Chapter XIX: The Theory of the Average as a Means of Approximation to the Truth (2)

It is not difficult to imagine an example which will aptly illustrate the case in point: at worst it may seem a little far-fetched. Conceive then that some firm in England received a hurried order to supply a portion of a machine, say a steam-engine, to customers at a distant place; and that it was absolutely essential that the work should be true to the tenth of an inch for it to be of any use. But conceive also that two specifications had been sent, resting on different measurements, in one of which the length of the requisite piece was described as sixty and in the other sixty-one inches. On the assumption of any ordinary law of error, whether of the binomial type or not, there can be no doubt that the firm would make the best of a very bad job by constructing a piece of 60 inches and a half: i.e. they would have a better chance of being within the requisite tenth of an inch by so doing, than by taking either of the two specifications at random and constructing it accurately to this. But if the law were of the kind indicated in our diagram,[11] then it seems equally certain that they would be less likely to be within the requisite narrow margin by so doing. As a mere question of probability,--that is, if such estimates were acted upon again and again,--there would be fewer failures encountered by simply choosing one of the conflicting measurements at random and working exactly to this, than by trusting to the average of the two.

This suggests some further reflections as to the taking of averages. We will turn now to another exceptional case, but one involving somewhat different considerations than those which have been just discussed. As before, it may be most conveniently introduced by commencing with an example.

32. Suppose then that two scouts were sent to take the calibre of a gun in a hostile fort,--we may conceive that the fort was to be occupied next day, and used against the enemy, and that it was important to have a supply of shot or shell,--and that the result is that one of them reports the calibre to be 8 inches and the other 9. Would it be wise to assume that the mean of these two, viz. 8-1/2 inches, was a likelier value than either separately?

The answer seems to be this. If we have reason to suppose that the possible calibres partake of the nature of a _continuous_ magnitude,--i.e. that all values, with certain limits, are to be considered as admissible, (an assumption which we always make in our ordinary inverse step from an observation or magnitude to the thing observed or measured)--then we should be justified in selecting the average as the likelier value. But if, on the other hand, we had reason to suppose that _whole_ inches are always or generally preferred, as is in fact the case now with heavy guns, we should do better to take, even at hazard, one of the two estimates set before us, and trust this alone instead of taking an average of the two.

33. The principle upon which we act here may be stated thus. Just as in the direct process of calculating or displaying the 'errors', whether in an algebraic formula or in a diagram, we generally assume that their possibility is continuous, i.e. that all intermediate values are possible; so, in the inverse process of determining the probable position of the original from the known value of two or more errors, we assume that that position is capable of falling at any point whatever between certain limits. In such an example as the above, where we know or suspect a discontinuity of that possibility of position, the value of the average may be entirely destroyed.

In the above example we were supposed to know that the calibre of the guns was likely to run in English inches or in some other recognized units. But if the battery were in China or Japan, and we knew nothing of the standards of length in use there, we could no longer appeal to this principle. It is doubtless highly probable that those calibres are not of the nature of continuously varying magnitudes; but in an entire ignorance of the standards actually adopted, we are to all intents and purposes in the same position as if they were of that continuous nature. When this is so the objections to trusting to the average would no longer hold good, and if we had only one opportunity, or a very few opportunities, we should do best to adhere to the customary practice.

34. When however we are able to collect and compare a large number of measurements of various objects, this consideration of the probable discontinuity of the objects we thus measure,--that is, their tendency to assume some one or other of a finite number of distinct magnitudes, instead of showing an equal readiness to adapt themselves to all intermediate values,--again assumes importance. In fact, given a sufficient number of measurable objects, we can actually deduce with much probability the standard according to which the things in question were made.

This is the problem which Mr Flinders Petrie has attacked with so much acuteness and industry in his work on _Inductive Metrology_, a work which, merely on the ground of its speculative interest, may well be commended to the student of Probability. The main principles on which the reasoning is based are these two:--(1) that all artificers are prone to construct their works according to round numbers, or simple fractions, of their units of measurement; and (2) that, aiming to secure this, they will stray from it in tolerable accordance with the law of error. The result of these two assumptions is that if we collect a very large number of measurements of the different parts and proportions of some ancient building,--say an Egyptian temple,--whilst no assignable length is likely to be permanently unrepresented, yet we find a marked tendency for the measurements to cluster about certain determinate points in our own, or any other standard scale of measurement. These points mark the length of the standard, or of some multiple or submultiple of the standard, employed by the old builders. It need hardly be said that there are a multitude of practical considerations to be taken into account before this method can be expected to give trustworthy results, but the leading principles upon which it rests are comparatively simple.

