with which he starts, viz. that the subjective value of any small
  increment (dx) is inversely proportional to the sum then possessed
  (x), and which leads at once to the logarithmic law above mentioned,
  is identical with one which is now familiar enough to every
  psychologist. It is what is commonly called Fechner's Law, which he
  has established by aid of an enormous amount of careful experiment
  in the case of a number of our simple sensations. But I do not
  believe that he has made any claim that such a law holds good in the
  far more intricate dependence of happiness upon wealth.

10. The formula expressive of this moral happiness is c log x/a; where
  x stands for the physical fortune possessed at the time, and a for
  that small value of it at which happiness is supposed to disappear:
  c being an arbitrary constant. Let two persons, whose fortune is x,
  risk y on an even bet. Then the balance, as regards happiness, must
  be drawn between
  c log x/a    and    1/2c log (x + y)/a + 1/2c log (x - y)/a,
  or    log x^{2} and log(x + y)(x - y),
  or    x^{2} and x^{2} - y^{2},
  the former of which is necessarily the greater.

11. This may be seen more clearly as follows. Suppose two pair of
  gamblers, each pair consisting of men possessing £50 and £30
  respectively. Now if we suppose the richer man to win in one case
  and the poorer in the other these two results will be a fair
  representation of the average; for there are only two alternatives
  and these will be equally frequent in the long run. It is obvious
  that we have had two fortunes of £50 and two of £30 converted into
  one of £20, two of £40, and one of £60. And this is clearly an
  increase of inequality.

CHAPTER XVI.

_THE APPLICATION OF PROBABILITY TO TESTIMONY._

1. On the principles which have been adopted in this
work, it becomes questionable whether several classes of
problems which may seem to have acquired a prescriptive
right to admission, will not have to be excluded from the
science of Probability. The most important, perhaps, of
these refer to what is commonly called the credibility of
testimony, estimated either at first hand and directly, or as
influencing a juryman, and so reaching us through his
sagacity and trustworthiness. Almost every treatise upon
the science contains a discussion of the principles according
to which credit is to be attached to combinations of the
reports of witnesses of various degrees of trustworthiness, or
the verdicts of juries consisting of larger or smaller numbers.
A great modern mathematician, Poisson, has written an
elaborate treatise expressly upon this subject; whilst a considerable
portion of the works of Laplace, De Morgan, and
others, is devoted to an examination of similar enquiries. It
would be presumptuous to differ from such authorities as
these, except upon the strongest grounds; but I confess that
the extraordinary ingenuity and mathematical ability which
have been devoted to these problems, considered as questions
in Probability, fails to convince me that they ought to have
been so considered. The following are the principal grounds
for this opinion.

2. It will be remembered that in the course of the
chapter on Induction we entered into a detailed investigation
of the process demanded of us when, instead of the
appropriate propositions from which the inference was to be
made being set before us, the _individual_ presented himself,
and the task was imposed upon us of selecting the requisite
groups or series to which to refer him. In other words, instead
of calculating the chance of an event from determinate
conditions of frequency of its occurrence (these being either
obtained by direct experience, or deductively inferred) we
have to _select_ the conditions of frequency out of a plurality
of more or less suitable ones. When the problem is presented
to us at such a stage as this, we may of course assume
that the preliminary process of obtaining the statistics
which are extended into the proportional propositions has
been already performed; we may suppose therefore that we
are already in possession of a quantity of such propositions,
our principal remaining doubt being as to which of them
we should then employ. This selection was shown to be
to a certain extent arbitrary; for, owing to the fact of the
individual possessing a large number of different properties,
he became in consequence a member of different series or
groups, which might present different averages. We must
now examine, somewhat more fully than we did before,
the practical conditions under which any difficulty arising
from this source ceases to be of importance.

3. One condition of this kind is very simple and obvious.
It is that the different statistics with which we are
presented should not in reality offer materially different
results, If, for instance, we were enquiring into the probability
of a man aged forty dying within the year, we might
if we pleased take into account the fact of his having red
hair, or his having been born in a certain county or town.
Each of these circumstances would serve to specialize the
individual, and therefore to restrict the limits of the statistics
which were applicable to his case. But the consideration of
such qualities as these would either leave the average precisely
as it was, or produce such an unimportant alteration
in it as no one would think of taking into account. Though
we could hardly say with certainty of any conceivable
characteristic that it has absolutely no bearing on the
result, we may still feel very confident that the bearing of
such characteristics as these is utterly insignificant. Of
course in the extreme case of the things most perfectly
suited to the Calculus of Probability, viz. games of pure
chance, these subsidiary characteristics are quite irrelevant.
Any further particulars about the characteristics of the cards
in a really fair pack, beyond those which are familiar to all
the players, would convey no information whatever about the
result.

Or again; although the different sets of statistics may
not as above give almost identical results, yet they may do
what practically comes to very much the same thing, that
is, arrange themselves into a small number of groups, all of
the statistics in any one group practically coinciding in their
results. If for example a consumptive man desired to insure
his life, there would be a marked difference in the statistics
according as we took his peculiar state of health into account
or not. We should here have two sets of statistics, so clearly
marked off from one another that they might almost rank
with the distinctions of natural kinds, and which would in
consequence offer decidedly different results. If we were
to specialize still further, by taking into account insignificant
qualities like those mentioned in the last paragraph, we
might indeed get more limited sets of statistics applicable
to persons still more closely resembling the individual in
question, but these would not differ sufficiently in their
results to make it worth our while to do so. In other words,
the different propositions which are applicable to the case in
point arrange themselves into a limited number of groups,
which, and which only, need be taken into account; whence
the range of choice amongst them is very much diminished
in practice.

4. The reasons for the conditions above described are
not difficult to detect. Where these conditions exist the
process of selecting a series or class to which to refer any
individual is very simple, and the selection is, for the particular
purposes of inference, final. In any case of insurance,
for example, the question we have to decide is of the very
simple kind; Is A. B. a man of a certain age? If so one in
fifty in his circumstances will die in the course of the year.
If any further questions have to be decided they would be of
the following description. Is A. B. a healthy man? Does he
follow a dangerous trade? But here too the classes in
question are but few, and the limits by which they are
bounded are tolerably precise; so that the reference of an
individual to one or other of them is easy. And when we
have once chosen our class we remain untroubled by any
further considerations; for since no other statistics are supposed
to offer a materially different average, we have no
occasion to take account of any other properties than those
already noticed.

The case of games of chance, already referred to, offers
of course an instance of these conditions in an almost ideal
state of perfection; the same circumstances which fit them
so eminently for the purposes of fair gambling, fitting them
equally to become examples in Probability. When a die is
to be thrown, all persons alike stand on precisely the same
footing of knowledge and of ignorance about the result; the
only data to which any one could appeal being that each
face turns up on an average once in six times.

5. Let us now examine how far the above conditions
are fulfilled in the case of problems which discuss what is
called the credibility of testimony. The following would be
a fair specimen of one of the elementary enquiries out of
which these problems are composed;--Here is a statement
made by a witness who lies once in ten times, what am I to
conclude about its truth? Objections might fairly be raised
against the possibility of thus assigning a man his place
upon a graduated scale of mendacity. This however we will
pass over, and will assume that the witness goes about the
world bearing stamped somehow on his face the appropriate
class to which he belongs, and consequently, the degree of
credit to which he has a claim on such general grounds.
But there are other and stronger reasons against the admissibility
of this class of problems.

