for instance, the results of measuring a large number of people who
belonged to two very heterogeneous races. The curve of facility would
here be of the kind indicated on p. 45, and if the numbers of the two
commingled races were equal it would display a pair of twin
peaks. Again the question arises, 'ought' we to involve the whole
range within the scope of a single average? The answer is that the
obligation depends upon the purpose we have in view. If we want to
compare that heterogeneous race, as a whole, with some other, or with
itself at some other time, we shall do well to average without
analysis. All statistics of population, as we have already seen
(v. p. 47), are forced to neglect a multitude of discriminating
characteristics of the kind in question. But if our object were to
interpret the causes of this abnormal error-curve we should do well to
break up the statistics into corresponding parts, and subject these to
analysis separately.
Similarly with the measurements of the ancient buildings. In this
case if all our various 'errors' were thrown together into one group
of statistics we should find that the resultant curve of facility
displayed, not two peaks only, but a succession of them; and these of
various magnitudes, corresponding to the frequency of occurrence of
each particular measurement. We _might_ take an average of the whole,
but hardly any rational purpose could be subserved in so doing;
whereas each separate point of maximum frequency of occurrence has
something significant to teach us.
37. One other peculiar case may be noticed in conclusion. Suppose a
distinctly asymmetrical, or lop-sided curve of facility, such as
this:--
[Figure: An asymmetric (lop-sided) distribution.]
Laws of error, of which this is a graphical representation, are, I
apprehend, far from uncommon. The curve in question, is, in fact, but
a slight exaggeration of that of barometrical heights as referred to
in the last chapter; when it was explained that in such cases the
mean, the median, and the maximum ordinate would show a mutual
divergence. The doubt here is not, as in the preceding instances,
whether or not a single average should be taken, but rather what kind
of average should be selected. As before, the answer must depend upon
the special purpose we have in view. For all ordinary purposes of
comparison between one time or place and another, any average will
answer, and we should therefore naturally take the arithmetical, as
the most familiar, or the median, as the simplest.
38. Cases might however arise under which other kinds of average could
justify themselves, with a momentary notice of which we may now
conclude. Suppose, for instance, that the question involved here were
one of desirability of climate. The ordinary mean, depending as it
does so largely upon the number and magnitude of extreme values, might
very reasonably be considered a less appropriate test than that of
judging simply by the relatively most frequent value: in other words,
by the maximum ordinate. And various other points of view can be
suggested in respect of which this particular value would be the most
suitable and significant.
In the foregoing case, viz. that of the weather curve, there was no
objective or 'true' value aimed at. But a curve closely resembling
this would be representative of that particular class of estimates
indicated by Mr Galton, and for which, as he has pointed out, the
_geometrical_ mean becomes the only appropriate one. In this case the
curve of facility ends abruptly at O: it resembles a much
foreshortened modification of the common exponential form. Its
characteristics have been discussed in the paper by Dr Macalister
already referred to, but any attempt to examine its properties here
would lead us into far too intricate details.
39. The general conclusion from all this seems quite in accordance
with the nature and functions of an average as pointed out in the last
chapter. Every average, it was urged, is but a single representative
intermediate value substituted for a plurality of actual values. It
must accordingly let slip the bulk of the information involved in
these latter. Occasionally, as in most ordinary measurements, the one
thing which it represents is obviously the thing we are in want of;
and then the only question can be, which mean will most accord with
the 'true' value we are seeking. But when, as may happen in most of
the common applications of statistics, there is really no 'true value'
of an objective kind behind the phenomena, the problem may branch out
in various directions. We may have a variety of purposes to work out,
and these may demand some discrimination as regards the average most
appropriate for them. Whenever therefore we have any doubt whether the
familiar arithmetical average is suitable for the purpose in hand we
must first decide precisely what that purpose is.
1. Mr Mansfield Merriman published in 1877 (_Trans. of the Connecticut
Acad._) a list of 408 writings on the subject of Least Squares.
2. In other words, we are to take the "centre of gravity" of the
shot-marks, regarding them as all of equal weight. This is, in
reality, the 'average' of all the marks, as the elementary
geometrical construction for obtaining the centre of gravity of a
system of points will show; but it is not familiarly so regarded. Of
course, when we are dealing with such cases as occur in Mensuration,
where we have to combine or reconcile three or more inconsistent
equations, some such rule as that of Least Squares becomes
imperative. No taking of an average will get us out of the
difficulty.
3. The only reason for supposing this exceptional shape is to secure
simplicity. The ordinary target, allowing errors in two dimensions,
would yield slightly more complicated results.
4. When first referred to, the _general_ form of this equation was
given (v. p. 29). The special form here assigned, in which
(h/sqrt{π}) is substituted for A, is commonly employed in
Probability, because the integral of y dx, between +infinity and
-infinity, becomes equal to unity. That is, the sum of all the
mutually exclusive possibilities is represented, as usual, by
unity. In this form of expression h is a quantity of the order
x^{-1}; for hx is to be a numerical quantity, standing as it does as
an index. The modulus, being the reciprocal of this, is of the same
order of quantities as the errors themselves. In fact, if we
multiply it by 0.4769... we have the so-called 'probable error.'
