278 
Peccavmius ! 
Formula (X) shows that even for small values of M we have a value only slightly 
less than the usual . 
Turning to the array in a sample of N we require the mean values of 
S{~^ and s(-\ , 
N. 
Substituting in (X) we find 
Jv^ V-M-^w, I7_r + -J 
For a normal distribution we have * 
^-^;l(^-8^-2i)-: (AA). 
Thus the usual value cr7(yV2?i^, will be about 2 "/^ in defect in an array of 25 in 
a sample of 1000. 
It will be realised therefore that if we do not take arrays of less than 25 in 
samples of 1000, the usual values of the mean standard-deviation of an array Up and 
the standard-deviation of these standard-deviations will not lead us badly astray. 
We have finally to ask what degree of weight we can give to the " probable error " 
of this standard-deviation, i.e. to •67449cto-,^^ . This is only determining how far 
o-,(^ follows the normal law of distribution, that is to say, how nearly are pB/ and 
pBo — 3 zero for the distribution of o-^^, these representing the first two /S-coefficients 
for this distribution. Before we do this, however, we may find from (Q), p. 276, 
the values of j/i?/, jfB^ for the /u., of samples of constant size M to the second order 
of approximation. They are 
"l + 1 //S, - 2 _ 3y8, -21,8, - 18^, +26^ 
3,84 -9a -18^1 + 6 J 
(BB), 
reducing with a Gaussian distribution tof 
+ (CO)' 
* The reader should note that "'cr,,^^ is not the same thing as if we put 
from (N) on p. 275. 
t These values are identical to the degree of approximation adopted with those given by " Student," 
in Biomctrika, Vol. vi, p. 4. Professor Tchouproff (Ibid., Vol. xii, p. 192) does scant justice to 
" Student." The only misprint we can see is that Ijii appears in the first term instead of !/«■', the 
power having probably been ' drawn ' in printing ; it re-appears in the next equation. Further " Student " 
gives not only Tchonproff's (19) but his (22); this as "Student" himself indicates (p. 4) involved 
Tchouproii's lengthy equations (20) and (21), which he refrained owing to their length from publishing. 
1 (0,-3/3,-60, + 2f 
MJJi — 