6. That which has been described in the previous
sections as the 'individual' which had to be assigned to an
appropriate class or series of statistics is, of course, in this
case, _a statement_. In the particular instance in question this
individual statement is already assigned to a class, that
namely of statements made by a witness of a given degree of
veracity; but it is clearly optional with us whether or not we
choose to confine our attention to this class in forming our
judgment; at least it would be optional whenever we were
practically called on to form an opinion. But in the case of
this statement, as in that of the mortality of the man whose
insurance we were discussing, there are a multitude of other
properties observable, besides the one which is supposed to
mark the given class. Just as in the latter there were
(besides his age), the place of his birth, the nature of his
occupation, and so on; so in the former there are (besides its
being a statement by a certain kind of witness), the fact of its
being uttered at a certain time and place and under certain
circumstances. At the time the statement is made all these
qualities or attributes of the statement are present to us, and
we clearly have a right to take into account as many of them
as we please. Now the question at present before us seems to
be simply this;--Are the considerations, which we might thus
introduce, as immaterial to the result in the case of the truth
of a statement of a witness, as the corresponding considerations
are in the case of the insurance of a life? There can
surely be no hesitation in the reply to such a question.
Under ordinary circumstances we soon know all that we can
know about the conditions which determine us in judging of
the prospect of a man's death, and we therefore rest content
with general statistics of mortality; but no one who heard a
witness speak would think of simply appealing to his figure
of veracity, even supposing that this had been authoritatively
communicated to us. The circumstances under which the
statement is made instead of being insignificant, are of overwhelming
importance. The appearance of the witness, the
tone of his voice, the fact of his having objects to gain,
together with a countless multitude of other circumstances
which would gradually come to light as we reflect upon the
matter, would make any sensible man discard the assigned
average from his consideration. He would, in fact, no more
think of judging in this way than he would of appealing to
the Carlisle or Northampton tables of mortality to determine
the probable length of life of a soldier who was already in
the midst of a battle.

7. It cannot be replied that under these circumstances
we still refer the witness to a class, and judge of his veracity
by an average of a more limited kind; that we infer, for example,
that of men who look and act like him under such
circumstances, a much larger proportion, say nine-tenths,
are found to lie. There is no appeal to a class in this way at
all, there is no immediate reference to statistics of any kind
whatever; at least none which we are conscious of using at
the time, or to which we should think of resorting for justification
afterwards. The decision seems to depend upon the
quickness of the observer's senses and of his apprehension
generally.

Statistics about the veracity of witnesses seem in fact to
be permanently as inappropriate as all other statistics occasionally
may be. We may know accurately the percentage
of recoveries after amputation of the leg; but what surgeon
would think of forming his judgment solely by such tables
when he had a case before him? We need not deny, of
course, that the opinion he might form about the patient's
prospects of recovery might ultimately rest upon the proportions
of deaths and recoveries he might have previously witnessed.
But if this were the case, these data are lying, as
one may say, obscurely in the background. He does not
appeal to them directly and immediately in forming his
judgment. There has been a far more important intermediate
process of apprehension and estimation of what is
essential to the case and what is not. Sharp senses, memory,
judgment, and practical sagacity have had to be called into
play, and there is not therefore the same direct conscious
and sole appeal to statistics that there was before. The
surgeon may have in his mind two or three instances in
which the operation performed was equally severe, but in
which the patient's constitution was different; the latter
element therefore has to be properly allowed for. There may
be other instances in which the constitution was similar, but
the operation more severe; and so on. Hence, although the
ultimate appeal may be to the statistics, it is not so directly;
their value has to be estimated through the somewhat hazy
medium of our judgment and memory, which places them
under a very different aspect.

8. Any one who knows anything of the game of whist
may supply an apposite example of the distinction here
insisted on, by recalling to mind the alteration in the nature
of our inferences as the game progresses. At the commencement
of the game our sole appeal is rightfully made to the
theory of Probability. All the rules upon which each player
acts, and therefore upon which he infers that the others will
act, rest upon the observed frequency (or rather upon the
frequency which calculation assures us will be observed) with
which such and such combinations of cards are found to
occur. Why are we told, if we have more than four trumps,
to lead them out at once? Because we are convinced, on
pure grounds of probability, capable of being stated in the
strictest statistical form, that in a majority of instances we
shall draw our opponent's trumps, and therefore be left with
the command. Similarly with every other rule which is
recognized in the early part of the play.

But as the play progresses all this is changed, and
towards its conclusion there is but little reliance upon any
rules which either we or others could base upon statistical
frequency of occurrence, observed or inferred. A multitude
of other considerations have come in; we begin to be influenced
partly by our knowledge of the character and
practice of our partner and opponents; partly by a rapid
combination of a multitude of judgments, founded upon
our observation of the actual course of play, the grounds
of which we could hardly realize or describe at the time
and which may have been forgotten since. That is, the
particular combination of cards, now before us, does not
readily fall into any well-marked class to which alone it can
reasonably be referred by every one who has the facts before
him.

9. A criticism somewhat resembling the above has
been given by Mill (_Logic,_ Bk. III. Chap. XVIII. §3) upon
the applicability of the theory of Probability to the credibility
of witnesses. But he has added other reasons which
do not appear to me to be equally valid; he says "common
sense would dictate that it is impossible to strike a general
average of the veracity, and other qualifications for true
testimony, of mankind or any class of them; and if it were
possible, such an average would be no guide, the credibility of
almost every witness being either below or above the average,"
The latter objection would however apply with equal force
to estimating the length of a man's life from tables of mortality;
for the credibility of different witnesses can scarcely
have a wider range of variation than the length of different
lives. If statistics of credibility could be obtained, and
could be conveniently appealed to when they were obtained,
they might furnish us in the long run with as accurate
inferences as any other statistics of the same general description.
These statistics would however in practice naturally
and rightly be neglected, because there can hardly fail
to be circumstances in each individual statement which would
more appropriately refer it to some new class depending on
different statistics, and affording a far better chance of our
being right in that particular case. In most instances of
the kind in question, indeed, such a change is thus produced
in the mode of formation of our opinion, that, as already
pointed out, the mental operation ceases to be in any proper
sense founded on appeal to statistics.[1]

10. The Chance problems which are concerned with testimony
are not altogether confined to such instances as those
hitherto referred to. Though we must, as it appears to me,
reject all attempts to estimate the credibility of any particular
witness, or to refer him to any assigned class in
respect of his trustworthiness, and consequently abandon as
unsuitable any of the numerous problems which start from
such data as 'a witness who is wrong once in ten times,'
yet it does not follow that testimony may not to a slight
extent be treated by our science in a somewhat different
manner. We may be quite unable to estimate, except in
the roughest possible way, the veracity of any particular
witness, and yet it may be possible to form some kind of
opinion upon the veracity of certain classes of witnesses;
to say, for instance, that Europeans are superior in this way
to Orientals. So we might attempt to explain why, and to
what extent, an opinion in which the judgments of ten persons,
say jurors, concur, is superior to one in which five only
concur. Something may also be done towards laying down
the principles in accordance with which we are to decide
whether, and why, extraordinary stories deserve less credence
than ordinary ones, even if we cannot arrive at any precise
and definite decision upon the point. This last question is
further discussed in the course of the next chapter.

11. The change of view in accordance with which it
follows that questions of the kind just mentioned need not
be entirely rejected from scientific consideration, presents itself
in other directions also. It has, for instance, been already
pointed out that the individual characteristics of any sick
man's disease would be quite sufficiently important in most
cases to prevent any surgeon from judging about his recovery
by a genuine and direct appeal to statistics, however such
considerations might indirectly operate upon his judgment.
But if an opinion had to be formed about a considerable
number of cases, say in a large hospital, statistics might
again come prominently into play, and be rightly recognized
as the principal source of appeal. We should feel able to
compare one hospital, or one method of treatment, with
another. The ground of the difference is obvious. It arises
from the fact that the characteristics of the individuals,
which made us so ready to desert the average when we had
to judge of them separately, do not produce the same disturbance
when were have to judge about a group of cases.
The averages then become the most secure and available
ground on which to form an opinion, and therefore Probability
again becomes applicable.