35. The case just considered is really nothing else than the recurrence, under a different application, of one which occupied our attention at a very early stage. We noticed (Chap. II.) the possibility of a curve of facility which instead of having a single vertex like that corresponding to the common law of error, should display two humps or vertices. It can readily be shown that this problem of the measurements of ancient buildings, is nothing more than the reopening of the same question, in a slightly more complex form, in reference to the question of the functions of an average.

Take a simple example. Suppose an instance in which great errors, of a certain approximate magnitude, are distinctly more likely to be committed than small ones, so that the curve of facility, instead of rising into one peak towards the centre, as in that of the familiar law of error, shows a depression or valley there. Imagine, in fact, two binomial curves, with a short interval between their centres. Now if we were to calculate the result of taking averages here we should find that this at once tends to fill up the valley; and if we went on long enough, that is, if we kept on taking averages of sufficiently large numbers, a peak would begin to arise in the centre. In fact the familiar single binomial curve would begin to make its appearance.

36. The question then at once suggests itself, ought we to do this? Shall we give the average free play to perform its allotted function of thus crowding things up towards the centre? To answer this question we must introduce a distinction. If that peculiar double-peaked curve had been, as it conceivably might, a true error-curve,--that is, if it had represented the divergences actually made in aiming at the real centre,--the result would be just what we should want. It would furnish an instance of the advantages to be gained by taking averages even in circumstances which were originally unfavourable. It is not difficult to suggest an appropriate illustration. Suppose a man firing at a mark from some sheltered spot, but such that the range crossed a broad exposed valley up or down which a strong wind was generally blowing. If the shot-marks were observed we should find them clustering about two centres to the right and left of the bullseye. And if the results were plotted out in a curve they would yield such a double-peaked curve as we have described. But if the winds were equally strong and prevalent in opposite directions, we should find that the averaging process redressed the consequent disturbance.

If however the curve represented, as it is decidedly more likely to do, some outcome of natural phenomena in which there was, so to say, a real double aim on the part of nature, it would be otherwise. Take, for instance, the results of measuring a large number of people who belonged to two very heterogeneous races. The curve of facility would here be of the kind indicated on p. 45, and if the numbers of the two commingled races were equal it would display a pair of twin peaks. Again the question arises, 'ought' we to involve the whole range within the scope of a single average? The answer is that the obligation depends upon the purpose we have in view. If we want to compare that heterogeneous race, as a whole, with some other, or with itself at some other time, we shall do well to average without analysis. All statistics of population, as we have already seen (v. p. 47), are forced to neglect a multitude of discriminating characteristics of the kind in question. But if our object were to interpret the causes of this abnormal error-curve we should do well to break up the statistics into corresponding parts, and subject these to analysis separately.

Similarly with the measurements of the ancient buildings. In this case if all our various 'errors' were thrown together into one group of statistics we should find that the resultant curve of facility displayed, not two peaks only, but a succession of them; and these of various magnitudes, corresponding to the frequency of occurrence of each particular measurement. We _might_ take an average of the whole, but hardly any rational purpose could be subserved in so doing; whereas each separate point of maximum frequency of occurrence has something significant to teach us.

37. One other peculiar case may be noticed in conclusion. Suppose a distinctly asymmetrical, or lop-sided curve of facility, such as this:--

[Figure: An asymmetric (lop-sided) distribution.]

Laws of error, of which this is a graphical representation, are, I apprehend, far from uncommon. The curve in question, is, in fact, but a slight exaggeration of that of barometrical heights as referred to in the last chapter; when it was explained that in such cases the mean, the median, and the maximum ordinate would show a mutual divergence. The doubt here is not, as in the preceding instances, whether or not a single average should be taken, but rather what kind of average should be selected. As before, the answer must depend upon the special purpose we have in view. For all ordinary purposes of comparison between one time or place and another, any average will answer, and we should therefore naturally take the arithmetical, as the most familiar, or the median, as the simplest.

