t]RROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 133 
L. 
a 
X 
Li 
a' 
0 
a' 
a 
0 
X 
a? 
(I - / 4s' 
and thence 
I, 
a 
X — (Lj -f- ax) 
L: 
a? 
0 
0 
a 
! 
0 
0 
i 
X — (Li -t- ax) 
0 
0 
a' (1 - a') / 42' - a' - ^a^x- 
The true value of X — (Li + ax), of course, is zero; so that the “error” in 
X — (Li + ax) is the difference between X as determined by direct observation of 
the value whose class-index is a, and Lj -|- ax, as determined by calculating the 
average and the standard deviation. This error is ^ — (w -j- xp) ; and therefore, if 
we write ^ = (o -\- xp the last table shows that the mean products of oj, p, 
and (f), taken in pairs, are zero. Hence we deduce the following conclusions :— 
(1.) The mean square of ^ is greater than the mean square of -f xp/' Hence, 
if we fix a class-index a, corresponding to abscissa x in the standard normal figure, 
and if X denote the unknown value of L whose class-index is a, the probable error 
in X as obtained by direct observation is greater! than the probable error in the 
value obtained by calculating the average and the standard deviation, and deducing 
X from the formula X = Li -[- ax. The following table, for instance, gives the 
probable errors in certain values which are often chosen for exhibiting the frequency- 
constants in any particular case :— 
* This shows that (1 — a?) j a? (I -t- Hence, if OH is the central ordinate, and MP 
any other ordinate, of a normal figure of parameter 2a, and if Aj and Aj are the areas into which the 
figure is divided by MP, the product A, A, is greater than MP" {a? + ^ OM'q. 
t The result, of course, only holds when we know that the distribution is normal. When we know 
nothing about it, the value corresponding to any particular class-index can only be obtained by direct 
observation. 