But although some resort to Probability may be admitted
in such cases as these, it nevertheless does not
appear to me that they can ever be regarded as particularly
appropriate examples to illustrate the methods and resources
of the theory. Indeed it is scarcely possible to resist the
conviction that the refinements of mathematical calculation
have here been pushed to lengths utterly unjustifiable, when
we bear in mind the impossibility of obtaining any corresponding
degree of accuracy and precision in the data from
which we have to start. To cite but one instance. It would
be hard to find a case in which love of consistency has prevailed
over common sense to such an extent as in the admission
of the conclusion that it is unimportant what are
the numbers for and against a particular statement, provided
the actual majority is the same. That is, the unanimous
judgment of a jury of eight is to count for the same as a
majority of ten to two in a jury of twelve. And yet this
conclusion is admitted by Poisson. The assumptions under
which it follows will be indicated in the course of the next
chapter.

Again, perfect independence amongst the witnesses or
jurors is an almost necessary postulate. But where can this
be secured? To say nothing of direct collusion, human
beings are in almost all instances greatly under the influence
of sympathy in forming their opinions. This influence, under
the various names of political bias, class prejudice, local
feeling, and so on, always exists to a sufficient degree to
induce a cautious person to make many of those individual
corrections which we saw to be necessary when we were
estimating the trustworthiness, in any given case, of a single
witness; that is, they are sufficient to destroy much, if not
all, of the confidence with which we resort to statistics and
averages in forming our judgment. Since then this Essay is
mainly devoted to explaining and establishing the general
principles of the science of Probability, we may very fairly
be excused from any further treatment of this subject, beyond
the brief discussions which are given in the next chapter.

1. It may be remarked also that there is another reason which tends to
  dissuade us from appealing to principles of Probability in the
  majority of the cases where testimony has to be estimated. It often,
  perhaps usually happens, that we are not absolutely forced to come
  to a decision; at least so far as the acquitting of an accused
  person may be considered as avoiding a decision. It may be of much
  greater importance to us to attain not merely truth on the average,
  but truth in each individual instance, so that we had rather not
  form an opinion at all than form one of which we can only say in its
  justification that it will tend to lead us right in the long run.

CHAPTER XVII.

_ON THE CREDIBILITY OF EXTRAORDINARY STORIES._

1. It is now time to recur for fuller investigation to an
enquiry which has been already briefly touched upon more
than once; that is, the validity of testimony to establish,
as it is frequently expressed, an otherwise improbable story.
It will be remembered that in a previous chapter (the
twelfth) we devoted some examination to an assertion by
Butler, which seemed to be to some extent countenanced
by Mill, that a great improbability before the proof might
become but a very small improbability after the proof. In
opposition to this it was pointed out that the different
estimates which we undoubtedly formed of the credibility
of the examples adduced, had nothing to do with the
fact of the event being past or future, but arose from a
very different cause; that the conception of the event
which we entertain at the moment (which is all that is then
and there actually present to us, and as to the correctness
of which as a representation of facts we have to make up our
minds) comes before us in two very different ways. In one
instance it was a mere guess of our own which we knew
from statistics would be right in a certain proportion of
cases; in the other instance it was the assertion of a witness,
and therefore the appeal was not now primarily to statistics of
the event, but to the trustworthiness of the witness. The conception,
or 'event' if we will so term it, had in fact passed
out of the category of guesses (on statistical grounds), into
that of assertions (most likely resting on some specific evidence),
and would therefore be naturally regarded in a very
different light.

2. But it may seem as if this principle would lead us
to somewhat startling conclusions. For, by transferring the
appeal from the frequency with which the event occurs to
the trustworthiness of the witness who makes the assertion,
is it not implied that the probability or improbability of an
assertion depends solely upon the veracity of the witness?
If so, ought not any story whatever to be believed when
it is asserted by a truthful person?

In order to settle this question we must look a little
more closely into the circumstances under which such
testimony is commonly presented to us. As it is of course
necessary, for clearness of exposition, to take a numerical
example, let us suppose that a given statement is made by
a witness who, on the whole and in the long run, is right
in what he says nine times out of ten.[1] Here then is an
average given to us, an average veracity that is, which
includes all the particular statements which the witness has
made or will make.

3. Now it has been abundantly shown in a former
chapter (Ch. IX. §§14-32) that the mere fact of a particular
average having been assigned, is no reason for our
being forced invariably to adhere to it, even in those cases
in which our most natural and appropriate ground of judgment
is found in an appeal to statistics and averages.
The general average may constantly have to be corrected
in order to meet more accurately the circumstances of particular
cases. In statistics of mortality, for instance; instead
of resorting to the wider tables furnished by people in
general of a given age, we often prefer the narrower tables
furnished by men of a particular profession, abode, or mode
of life. The reader may however be conveniently reminded
here that in so doing we must not suppose that we are able,
by any such device, in any special or peculiar way to secure
truth. The general average, if persistently adhered to
throughout a sufficiently wide and varied experience, would
in the long run tend to give us the truth; all the advantage
which the more special averages can secure for us is to give
us the same tendency to the truth with fewer and slighter
aberrations.

4. Returning then to our witness, we know that if
we have a very great many statements from him upon all
possible subjects, we may feel convinced that in nine out of
ten of these he will tell us the truth, and that in the tenth
case he will go wrong. This is nothing more than a matter
of definition or consistency. But cannot we do better than
thus rely upon his general average? Cannot we, in almost
any given case, specialize it by attending to various characteristic
circumstances in the nature of the statement
which he makes; just as we specialize his prospects of
mortality by attending to circumstances in his constitution
or mode of life?

Undoubtedly we may do this; and in any of the practical
contingencies of life, supposing that we were at all guided
by considerations of this nature, we should act very foolishly
if we did not adopt some such plan. Two methods of thus
correcting the average may be suggested: one of them being
that which practical sagacity would be most likely to employ,
the other that which is almost universally adopted by writers
on Probability. The former attempts to make the correction
by the following considerations: instead of relying upon the
witness' general average, we assign to it a sort of conjectural
correction to meet the case before us, founded on our experience
or observation; that is, we appeal to experience to
establish that stories of such and such a kind are more or
less likely to be true, as the case may be, than stories in
general. The other proceeds upon a different and somewhat
more methodical plan. It is here endeavoured to
show, by an analysis of the nature and number of the
sources of error in the cases in question, that such and such
kinds of stories must be more or less likely to be correctly
reported, and this in certain numerical proportions.

5. Before proceeding to a discussion of these methods
a distinction must be pointed out to which writers upon the
subject have not always attended, or at any rate to which
they have not generally sufficiently directed their readers'
attention.[2] There are, broadly speaking, two different ways
in which we may suppose testimony to be given. It may, in
the first place, take the form of a reply to an alternative
question, a question, that is, framed to be answered by _yes_
or _no_. Here, of course, the possible answers are mutually
contradictory, so that if one of them is not correct the other
must be so:--Has A happened, yes or no? The common
mode of illustrating this kind of testimony numerically is by
supposing a lottery with a prize and blanks, or a bag of balls
of two colours only, the witness knowing that there are only
two, or at any rate being confined to naming one or other of
them. If they are black and white, and he errs when black
is drawn, he must say 'white,' The reason for the prominence
assigned to examples of this class is, probably, that
they correspond to the very important case of verdicts of
juries; juries being supposed to have nothing else to do than
to say 'guilty' or 'not guilty.'