38. Cases might however arise under which other kinds of average could justify themselves, with a momentary notice of which we may now conclude. Suppose, for instance, that the question involved here were one of desirability of climate. The ordinary mean, depending as it does so largely upon the number and magnitude of extreme values, might very reasonably be considered a less appropriate test than that of judging simply by the relatively most frequent value: in other words, by the maximum ordinate. And various other points of view can be suggested in respect of which this particular value would be the most suitable and significant.

In the foregoing case, viz. that of the weather curve, there was no objective or 'true' value aimed at. But a curve closely resembling this would be representative of that particular class of estimates indicated by Mr Galton, and for which, as he has pointed out, the _geometrical_ mean becomes the only appropriate one. In this case the curve of facility ends abruptly at O: it resembles a much foreshortened modification of the common exponential form. Its characteristics have been discussed in the paper by Dr Macalister already referred to, but any attempt to examine its properties here would lead us into far too intricate details.

39. The general conclusion from all this seems quite in accordance with the nature and functions of an average as pointed out in the last chapter. Every average, it was urged, is but a single representative intermediate value substituted for a plurality of actual values. It must accordingly let slip the bulk of the information involved in these latter. Occasionally, as in most ordinary measurements, the one thing which it represents is obviously the thing we are in want of; and then the only question can be, which mean will most accord with the 'true' value we are seeking. But when, as may happen in most of the common applications of statistics, there is really no 'true value' of an objective kind behind the phenomena, the problem may branch out in various directions. We may have a variety of purposes to work out, and these may demand some discrimination as regards the average most appropriate for them. Whenever therefore we have any doubt whether the familiar arithmetical average is suitable for the purpose in hand we must first decide precisely what that purpose is.

1. Mr Mansfield Merriman published in 1877 (_Trans. of the Connecticut
Acad._) a list of 408 writings on the subject of Least Squares.

2. In other words, we are to take the "centre of gravity" of the
shot-marks, regarding them as all of equal weight. This is, in
reality, the 'average' of all the marks, as the elementary
geometrical construction for obtaining the centre of gravity of a
system of points will show; but it is not familiarly so regarded. Of
course, when we are dealing with such cases as occur in Mensuration,
where we have to combine or reconcile three or more inconsistent
equations, some such rule as that of Least Squares becomes
imperative. No taking of an average will get us out of the
difficulty.

3. The only reason for supposing this exceptional shape is to secure
simplicity. The ordinary target, allowing errors in two dimensions,
would yield slightly more complicated results.

4. When first referred to, the _general_ form of this equation was
given (v. p. 29). The special form here assigned, in which
(h/sqrt{π}) is substituted for A, is commonly employed in
Probability, because the integral of y dx, between +infinity and
-infinity, becomes equal to unity. That is, the sum of all the
mutually exclusive possibilities is represented, as usual, by
unity. In this form of expression h is a quantity of the order
x^{-1}; for hx is to be a numerical quantity, standing as it does as
an index. The modulus, being the reciprocal of this, is of the same
order of quantities as the errors themselves. In fact, if we
multiply it by 0.4769... we have the so-called 'probable error.'

5. See, for the explanation of this, and of the graphical method of
illustrating it, the note on p. 29.

6. Broadly speaking, we may say that the above remarks hold good of
any law of frequency of error in which there are actual limits,
however wide, to the possible magnitude of an error. If there are no
limits to the possible errors, this characteristic of an average to
heap its results up towards the centre will depend upon
circumstances. When, as in the exponential curve, the approximation
to the base, as asymptote, is exceedingly rapid,--that is, when the
extreme errors are relatively very few,--it still holds good. But if
we were to take as our law of facility such an equation as
y = π/(1 + x^{2}), (as hinted by De Morgan and noted by Mr Edgeworth:
_Camb. Phil. Trans._ vol. X. p. 184, and vol. XIV. p. 160) it does
not hold good. The result of averaging is to _diminish_ the tendency
to cluster towards the centre.

7. The reader will find the proofs of these and other similar formulæ
in Galloway _on Probability_, and in Airy _on Errors_.

8. The formula commonly used for the E.M.S. in this case is
(sum e^{2})/(n - 1) and not (sum e^{2})/n. The difference is trifling,
unless n be small; the justification has been offered for it that
since the sum of the squares measured from the true centre is a
minimum (that centre being the ultimate arithmetical mean) the sum
of the squares measured from the somewhat incorrectly assigned
centre will be somewhat larger.

