PROF. K. PEARSON ON THE MATHEMATICAL THEORY OP EVOLUTION. 109 
Now, ]et an error 8// occur in the frequency y corresponding to x, and let Scr.,,. be 
the error of a, when calculated from Mv,.; then by the above result, 
a:-'- 8y hx = (2r - 1) (2r - 3) . . . 5.3.1, 2ro-"'-' X So- 2 ,c. 
Comparing this with the error Sera arising in calculating cr from the second moment 
in the usual manner, we have 
h(T.^r 
8 a:, 
When X is small Scto will be very great as compared with Scr.^,, and the high moment 
has a great advantage. This advantage is maintained until 
a; = o- [i- (2r — 1) (2r - 3) . . . 5.3.1]’'<“'-% 
= 2'45(t for the fourth moment, 
= 2‘59cr for tlie sixth moment. 
,..(2r-^l)(2r-3) 
5.3.1 
But the probability of an organ 2'59cr is less than 1 in the 100, and of 2’45 about 
2 in the 100. Hence we may take it that errors for which the 4th or 6th moments 
give a worse result than the 2nd moment for cr are improbable, while errors for which 
they give a much better result than the 2nd moment are very probable. Take, how¬ 
ever, practically the worst case, an error occurring in the frec|uency of an organ corre¬ 
sponding to 3 (t, an error only likely to occur about three times in the thousand errors 
supposing errors distributed as normal frequencies. We find 
Scr^, = I'o Scr', 
ScTg = 1'8 Scr'. 
The errors from the fourth and sixth moments are thus only P5 and 1'8 times the 
errors from the second moment, but errors from the second moment greater than 
6 Scr^, and 45 Sctq are given whenever x is less than cr, or in more than 68 per cent, of 
cases. It would thus appear that an error which will put a high moment at a great 
disadvantage as compared with a low moment is extremely rare; while, on the con¬ 
trary, errors which put a low moment at a great disadvantage as compared with a 
high moment are very frequent. 
As a type of the sort of differences we obtain from working with low and high 