On the other hand, the testimony may take the form of a
more original statement or piece of information. Instead of
saying, Did A happen? we may ask, What happened? Here
if the witness speaks truth he must be supposed, as before,
to have but one way of doing so; for the occurrence of some
specific event was of course contemplated. But if he errs he
has many ways of going wrong, possibly an infinite number.
Ordinarily however his possible false statements are assumed
to be limited in number, as must generally be more or less
the result in practice. This case is represented numerically
by supposing the balls in the bag not to be of two colours
only, but to be all distinct from each other; say by their
being all numbered successively. It may of course be objected
that a large number of the statements that are made
in the world are not in any way answers to questions, either
of the alternative or of the open kind. For instance, a man
simply asserts that he has drawn the seven of spades from a
pack of cards; and we do not know perhaps whether he had
been asked 'Has that card been drawn?' or 'What card has
been drawn?' or indeed whether he had been asked anything
at all. Still more might this be so in the case of any ordinary
historical statement.

This objection is quite to the point, and must be recognized
as constituting an additional difficulty. All that we
can do is to endeavour, as best we may, to ascertain, from
the circumstances of the case, what number of alternatives
the witness may be supposed to have had before him. When
he simply testifies to some matter well known to be in dispute,
and does not go much into detail, we may fairly consider
that there were practically only the two alternatives
before him of saying 'yes' or 'no.' When, on the other hand,
he tells a story of a more original kind, or (what comes to
much the same thing) goes into details, we must regard him
as having a wide comparative range of alternatives before
him.

These two classes of examples, viz. that of the black and
white balls, in which only one form of error is possible, and
the numbered balls, in which there may be many forms of
error, are the only two which we need notice. In practice it
would seem that they may gradually merge into each other,
according to the varying ways in which we choose to frame
our question. Besides asking, Did you see A strike B? and,
What did you see? we may introduce any number of intermediate
leading questions, as, What did A do? What did
he do to B? and so on. In this way we may gradually narrow
the possible openings to wrong statement, and so approach
to the direct alternative question. But it is clear that all
these cases may be represented numerically by a supposed
diminution in the number of the balls which are thus distinguished
from each other.

6. Of the two plans mentioned in §4 we will begin
with the latter, as it is the only methodical and scientific one
which has been proposed. Suppose that there is a bag with
1000 balls, only one of which is white, the rest being all
black. A ball is drawn at random, and our witness whose
veracity is 9/10 reports that the white ball was drawn. Take
a great many of his statements upon this particular subject,
say 10,000; that is, suppose that 10,000 balls having been
successively drawn out of this bag, or bags of exactly the
same kind, he makes his report in each case. His 10,000
statements being taken as a fair sample of his general average,
we shall find, by supposition, that 9 out of every 10 of
them are true and the remaining one false. What will be
the nature of these false statements? Under the circumstances
in question, he having only one way of going wrong,
the answer is easy. In the 10,000 drawings the white ball
would come out 10 times, and therefore be rightly asserted
9 times, whilst on the one of these occasions on which he
goes wrong he has nothing to say but 'black.' So with the
9990 occasions on which black is drawn; he is right and
says black on 8991 of them, and is wrong and therefore says
white on 999 of them. On the whole, therefore, we conclude
that out of every 1008 times on which he says that white is
drawn he is wrong 999 times and right only 9 times. That
is, his special veracity, as we may term it, for cases of this
description, has been reduced from 9/10 to 9/1008. As it would
commonly be expressed, the latter fraction represents the
chance that this particular statement of his is true.[3]

7. We will now take the case in which the witness
has many ways of going wrong, instead of merely one. Suppose
that the balls were all numbered, from 1 to 1,000, and
the witness knows this fact. A ball is drawn, and he tells
me that it was numbered 25, what are the odds that he is
right? Proceeding as before, in 10,000 drawings this ball
would be obtained 10 times, and correctly named 9 times.
But on the 9990 occasions on which it was not drawn there
would be a difference, for the witness has now many openings
for error before him. It is, however, generally considered
reasonable to assume that his errors will all take the form of
announcing wrong numbers; and that, there being no apparent
reason why he should choose one number rather than another,
he will be likely to announce all the wrong ones equally
often. Hence his 999 errors, instead of all leading him now
back again to one spot, will be uniformly spread over as
many distinct ways of going wrong. On one only of these
occasions, therefore, will he mention 25 as having been
drawn. It follows therefore that out of every 10 times that
he names 25 he is right 9 times; so that in this case his
average or general truthfulness applies equally well to the
special case in point.

8. With regard to the truth of these conclusions, it
must of course be admitted that if we grant the validity of
the assumptions about the limits within which the blundering
or mendacity of the witness are confined, and the complete
impartiality with which his answers are disposed within
those limits, the reasoning is perfectly sound. But are not
these assumptions extremely arbitrary, that is, are not our
lotteries and bags of balls rendered perfectly precise in many
respects in which, in ordinary life, the conditions supposed to
correspond to them are so vague and uncertain that no such
method of reasoning becomes practically available? Suppose
that a person whom I have long known, and of whose measure
of veracity and judgment I may be supposed therefore
to have acquired some knowledge, informs me that there is
something to my advantage if I choose to go to certain
trouble or expense in order to secure it. As regards the
general veracity of the witness, then, there is no difficulty;
we suppose that this is determined for us. But as regards
his story, difficulty and vagueness emerge at every point.
What is the number of balls in the bag here? What in fact
are the nature and contents of the bag out of which we suppose
the drawing to have been made? It does not seem
that the materials for any rational judgment exist here.
But if we are to get at any such amended figure of veracity
as those attained in the above example, these questions must
necessarily be answered with some degree of accuracy; for
the main point of the method consists in determining how
often the event must be considered _not_ to happen, and thence
inferring how often the witness will be led wrongly to assert
that it has happened.

It is not of course denied that considerations of the kind
in question have some influence upon our decision, but only
that this influence could under any ordinary circumstances
be submitted to numerical determination. We are doubtless
liable to have information given to us that we have
come in for some kind of fortune, for instance, when no
such good luck has really befallen us; and this not once
only but repeatedly. But who can give the faintest intimation
of the nature and number of the occasions on which,
a blank being thus really drawn, a prize will nevertheless
be falsely announced? It appears to me therefore that
numerical results of any practical value can seldom, if ever,
be looked for from this method of procedure.

9. Our conclusion in the case of the lottery, or, what
comes to the same thing, in the case of the bag with black
and white balls, has been questioned or objected to[4] on the
ground that it is contrary to all experience to suppose that
the testimony of a moderately good witness could be so
enormously depreciated under such circumstances. I should
prefer to base the objection on the ground that experience
scarcely ever presents such circumstances as those supposed;
but if we postulate their existence the given conclusion seems
correct enough. Assume that a man is merely required to
say _yes_ or _no_; assume also a group or succession of cases in
which _no_ should rightly be said very much oftener than
_yes_. Then, assuming almost any general truthfulness of the
witness, we may easily suppose the rightful occasions for
denial to be so much the more frequent that a majority of
his affirmative answers will actually occur as false 'noes'
rather than as correct 'ayes.' This of course lowers the
average value of his 'ayes,' and renders them comparatively
untrustworthy.

