whatever it might be, we might get quite different results. Accordingly some comparatively natural unit must have been chosen for us in which we are bound to reckon, just as in the circular measurement of an angle as distinguished from that by degrees.
The answer is that the unit here is the modulus, and that to put ‘t = 1.5’ is to say, ‘suppose the error half as great again as the modulus’; the modulus itself being an error of a certain assignable magnitude depending upon the nature of the measurements or observations in question. We shall see this better if we put the integral in the form 2/√π ∫0hxe−h2x2 d(hx); which is precisely equivalent, since the value of a definite integral is independent of the particular variable employed. Here hx is the same as x : 1/h; i.e.[ TN: space] it is the ratio of x to 1/h, or x measured in terms of 1/h. But 1/h is the modulus in the equation (y = h/√πe−h2x2) for the law of error. In other words the numerical value of an error in this formula, is the number of times, whole or fractional, which it contains the modulus.
1 This kind of mean is called by Fechner and others the “dichteste Werth.” The most appropriate appeal to it that I have seen is by Prof.[ TN: space] Lexis (Massenerscheinungen, p. 42) where he shows that it indicates clearly a sort of normal length of human life, of about 70 years; a result which is almost entirely masked when we appeal to the arithmetical average.
This mean ought to be called the ‘probable’ value (a name however in possession of another) on the ground that it indicates the point of likeliest occurrence; i.e.[** TN: space] if we compare all the indefinitely small and equal units of variation, the one corresponding to this will tend to be most frequently represented.
2 A diagram illustrative of this number of results was given in Nature (Sept. 1, 1887). In calculating, as above, the different means, I may remark that the original results were given to three decimal places; but, in classing them, only one place was noted. That is, 29.9 includes all values between 29.900 and 29.999. Thus the value most frequently entered in my tables was 30.0, but on the usual principles of interpolation this is reckoned as 30.05.
3 There is some ambiguity in the phraseology in use here. Thus Airy commonly uses the expression ‘Error of Mean Square’ to represent, as
here, √ ∑e2/n. Galloway commonly speaks of the ‘Mean Square of the Errors’ to represent ∑e2/n. I shall adhere to the former usage and represent it briefly by E.M.S. Still more unfortunate (to my thinking) is the employment, by Mr Merriman and others, of the expression ‘Mean Error,’ (widely in use in its more natural signification,) as the equivalent of this E.M.S.
The technical term ‘Fluctuation’ is applied by Mr F. Y. Edgeworth to the expression 2∑e2/n.
4 Practically, of course, we should allow for the expansion or contraction. But for purposes of logical explanation we may conveniently take this variation as a specimen of one of those disturbances which may be neutralised by resort to an average.
5 More strictly multinomial: the relative frequency of the different numbers being indicated by the coefficients of the powers of x in the development of
6 By Mr Merriman, in his work on Least Squares.