assigned with certainty, we want what is always required in the Theory of Chances, viz.[** TN: space] some estimate of their probable truth. That is, after making the best assignment we can as to the value of these elements, we want also to assign numerically the ‘probable error’ committed in such assignment. Nothing more than this can be attained in Probability, but nothing less than this should be set
§ 22. (1) As regards the first of these questions, the answer is very simple. Whether the number of measurements or observations be few or many, we must make the assumption that their average is the point we want; that is, that the average of the few will coincide with the ultimate average. This is the best, in fact the only assumption we can make. We should adopt this plan, of course, in the extreme case of there being only one value before us, by just taking that one; and our confidence increases slowly with the number of values before us. The only difference therefore here between knowledge resting upon such data, and knowledge resting upon complete data, lies not in the result obtained but in the confidence with which we entertain it.
§ 23. (2) As regards the second question, viz.[** TN: space] the determination of the modulus or degree of dispersion about the mean, much the same may be said. That is, we adopt the same rule for the determination of the E.M.S. (error of mean square) by which the modulus is assigned, as we should
adopt if we possessed full Information. Or rather we are confined to one of the rules given on p. 473, viz.[ TN: space] the second, for by supposition we have neither the à priori knowledge which would be able to supply the first, nor a sufficient number of observations to justify the third. That is, we reckon the errors, measured from the average, and calculate their mean square: twice this is equal to the square of the modulus of the probable curve of facility.[8]
§ 24. (3) The third question demands for its solution somewhat advanced mathematics; but the results can be indicated without much difficulty. A popular way of stating our requirement would be to say that we want to know how likely it is that the mean of the few, which we have thus accepted, shall coincide with the true mean. But this would be to speak loosely, for the chances are of course indefinitely great against such precise coincidence. What we really do is to assign the ‘probable error’; that is, to assign a limit which it is as likely as not that the discrepancy between the inferred mean and the true mean should exceed.[9] To take a numerical example: suppose we had made several
measurements of a wall with a tape, and that the average of these was 150 feet. The scrupulous surveyor would give us this result, with some such correction as this added,—‘probable error 3 inches’. All that this means is that we may assume that the true value is 150 feet, with a confidence that in half the cases (of this description) in which we did so, we should really be within three inches of the truth.
The expression for this probable error is a simple multiple of the modulus: it is the modulus multiplied by 0.4769…. That it should be some function of the modulus, or E.M.S., seems plausible enough; for the greater the errors,—in other words the wider the observed discrepancy amongst our measurements,—the less must be the confidence we can feel in the accuracy of our determination of the mean. But, of course, without mathematics we should be quite unable to attempt any numerical assignment.
§ 25. The general conclusion therefore is that the determination of the curve of facility,—and therefore ultimately of every conclusion which rests upon a knowledge of this curve,—where only a few observations are available, is of just the same kind as where an infinity are available. The rules for obtaining it are the same, but the confidence with which it can be accepted is less.