Consider the following example. I have a gardener whom
I trust as to all ordinary matters of fact. If he were to
tell me some morning that my dog had run away I should
fully believe him. He tells me however that the dog has
gone mad. Surely I should accept the statement with much
hesitation, and on the grounds indicated above. It is not
that he is more likely to be wrong when the dog _is_ mad;
but that experience shows that there are other complaints
(e.g. fits) which are far more common than madness, and
that most of the assertions of madness are erroneous assertions
referring to these. This seems a somewhat parallel
case to that in which we find that most of the assertions
that a white ball had been drawn are really false assertions
referring to the drawing of a black ball. Practically I do
not think that any one would feel a difficulty in thus exorbitantly
discounting some particular assertion of a witness
whom in most other respects he fully trusted.

10. There is one particular case which has been regarded
as a difficulty in the way of this treatment of the
problem, but which seems to me to be a decided confirmation
of it; always, be it understood, within the very narrow
and artificial limits to which we must suppose ourselves to
be confined. This is the case of a witness whose veracity is
just one-half; that is, one who, when a mere _yes_ or _no_ is
demanded of him, is as often wrong as right. In the case of
any other assigned degree of veracity it is extremely difficult
to get anything approaching to a confirmation from practical
judgment and experience. We are not accustomed to
estimate the merits of witnesses in this way, and hardly appreciate
what is meant by his numerical degree of truthfulness.
But as regards the man whose veracity is one-half, we
are (as Mr C. J. Monro has very ingeniously suggested) only
too well acquainted with such witnesses, though under a
somewhat different name; for this is really nothing else than
the case of a person confidently answering a question about
a subject-matter of which he knows nothing, and can therefore
only give a mere guess.

Now in the case of the lottery with one prize, when the
witness whose veracity is one-half tells us that we have
gained the prize, we find on calculation that his testimony
goes for absolutely nothing; the chances that we have got
the prize are just the same as they would be if he had never
opened his lips, viz. 1/1000. But clearly this is what ought
to be the result, for the witness who knows nothing about
the matter leaves it exactly as he found it. He is indeed,
in strictness, scarcely a witness at all; for the natural function
of a witness is to examine the matter, and so to add
confirmation, more or less, according to his judgment and
probity, but at any rate to offer an improvement upon the
mere guesser. If, however, we will give heed to his mere
guess we are doing just the same thing as if we were to guess
ourselves, in which case of course the odds that we are right
are simply measured by the frequency of occurrence of the
events.

We cannot quite so readily apply the same rule to the
other case, namely to that of the numbered balls, for there
the witness who is right every other time may really be a
very fair, or even excellent, witness. If he has many ways
of going wrong, and yet is right in half his statements, it is
clear that he must have taken some degree of care, and cannot
have merely guessed. In a case of _yes_ or _no_, any one can
be right every other time, but it is different where truth
is single and error is manifold. To represent the case of a
simply worthless witness when there were 1000 balls and
the drawing of one assigned ball was in question, we should
have to put his figure of veracity at 1/1000. If this were
done we should of course get a similar result.

11. It deserves notice therefore that the figure of
veracity, or fraction representing the general truthfulness
of a witness, is in a way relative, not absolute; that is, it
depends upon, and varies with, the general character of the
answer which he is supposed to give. Two witnesses of
equal intrinsic veracity and worth, one of whom confined
himself to saying _yes_ and _no_, whilst the other ventured to
make more original assertions, would be represented by
different fractions; the former having set himself a much
easier task than the latter. The real caution and truthfulness
of the witness are only one factor, therefore, in his
actual figure of veracity; the other factor consists of the
nature of his assertions, as just pointed out. The ordinary
plan therefore, in such problems, of assigning an average
truthfulness to the witness, and accepting this alike in the
case of each of the two kinds of answers, though convenient,
seems scarcely sound. This consideration would however
be of much more importance were not the discussions upon
the subject mainly concerned with only one description of
answer, namely that of the 'yes or no' kind.

12. So much for the methodical way of treating such
a problem. The way in which it would be taken in hand by
those who had made no study of Probability is very different.
It would, I apprehend, strike them as follows. They would
say to themselves, Here is a story related by a witness who
tells the truth, say, nine times out of ten. But it is a story
of a kind which experience shows to be very generally made
untruly, say 99 times out of 100. Having then these opposite
inducements to belief, they would attempt in some way
to strike a balance between them. Nothing in the nature of
a strict rule could be given to enable them to decide how
they might escape out of the difficulty. Probably, in so far
as they did not judge at haphazard, they would be guided
by still further resort to experience, or unconscious recollections
of its previous teachings, in order to settle which
of the two opposing inductions was better entitled to carry
the day in the particular case before them. The reader
will readily see that any general solution of the problem,
when thus presented, is impossible. It is simply the now
familiar case (Chap. IX. §§14-32) of an individual which
belongs equally to two distinct, or even, in respect of their
characteristics, opposing classes. We cannot decide off-hand
to which of the two its characteristics most naturally and
rightly refer it. A fresh induction is needed in order to
settle this point.

13. Rules have indeed been suggested by various
writers in order to extricate us from the difficulty. The controversy
about miracles has probably been the most fertile
occasion for suggestions of this kind on one side or the
other. It is to this controversy, presumably, that the phrase
is due, so often employed in discussions upon similar subjects,
'a contest of opposite improbabilities.' What is meant
by such an expression is clearly this: that in forming a
judgment upon the truth of certain assertions we may find
that they are comprised in two very distinct classes, so that,
according as we regarded them as belonging to one or the
other of these distinct classes, our opinion as to their truth
would be very different. Such an assertion belongs to one
class, of course, by its being a statement of a particular
witness, or kind of witness; it belongs to the other by its
being a particular kind of story, one of what is called an
improbable nature. Its belonging to the former class is so
far favourable to its truth, its belonging to the latter is so far
hostile to its truth. It seems to be assumed, in speaking of
a contest of opposite improbabilities, that when these different
sources of conviction co-exist together, they would each in
some way retain their probative force so as to produce a
contest, ending generally in a victory to one or other of
them. Hume, for instance, speaks of our _deducting_ one
probability from the other, and apportioning our belief to
the remainder.[5] Thomson, in his _Laws of Thought_, speaks
of one probability as entirely superseding the other.

14. It does not appear to me that the slightest philosophical
value can be attached to any such rules as these.
They doubtless may, and indeed will, hold in individual
cases, but they cannot lay claim to any generality. Even
the notion of a contest, as any necessary ingredient in the
case, must be laid aside. For let us refer again to the way
in which the perplexity arises, and we shall readily see, as
has just been remarked, that it is nothing more than a particular
exemplification of a difficulty which has already been
recognized as incapable of solution by any general _à priori_
method of treatment. All that we are supposed to have before
us is a statement. On this occasion it is made by a witness
who lies, say, once in ten times in the long run; that is, who
mostly tells the truth. But on the other hand, it is a statement
which experience, derived from a variety of witnesses on
various occasions, assures us is mostly false; stated numerically
it is found, let us suppose, to be false 99 times in a hundred.

Now, as was shown in the chapter on Induction, we are
thus brought to a complete dead lock. Our science offers no
principles by which we can form an opinion, or attempt to
decide the matter one way or the other; for, as we found,
there are an indefinite number of conclusions which are all
equally possible. For instance, all the witness' extraordinary
assertions may be true, or they may all be false, or they may
be divided into the true and the false in any proportion
whatever. Having gone so far in our appeal to statistics as
to recognize that the witness is generally right, but that his
story is generally false, we cannot stop there. We ought to
make still further appeal to experience, and ascertain how it
stands with regard to his stories when they are of that
particular nature: or rather, for this would be to make a
needlessly narrow reference, how it stands with regard to
stories of that kind when advanced by witnesses of his
general character, position, sympathies, and so on.[6]

15. That extraordinary stories are in many cases, probably
in a great majority of cases, less trustworthy than
others must be fully admitted. That is, if we were to make
two distinct classes of such stories respectively, we should
find that the same witness, or similar witnesses, were proportionally
more often wrong when asserting the former than
when asserting the latter. But it does not by any means
appear to me that this must always be the case. We may
well conceive, for instance, that with some people the mere
fact of the story being of a very unusual character may make
them more careful in what they state, so as actually to add
to their veracity. If this were so we might be ready to
accept their extraordinary stories with even more readiness
than their ordinary ones.