5. See, for the explanation of this, and of the graphical method of
illustrating it, the note on p. 29.
6. Broadly speaking, we may say that the above remarks hold good of
any law of frequency of error in which there are actual limits,
however wide, to the possible magnitude of an error. If there are no
limits to the possible errors, this characteristic of an average to
heap its results up towards the centre will depend upon
circumstances. When, as in the exponential curve, the approximation
to the base, as asymptote, is exceedingly rapid,--that is, when the
extreme errors are relatively very few,--it still holds good. But if
we were to take as our law of facility such an equation as
y = π/(1 + x^{2}), (as hinted by De Morgan and noted by Mr Edgeworth:
_Camb. Phil. Trans._ vol. X. p. 184, and vol. XIV. p. 160) it does
not hold good. The result of averaging is to _diminish_ the tendency
to cluster towards the centre.
7. The reader will find the proofs of these and other similar formulæ
in Galloway _on Probability_, and in Airy _on Errors_.
8. The formula commonly used for the E.M.S. in this case is
(sum e^{2})/(n - 1) and not (sum e^{2})/n. The difference is trifling,
unless n be small; the justification has been offered for it that
since the sum of the squares measured from the true centre is a
minimum (that centre being the ultimate arithmetical mean) the sum
of the squares measured from the somewhat incorrectly assigned
centre will be somewhat larger.
9. It appears to me that in strict logical propriety we should like to
know the probable error committed in _both_ the assignments of the
preceding two sections. But the profound mathematicians who have
discussed this question, and who alone are competent to treat it,
have mostly written with the practical wants of Astronomy in view;
and for this purpose it is sufficient to take account of the one
great desideratum, viz. the true values sought. Accordingly the only
rules commonly given refer to the probable error of the mean.
10. i.e. as distinguished from acting upon them indirectly. This
latter proceeding, as explained in the chapter on Randomness, may
result in giving a non-uniform distribution.
11. There is no difficulty in conceiving circumstances under which a
law very closely resembling this would prevail. Suppose, e.g., that
one of the two measurements had been made by a careful and skilled
mechanic, and the other by a man who to save himself trouble had put
in the estimate at random (within certain limits),--the firm having
a knowledge of this fact but being of course unable to assign the
two to their authors,--we should get very much such a Law of Error
as is supposed above.
INDEX.
Accidents 342
Airy, G. B. 447, 484
Anticipations, tacit 287
Arbuthnott 258
Aristotle 205, 307
Average
arithmetical 437
geometrical 439
median 442
consequences of 482
necessary results of 457
uses of 439, 489
Babbage 343
Bags and balls 180, 411
Belief
correctness of 125, 131, 178
gradations of 139
growth of 199
language of 143
measurement of 119, 125, 146
quantity of 133
test of 140, 149, 294
undue 129
vagueness of 127
Bentham 319, 323
Bernoulli 91, 117, 389
Bertillon 435
Births, male and female 90, 258, 263
Boat race, Oxford and Cambridge 339
Boole 183
Buckle 237
Buffon 153, 205, 352, 389
Burgersdyck 311
Butler 209, 281, 333, 366
Carlisle Tables 169
Casual, meaning of 245
Causation
need of 237
proof of 244
Centre of gravity 467
Certainty, in Law 324
reasonable 327
hypothetical 210
Chance
and Causation 244
Creation 258
Design 256
Genius 353
neglect of small 363
selections 338
Chauvenet 352
Classification, numerical scheme of 48
Coincidences 245
Combinations and Permutations 87
Communism 375, 392
Conceptualism 275
Conflict of chances 418
Consumptives, insurance of 227
Cournot 245, 255, 338
Crackanthorpe 312, 320
Craig, J. 192
Crofton, M. W. 61, 101, 104
Dante 285
Deflection
causes of 57
from aim 38
De Morgan 83, 106, 119, 122, 135, 177, 179, 197, 236, 247, 296, 308,
350, 379, 382, 483
De Ros trial 255
Digits, random 111, 114
Discontinuity 116
Distribution, random 106
Diagrams 29, 45, 118, 443, 476, 481, 493, 501
Dialectic 302, 320
Donkin 123, 188, 283
Duration of life 15, 441
Düsing 259
Ebbinghaus 199
Edgeworth, F. Y. 34, 119, 256, 339, 393, 435, 483
Ellis, L. 9
Epidemics 62
Error, law of 29