9. It appears to me that in strict logical propriety we should like to
know the probable error committed in _both_ the assignments of the
preceding two sections. But the profound mathematicians who have
discussed this question, and who alone are competent to treat it,
have mostly written with the practical wants of Astronomy in view;
and for this purpose it is sufficient to take account of the one
great desideratum, viz. the true values sought. Accordingly the only
rules commonly given refer to the probable error of the mean.

10. i.e. as distinguished from acting upon them indirectly. This
latter proceeding, as explained in the chapter on Randomness, may
result in giving a non-uniform distribution.

11. There is no difficulty in conceiving circumstances under which a
law very closely resembling this would prevail. Suppose, e.g., that
one of the two measurements had been made by a careful and skilled
mechanic, and the other by a man who to save himself trouble had put
in the estimate at random (within certain limits),--the firm having
a knowledge of this fact but being of course unable to assign the
two to their authors,--we should get very much such a Law of Error
as is supposed above.

INDEX.

Accidents 342

Airy, G. B. 447, 484

Anticipations, tacit 287

Arbuthnott 258

Aristotle 205, 307

Average
arithmetical 437
geometrical 439
median 442
consequences of 482
necessary results of 457
uses of 439, 489

Babbage 343

Bags and balls 180, 411

Belief
correctness of 125, 131, 178
gradations of 139
growth of 199
language of 143
measurement of 119, 125, 146
quantity of 133
test of 140, 149, 294
undue 129
vagueness of 127

Bentham 319, 323

Bernoulli 91, 117, 389

Bertillon 435

Births, male and female 90, 258, 263

Boat race, Oxford and Cambridge 339

Boole 183

Buckle 237

Buffon 153, 205, 352, 389

Burgersdyck 311

Butler 209, 281, 333, 366

Carlisle Tables 169

Casual, meaning of 245

Causation
need of 237
proof of 244

Centre of gravity 467

Certainty, in Law 324
reasonable 327
hypothetical 210

Chance
and Causation 244
Creation 258
Design 256
Genius 353
neglect of small 363
selections 338

Chauvenet 352

Classification, numerical scheme of 48

Coincidences 245

Combinations and Permutations 87

Communism 375, 392

Conceptualism 275

Conflict of chances 418

Consumptives, insurance of 227

Cournot 245, 255, 338

Crackanthorpe 312, 320

Craig, J. 192

Crofton, M. W. 61, 101, 104

Dante 285

Deflection
causes of 57
from aim 38

De Morgan 83, 106, 119, 122, 135, 177, 179, 197, 236, 247, 296, 308, 350, 379, 382, 483

De Ros trial 255

Digits, random 111, 114

Discontinuity 116

Distribution, random 106

Diagrams 29, 45, 118, 443, 476, 481, 493, 501

Dialectic 302, 320

Donkin 123, 188, 283

Duration of life 15, 441

Düsing 259

Ebbinghaus 199

Edgeworth, F. Y. 34, 119, 256, 339, 393, 435, 483

Ellis, L. 9

Epidemics 62

Error, law of 29
asymmetrical 34, 441, 443
binomial 37, 457, 469, 480
geometrical 34, 502
heterogeneous 45
production of 36

Error
mean 446
probable 446, 472, 488
of mean square 447, 488

Escapes, narrow 341

Expectation, moral 388

Experience and probability 74

Exponential curve 29

Extraordinary
sense of 159, 423
stories 407, 421

Fallacies in Logic and Probability 367

Fatalism 243

Fechner 34, 389, 435, 441

Fluctuation 448
unlimited 73

Forbes, J. D. 188, 262

Formal Logic 123

Formal and Material treatment 86

Free will 240

Galloway 248, 448, 484

Galton, F. 33, 50, 70, 318, 442, 451, 473, 502

Gambling
and Insurance 370
disadvantage of 384
final results of 385, 391

Godfray, H. 99

Grote, G. 307

Guy 6

Hamilton, W. 266, 297

Happiness, human 382

Heads and Tails 77

Heredity 50, 357

Herschel 30, 466

Houdin 361

Hume 236, 419, 433

Hypotheses 268

Immediate inferences 121

Independent events 175, 246

Induction
and Probability 194, 201, 208, 233, 358
difficulty of 213
pure 200

Inequality of wealth 382

Inference, rules of 167

Inoculation 374

Insurance
justification of 149
difficulties of 221
life 151
peculiar case 224
theory of 372
varieties of 374