Such a supposition as that just made does not seem to me
by any means forced. Put such a case as this: let us suppose
that two persons, one of them a man of merely ordinary
probity and intelligence, the other a scientific naturalist,
make a statement about some common event. We believe
them both. Let them now each report some extraordinary
_lusus naturæ_ or monstrosity which they profess to have seen.
Most persons, we may presume, would receive the statement
of the naturalist in this latter case almost as readily as in
the former: whereas when the same story came from the
unscientific observer it would be received with considerable
hesitation. Whence arises the difference? From the conviction
that the naturalist will be far more careful, and
therefore to the full as accurate, in matters of this kind as
in those of the most ordinary description, whereas with the
other man we feel by no means the same confidence. Even
if any one is not prepared to go this length, he will probably
admit that the difference of credit which he would attach to
the two kinds of story, respectively, when they came from
the naturalist, would be much less than what it would be
when they came from the other man.

16. Whilst we are on this part of the subject, it must
be pointed out that there is considerable ambiguity and
consequent confusion about the use of the term 'an extraordinary
story.' Within the province of pure Probability it
ought to mean simply a story which asserts an _unusual_ event.
At least this is the view which has been adopted and maintained,
it is hoped consistently, throughout this work. So
long as we adhere to this sense we know precisely what we
mean by the term. It has a purely objective reference; it
simply connotes a very low degree of relative statistical
frequency, actual or prospective. Out of a great number of
events we suppose a selection of some particular kind to be
contemplated, which occurs relatively very seldom, and this
is termed an unusual or extraordinary event. It follows, as
was abundantly shown in a former chapter, that owing to
the rarity of the event we are very little disposed to expect
its occurrence in any given case. Our guess about it, in case
we thus anticipated it, would very seldom be justified, and
we are therefore apt to be much surprised when it does
occur. This, I take it, is the only legitimate sense of 'extraordinary'
so far as Probability is concerned.

But there is another and very different use of the word,
which belongs to Induction, or rather to the science of
evidence in general, more than to that limited portion of it
termed Probability. In this sense the 'extraordinary,' and
still more the 'improbable,' event is not merely one of
extreme statistical rarity, which we could not expect to
guess aright, but which on moderate evidence we may pretty
readily accept; it is rather one which possesses, so to say, an
actual evidence-resisting power. It may be something which
affects the credibility of the witness at the fountain-head,
which makes, that is, his statements upon such a subject
essentially inferior to those on other subjects. This is the
case, for instance, with anything which excites his prejudices
or passions or superstitions. In these cases it would seem
unreasonable to attempt to estimate the credibility of the
witness by calculating (as in §6) how often his errors would
mislead us through his having been wrongly brought to an
affirmation instead of adhering correctly to a negation. We
should rather be disposed to put our correction on the witness'
average veracity at once.

17. In true Probability, as has just been remarked,
every event has its own definitely recognizable degree of
frequency of occurrence. It may be excessively rare, rare to
any extreme we like to postulate, but still every one who
understands and admits the data upon which its occurrence
depends will be able to appreciate within what range of
experience it may be expected to present itself. We do not
expect it in any individual case, nor within any brief range,
but we do confidently expect it within an adequately extensive
range. How therefore can miraculous stories be similarly
taken account of, when the disputants, on one side at
least, are not prepared to admit their actual occurrence anywhere
or at any time? How can any arrangement of bags
and balls, or other mechanical or numerical illustrations of
unlikely events, be admitted as fairly illustrative of miraculous
occurrences, or indeed of many of those which come
under the designation of 'very extraordinary' or 'highly
improbable'? Those who contest the occurrence of a particular
miracle, as reported by this or that narrator, do not
admit that miracles are to be confidently expected sooner or
later. It is not a question as to whether what must happen
sometimes has happened some particular time, and therefore
no illustration of the kind can be regarded as apposite.

How unsuitable these merely rare events, however excessive
their rarity may be, are as examples of miraculous
events, will be evident from a single consideration. No one,
I presume, who admitted the occasional occurrence of an exceedingly
unusual combination, would be in much doubt if
he considered that he had actually seen it himself.[7] On the
other hand, few men of any really scientific turn would
readily accept a miracle even if it appeared to happen under
their very eyes. They might be staggered at the time, but
they would probably soon come to discredit it afterwards,
or so explain it as to evacuate it of all that is meant by
miraculous.

18. It appears to me therefore, on the whole, that
very little can be made of these problems of testimony in
the way in which it is generally intended that they should
be treated; that is, in obtaining specific rules for the estimation
of the testimony under any given circumstances.
Assuming that the veracity of the witness can be measured,
we encounter the real difficulty in the utter impossibility of
determining the limits within which the failures of the event
in question are to be considered to lie, and the degree of
explicitness with which the witness is supposed to answer
the enquiry addressed to him; both of these being characteristics
of which it is necessary to have a numerical estimate
before we can consider ourselves in possession of the
requisite data.

Since therefore the practical resource of most persons,
viz. that of putting a direct and immediate correction, of
course of a somewhat conjectural nature, upon the general
trustworthiness of the witness, by a consideration of the
nature of the circumstances under which his statement is
made, is essentially unscientific and irreducible to rule; it
really seems to me that there is something to be said in
favour of the simple plan of trusting in all cases alike to
the witness' general veracity.[8] That is, whether his story
is ordinary or extraordinary, we may resolve to put it on
the same footing of credibility, provided of course that the
event is fully recognized as one which does or may occasionally
happen. It is true that we shall thus go constantly
astray, and may do so to a great extent, so that if there
were any rational and precise method of specializing his
trustworthiness, according to the nature of his story, we
should be on much firmer ground. But at least we may
thus know what to expect on the average. Provided we
have a sufficient number and variety of statements from
him, and always take them at the same constant rate or
degree of trustworthiness, we may succeed in balancing and
correcting our conduct in the long run so as to avoid any
ruinous error.

19. A few words may now be added about the combination
of testimony. No new principles are introduced here,
though the consequent complication is naturally greater. Let
us suppose two witnesses, the veracity of each being 9/10.
Now suppose 100 statements made by the pair; according to
the plan of proceeding adopted before, we should have them
both right 81 times and both wrong once, in the remaining
18 cases one being right and the other wrong. But since
they are both supposed to give the same account, what we
have to compare together are the number of occasions on
which they agree and are right, and the total number on which
they agree whether right or wrong. The ratio of the former
to the latter is the fraction which expresses the trustworthiness
of their combination of testimony in the case in question.