asymmetrical 34, 441, 443
binomial 37, 457, 469, 480
geometrical 34, 502
heterogeneous 45
production of 36
Error
mean 446
probable 446, 472, 488
of mean square 447, 488
Escapes, narrow 341
Expectation, moral 388
Experience and probability 74
Exponential curve 29
Extraordinary
sense of 159, 423
stories 407, 421
Fallacies in Logic and Probability 367
Fatalism 243
Fechner 34, 389, 435, 441
Fluctuation 448
unlimited 73
Forbes, J. D. 188, 262
Formal Logic 123
Formal and Material treatment 86
Free will 240
Galloway 248, 448, 484
Galton, F. 33, 50, 70, 318, 442, 451, 473, 502
Gambling
and Insurance 370
disadvantage of 384
final results of 385, 391
Godfray, H. 99
Grote, G. 307
Guy 6
Hamilton, W. 266, 297
Happiness, human 382
Heads and Tails 77
Heredity 50, 357
Herschel 30, 466
Houdin 361
Hume 236, 419, 433
Hypotheses 268
Immediate inferences 121
Independent events 175, 246
Induction
and Probability 194, 201, 208, 233, 358
difficulty of 213
pure 200
Inequality of wealth 382
Inference, rules of 167
Inoculation 374
Insurance
justification of 149
difficulties of 221
life 151
peculiar case 224
theory of 372
varieties of 374
Inverse probability 179, 196, 249
Irregularity, absolute and relative 6
Jacobs, J. 199
Jackson, J. G. 253
Jevons 37, 83, 136, 198, 201, 209, 247
Kant 310, 317
Keckermann 298, 316
Kinds, natural 55
Krug 324
Lambert 309
Language of Chance 159
Laplace 89, 120, 197, 237, 424
Law
absence of 101
empirical 160
of causation 206
Least squares 41, 467
Leibnitz 309, 320
Letters
lost 162, 368
misdirected 67, 237, 241
Lexis, W. 263, 441
Limit
conception of 18, 109, 164
of possible fluctuation 32
Lines, random 113
Lister's method 187
Lotteries 128
Lunn, J. R. 248
Likely, equally 77, 183
McAlister, D. 34, 187, 502
Mansel, H. L. 299, 301, 320
Martingale 343
Material and Formal Logic 265
Maximum ordinate 441, 455
Measurement of
Belief 119
Memory 192
Mental qualities, measurement of 49
Merriman, M. 352, 448, 460, 465
Mill, J. S. 131, 207, 266, 282, 402
Milton, chance production of 353
Miracles 428
Michell, J. 260
Modality 295
divisions of 307
false 297
formal 298
in Law 319
Modulus 464, 472, 484
Monro, C. J. 325, 416
Names, reference of 270
Nations, comparison of 51
Natural Kinds 55, 63, 71
Necessary and impossible matter 310
Objects and agencies 53
Occam 314
Paley 433
Penny, tosses of 144
Petrie, F. 498
Petersburg Problem 19, 154
Poisson 405
Prantl 311
Presumption, legal 329
Prévost 348
Probability
definition of 165
relative 290
integral 463
Probable
facts 269
value 441
error 446, 472
Problem, Three point 104
Proctor, R. A. 262, 378
Prophecies, suicidal 226
Providence 89, 431
Propositions, proportional 2
Psychical research 256
Pyramid, the great 251
π, digits in 111, 247
Quartiles 446
Quetelet 23, 30, 43, 91, 259, 330, 348, 454
Randomness
etymology of 96
in firing 98
proof of 107
Rare events 349
Realism 92
Reason, sufficient 82
Residuals 460
Roberts, C. 25
Rod
broken at random 98
thrown at random 103
Rules
Inductive and Deductive 176
of Succession 191
conflict of 222
plurality of 217
Series
definite proportions in 11
fixed and variable 16
ideal 95
peculiar 12
Shanks 248
Skeat, W. W. 96
Smiglecius 306, 316
Smyth, P. 251
Socialism 392
Spiritualism 365
Stars, random arrangement of 108, 260
Statistics
by Intercomparison 473
unconscious appeal to 400
Statistical Journal 6
Stature
human 25, 471
French and English 44
Stephen, J. F. 282, 323, 326
Stewart, D. 209, 237
Subjective and objective terms 160
Succession
long 360
Rule of 190, 362
Suffield, G. 248
Suicides 67, 237
Surnames, extinction of 387
Surprise, emotion of 157
Syllogisms, pure and modal 316
Taylor 329
Testimony
single 411
combined 426
two kinds of 409
worthless 416
Thomson, W. 153, 314, 419
Time
influence of 191
in Probability 279
Todhunter 415
Tontines 380
Triangle, random 103
Tucker, A. 127
Types
existence of 42, 60, 453
fixed and fluctuating 64, 93
Ueberweg 311
Uncertainty in life 370
Uniformity 240
Units of calculation 464
Voluntary agency 65, 68, 85
Watford 374
Wallis, J. 312
Watson, H. W. 387
Whately 297, 307
Whist 401
Whitworth, W. A. 87, 183, 384
Wilson, J. M. 104
Witnesses, independent 405
Wolf 309
Woolhouse 101
CAMBRIDGE: PRINTED BY C.J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS.
End of Project Gutenberg's The Logic of Chance, 3rd edition, by John Venn