Inverse probability 179, 196, 249

Irregularity, absolute and relative 6

Jacobs, J. 199

Jackson, J. G. 253

Jevons 37, 83, 136, 198, 201, 209, 247

Kant 310, 317

Keckermann 298, 316

Kinds, natural 55

Krug 324

Lambert 309

Language of Chance 159

Laplace 89, 120, 197, 237, 424

Law
absence of 101
empirical 160
of causation 206

Least squares 41, 467

Leibnitz 309, 320

Letters
lost 162, 368
misdirected 67, 237, 241

Lexis, W. 263, 441

Limit
conception of 18, 109, 164
of possible fluctuation 32

Lines, random 113

Lister's method 187

Lotteries 128

Lunn, J. R. 248

Likely, equally 77, 183

McAlister, D. 34, 187, 502

Mansel, H. L. 299, 301, 320

Martingale 343

Material and Formal Logic 265

Maximum ordinate 441, 455

Measurement of
Belief 119
Memory 192

Mental qualities, measurement of 49

Merriman, M. 352, 448, 460, 465

Mill, J. S. 131, 207, 266, 282, 402

Milton, chance production of 353

Miracles 428

Michell, J. 260

Modality 295
divisions of 307
false 297
formal 298
in Law 319

Modulus 464, 472, 484

Monro, C. J. 325, 416

Names, reference of 270

Nations, comparison of 51

Natural Kinds 55, 63, 71

Necessary and impossible matter 310

Objects and agencies 53

Occam 314

Paley 433

Penny, tosses of 144

Petrie, F. 498

Petersburg Problem 19, 154

Poisson 405

Prantl 311

Presumption, legal 329

Prévost 348

Probability
definition of 165
relative 290
integral 463

Probable
facts 269
value 441
error 446, 472

Problem, Three point 104

Proctor, R. A. 262, 378

Prophecies, suicidal 226

Providence 89, 431

Propositions, proportional 2

Psychical research 256

Pyramid, the great 251

π, digits in 111, 247

Quartiles 446

Quetelet 23, 30, 43, 91, 259, 330, 348, 454

Randomness
etymology of 96
in firing 98
proof of 107

Rare events 349

Realism 92

Reason, sufficient 82

Residuals 460

Roberts, C. 25

Rod
broken at random 98
thrown at random 103

Rules
Inductive and Deductive 176
of Succession 191
conflict of 222
plurality of 217

Series
definite proportions in 11
fixed and variable 16
ideal 95
peculiar 12

Shanks 248

Skeat, W. W. 96

Smiglecius 306, 316

Smyth, P. 251

Socialism 392

Spiritualism 365

Stars, random arrangement of 108, 260

Statistics
by Intercomparison 473
unconscious appeal to 400

Statistical Journal 6

Stature
human 25, 471
French and English 44

Stephen, J. F. 282, 323, 326

Stewart, D. 209, 237

Subjective and objective terms 160

Succession
long 360
Rule of 190, 362

Suffield, G. 248

Suicides 67, 237

Surnames, extinction of 387

Surprise, emotion of 157

Syllogisms, pure and modal 316

Taylor 329

Testimony
single 411
combined 426
two kinds of 409
worthless 416

Thomson, W. 153, 314, 419

Time
influence of 191
in Probability 279

Todhunter 415

Tontines 380

Triangle, random 103

Tucker, A. 127

Types
existence of 42, 60, 453
fixed and fluctuating 64, 93

Ueberweg 311

Uncertainty in life 370

Uniformity 240

Units of calculation 464

Voluntary agency 65, 68, 85

Watford 374

Wallis, J. 312

Watson, H. W. 387

Whately 297, 307

Whist 401

Whitworth, W. A. 87, 183, 384

Wilson, J. M. 104

Witnesses, independent 405

Wolf 309

Woolhouse 101

CAMBRIDGE: PRINTED BY C.J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS.

End of Project Gutenberg's The Logic of Chance, 3rd edition, by John Venn