of the errors will exceed ‘one and a half.’ But when we ask, ‘one and a half’ what?
the answer would not always be very ready. As usual, the main difficulty of the beginner is not to manipulate the formulæ, but to be quite clear about his units.
It will be seen at once that this case differs from the preceding in that we cannot now choose our unit as we please. Where, as here, there is only one variable (t), if we were allowed to select our own unit, the inch, foot, or 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.
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.
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.
if we compare all the indefinitely small and equal units of variation, the one corresponding to this will tend to be most frequently represented.