In attempting to decide this point the only difficulty is
in determining how often they will be found to agree when
they are both wrong, for clearly they must agree when they
are both right. This enquiry turns of course upon the number
of ways in which they can succeed in going wrong.
Suppose first the case of a simple _yes_ or _no_ (as in §6), and
take the same example, of a bag with 1000 balls, in which
one only is white. Proceeding as before, we should find that
out of 100,000 drawings (the number required in order to
obtain a complete cycle of all possible occurrences, as well as
of all possible reports about them) the two witnesses agree
in a correct report of the appearance of white in 81, and
agree in a wrong report of it in 999. The Probability therefore
of the story when so attested is 81/1080; the fact therefore
of two such witnesses of equal veracity having concurred
makes the report nearly 9 times as likely as when it rested
upon the authority of only one of them.[9]

20. When however the witnesses have many ways of
going wrong, the fact of their agreeing makes the report far
more likely to be true. For instance, in the case of the 1000
numbered balls, it is very unlikely that when they both mistake
the number they should (without collusion) happen to
make the same misstatement. Whereas, in the last case,
every combined misstatement necessarily led them both to
the assertion that the event in question had happened, we
should now find that only once in 999 × 999 times would
they both be led to assert that _some given number_ (say, as
before, 25) had been drawn. The odds in favour of the
event in fact now become 80919/80920, which are enormously
greater than when there was only one witness.

It appears therefore that when two, and of course still
more when many, witnesses agree in a statement in a matter
about which they might make many and various errors, the
combination of their favourable testimony adds enormously
to the likelihood of the event; provided always that there
is no chance of collusion. And in the extreme case of the
opportunities for error being, as they well may be, practically
infinite in number, such combination would produce almost
perfect certainty. But then this condition, viz. absence of
collusion, very seldom can be secured. Practically our main
source of error and suspicion is in the possible existence of
some kind of collusion. Since we can seldom entirely get
rid of this danger, and when it exists it can never be submitted
to numerical calculation, it appears to me that combination
of testimony, in regard to detailed accounts, is yet
more unfitted for consideration in Probability than even that
of single testimony.

21. The impossibility of any adequate or even appropriate
consideration of the credibility of miraculous stories
by the rules of Probability has been already noticed in §17.
But, since the grounds of this impossibility are often very
insufficiently appreciated, a few pages may conveniently be
added here with a view to enforcing this point. If it be
regarded as a digression, the importance of the subject and
the persistency with which various writers have at one time
or another attempted to treat it by the rules of our science
must be the excuse for entering upon it.

A necessary preliminary will be to decide upon some definition
of a miracle. It will, we may suppose, be admitted by
most persons that in calling a miracle 'a suspension of a law
of causation,' we are giving what, though it may not amount
to an adequate definition, is at least true as a description.
It is true, though it may not be the whole truth. Whatever
else the miracle may be, this is its physical aspect: this is the
point at which it comes into contact with the subject-matter
of science. If it were not considered that any suspension of
causation were involved, the event would be regarded merely
as an ordinary one to which some special significance was
attached, that is, as a type or symbol rather than a miracle.
It is this aspect moreover of the miracle which is now exposed
to the main brunt of the attack, and in support of
which therefore the defence has generally been carried on.

Now it is obvious that this, like most other definitions or
descriptions, makes some assumption as to matters of fact,
and involves something of a theory. The assumption clearly
is, that laws of causation prevail universally, or almost universally,
throughout nature, so that infractions of them are
marked and exceptional. This assumption is made, but it
does not appear that anything more than this is necessarily
required; that is, there is nothing which need necessarily
make us side with either of the two principal schools which
are divided as to the nature of these laws of causation. The
definition will serve equally well whether we understand by
_law_ nothing more than uniformity of antecedent and consequent,
or whether we assert that there is some deeper and
more mysterious tie between the events than mere sequence.
The use of the term 'causation' in this minimum of signification
is common to both schools, though the one might
consider it inadequate; we may speak, therefore, of 'suspensions
of causation' without committing ourselves to either.

22. It should be observed that the aspect of the question
suggested by this definition is one from which we can
hardly escape. Attempts indeed have been sometimes made
to avoid the necessity of any assumption as to the universal
prevalence of law and order in nature, by defining a miracle
from a different point of view. A miracle may be called, for
instance, 'an immediate exertion of creative power,' 'a sign
of a revelation,' or, still more vaguely, an 'extraordinary
event.' But nothing would be gained by adopting any such
definitions as these. However they might satisfy the theologian,
the student of physical science would not rest content
with them for a moment. He would at once assert his own
belief, and that of other scientific men, in the existence of
universal law, and enquire what was the connection of the
definition with this doctrine. An answer would imperatively
be demanded to the question, Does the miracle, as you have
described it, imply an infraction of one of these laws, or does
it not? And an answer must be given, unless indeed we
reject his assumption by denying our belief in the existence
of this universal law, in which case of course we put ourselves
out of the pale of argument with him. The necessity
of having to recognize this fact is growing upon men day by
day, with the increased study of physical science. And since
this aspect of the question has to be met some time or other,
it is as well to place it in the front. The difficulty, in its
scientific form, is of course a modern one, for the doctrine out
of which it arises is modern. But it is only one instance, out
of many that might be mentioned, in which the growth of
some philosophical conception has gradually affected the
nature of the dispute, and at last shifted the position of the
battle-ground, in some discussion with which it might not at
first have appeared to have any connection whatever.

23. So far our path is plain. Up to this point disciples
of very different schools may advance together; for in laying
down the above doctrine we have carefully abstained from
implying or admitting that it contains the whole truth. But
from this point two paths branch out before us, paths as
different from each other in their character, origin, and
direction, as can well be conceived. As this enquiry is only
a digression, we may confine ourselves to stating briefly what
seem to be the characteristics of each, without attempting to
give the arguments which might be used in their support.

(I.) On the one hand, we may assume that this principle
of causation is the ultimate one. By so terming it, we do
not mean that it is one from which we consciously start in
our investigations, as we do from the axioms of geometry,
but rather that it is the final result towards which we find
ourselves drawn by a study of nature. Finding that,
throughout the scope of our enquiries, event follows event in
never-failing uniformity, and finding moreover (some might
add) that this experience is supported or even demanded by
a tendency or law of our nature (it does not matter here how
we describe it), we may come to regard this as the one
fundamental principle on which all our enquiries should rest.

(II.) Or, on the other hand, we may admit a class of
principles of a very different kind. Allowing that there is
this uniformity so far as our experience extends, we may yet
admit what can hardly be otherwise described than by
calling it a Superintending Providence, that is, a Scheme or
Order, in reference to which Design may be predicated
without using merely metaphorical language. To adopt an
aptly chosen distinction, it is not to be understood as _over-ruling_
events, but rather as _underlying_ them.

24. Now it is quite clear that according as we come to
the discussion of any particular miracle or extraordinary
story under one or other of these prepossessions, the question
of its credibility will assume a very different aspect. It
is sometimes overlooked that although a difference about
_facts_ is one of the conditions of a _bonâ fide_ argument, a difference
which reaches to ultimate principles is fatal to all
argument. The possibility of present conflict is banished in
such a case as absolutely as that of future concord. A large
amount of popular literature on the subject of miracles
seems to labour under this defect. Arguments are stated
and examined for and against the credibility of miraculous
stories without the disputants appearing to have any
adequate conception of the chasm which separates one side
from the other.

25. The following illustration may serve in some
degree to show the sort of inconsistency of which we are
speaking. A sailor reports that in some remote coral island
of the Pacific, on which he had landed by himself, he had
found a number of stones on the beach disposed in the exact
form of a cross. Now if we conceive a debate to arise about
the truth of his story, in which it is attempted to decide the
matter simply by considerations about the validity of testimony,
without introducing the question of the existence of
inhabitants, and the nature of their customs, we shall have
some notion of the unsatisfactory nature of many of the
current arguments about miracles. All illustrations of this
subject are imperfect, but a case like this, in which a supposed
trace of human agency is detected interfering with the
orderly sequence of other and non-intelligent natural causes,
is as much to the point as any illustration can be. The
thing omitted here from the discussion is clearly the one important
thing. If we suppose that there is no inhabitant, we
shall probably disbelieve the story, or consider it to be
grossly exaggerated. If we suppose that there are inhabitants,
the question is at once resolved into a different and
somewhat more intricate one. The credibility of the witness
is not the only element, but we should necessarily have to
take into consideration the character of the supposed inhabitants,
and the object of such an action on their part.

26. Considerations of this character are doubtless
often introduced into the discussion, but it appears to me
that they are introduced to a very inadequate extent. It is
often urged, after Paley, 'Once believe in a God, and miracles
are not incredible.' Such an admission surely demands
some modification and extension. It should rather be stated
thus, Believe in a God whose working may be traced throughout
the whole moral and physical world. It amounts, in
fact, to this;--Admit that there may be a _design_ which we
can trace somehow or other in the course of things; admit
that we are not wholly confined to tracing the connection of
events, or following out their effects, but that we can form
some idea, feeble and imperfect though it be, of a _scheme_.[10]
Paley's advice sounds too much like saying, Admit that there
are fairies, and we can account for our cups being cracked.
The admission is not to be made in so off-hand a manner.
To any one labouring under the difficulty we are speaking
of, this belief in a God almost out of any constant relation to
nature, whom we then imagine to occasionally manifest himself
in a perhaps irregular manner, is altogether impossible.
The only form under which belief in the Deity can gain entrance
into his mind is as the controlling Spirit of an infinite
and orderly system. In fact, it appears to me, paradoxical
as the suggestion may appear, that it might even be more
easy for a person thoroughly imbued with the spirit of Inductive
science, though an atheist, to believe in a miracle
which formed a part of a vast system, than for such a person,
as a theist, to accept an isolated miracle.

27. It is therefore with great prudence that Hume,
and others after him, have practically insisted on commencing
with a discussion of the credibility of the single miracle,
treating the question as though the Christian Revelation
could be adequately regarded as a succession of such events.
As well might one consider the living body to be represented
by the aggregate of the limbs which compose it. What is to
be complained of in so many popular discussions on the subject
is the entire absence of any recognition of the different
ground on which the attackers and defenders of miracles are
so often really standing. Proofs and illustrations are produced
in endless number, which involving, as they almost all
do in the mind of the disputants on one side at least, that
very principle of causation, the absence of which in the case
in question they are intended to establish, they fail in the
single essential point. To attempt to induce any one to disbelieve
in the existence of physical causation, in a given
instance, by means of illustrations which to him seem only
additional examples of the principle in question, is like trying
to make a dam, in order to stop the flow of a river, by
shovelling in snow. Such illustrations are plentiful in times
of controversy, but being in reality only modified forms of
that which they are applied to counteract, they change their
shape at their first contact with the disbeliever's mind, and
only help to swell the flood which they were intended to
check.

1. Reasons were given in the last chapter against the propriety of
  applying the rules of Probability with any strictness to such
  examples as these. But although all approach to numerical accuracy
  is unattainable, we do undoubtedly recognize in ordinary life a
  distinction between the credibility of one witness and another; such
  a rough practical distinction will be quite sufficient for the
  purposes of this chapter. For convenience, and to illustrate the
  theory, the examples are best stated in a numerical form, but it is
  not intended thereby to imply that any such accuracy is really
  attainable in practice.

2. I must plead guilty to this charge myself, in the first edition of
  this work. The result was to make the treatment of this part of the
  subject obscure and imperfect, and in some respects erroneous.

3. The generalized algebraical form of this result is as follows. Let
  p be the _à priori_ probability of an event, and x be the
  credibility of the witness. Then, if he asserts that the event
  happened, the probability that it really did happen is
  px/(px + (1 - p)(1 - x)); whilst if he asserts that it did _not_
  happen the probability that it did happen is
  p(1 - x)/(p(1 - x) + (1 - p)x).

  In illustration of some remarks to be presently made, the reader
  will notice that on making either of these expressions = p, we
  obtain in each case x = 1/2. That is, a witness whose veracity = 1/2
  leaves the _à priori_ probability of an event (of this kind)
  unaffected.

  If, on the other hand, we make these expressions equal to x and 1 - x
  respectively, we obtain in each case p = 1/2. That is, when an event
  (of this kind) is as likely to happen as not, the ordinary veracity
  of the witness in respect of it remains unaffected.

4. Todhunter's _History_, p. 400. _Philosophical Magazine_, July, 1864.

5. "When therefore these two kinds of experience are contrary, we have
  nothing to do but subtract the one from the other, and embrace an
  opinion, either on one side or the other, with that assurance which
  arises from the remainder." (Essay on Miracles.)

6. Considerations of this kind have indeed been introduced into the
  mathematical treatment of the subject. The common algebraical
  solution of the problem in §5 (to begin with the simplest case) is
  of course as follows. Let p be the antecedent probability of the
  event, and t the measure of the truthfulness of the witness; then
  the chance of his statement being true is pt/(pt + (1 - p)(1 - t)).
  This supposes him to lie as much when the event does not happen as
  when it does. But we may meet the cases supposed in the text by
  assuming that t' is the measure of his veracity when the event does
  not happen, so that the above formula becomes pt/(pt + (1 - p)(1 - t')).
  Here t' and t measure respectively his trustworthiness in usual and
  unusual events. As a formal solution this certainly meets the
  objections stated above in §§14 and 15. The determination however
  of t' would demand, as I have remarked, continually renewed appeal
  to experience. In any case the practical methods which would be
  adopted, if any plans of the kind indicated above were resorted to,
  seem to me to differ very much from that adopted by the
  mathematicians, in their spirit and plan.

7. Laplace, for instance (_Essai_, ed. 1825, p. 149), says that if we
  saw 100 dies (known of course to be fair ones) all give the same
  face, we should be bewildered at the time, and need confirmation
  from others, but that, after due examination, no one would feel
  obliged to postulate hallucination in the matter. But the chance of
  this occurrence is represented by a fraction whose numerator is 1,
  and denominator contains 77 figures, and is therefore utterly
  inappreciable by the imagination. It must be admitted, though, that
  there is something hypothetical about such an example, for we could
  not really know that the dies were fair with a confidence even
  distantly approaching such prodigious odds. In other words, it is
  difficult here to keep apart those different aspects of the question
  discussed in Chap. XIV. §§28-33.

8. In the first edition this was stated, as it now seems to me, in
  decidedly too unqualified a manner. It must be remembered, however,
  that (as was shown in §7) this plan is really the best theoretical
  one which can be adopted in certain cases.

9. It is on this principle that the remarkable conclusion mentioned on
  p. 405 is based. Suppose an event whose probability is p; and that,
  of a number of witnesses of the same veracity (y), m assert that it
  happened, and n deny this. Generalizing the arithmetical reasoning
  given above we see that the chance of the event being asserted
  varies as

  py^{m} (1 - y)^{n} + (1 - p) y^{n} (1 - y)^{m};

  (viz. as the chance that the event happens, and that m are right and
  n are wrong; _plus_ the chance that it does not happen, and that n
  are right and m are wrong). And the chance of its being rightly
  asserted as py^{m} (1 - y)^{n}. Therefore the chance that when we
  have an assertion before us it is a true one is

               py^{m} (1 - y)^{n}
  ----------------------------------------------
  py^{m} (1 - y)^{n} + (1 - p) y^{n} (1 - y)^{m},

  which is equal to

               py^{m-n}
  --------------------------------
  py^{m-n} + (1 - p) (1 - y)^{m-n}.

  But this last expression represents the probability of an assertion
  which is unanimously supported by m - n such witnesses.

10. The stress which Butler lays upon this notion of a scheme is, I
  think, one great merit of his _Analogy_.

CHAPTER XVIII.

_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