H-l = 
r = -591,825, 
f^2' = 
•368,739, 
1^3' = 
•238;293, 
f^i = 
•158,510. 
|t^2 = 
•018,482, 
•135,950, 
= 
- -001,812,380, 
/' 1 = 
•001,279,141. 
^1 = 
•520,265, 
^2 = 
3-744,573. 
H. B. SoPER, A. W. Young, B. M. Cave, A. Lee, K. Pearson 337 
Illustration. Samples of 25 are taken out of a population in which two variates 
have the correlation p = -6. Determination of the nature of the distribution of r 
in these samples. 
Here n = 25, and with p = -6 we find from (xx), (xxi), (xxv) and (xxvi) the 
values * 
Further 
giving 
The distribution is thus very far from normal. 
Hence by the formula f : 
Distance from mean to mode = ^^-.^^s Wts (xxvn). 
2 (5^2 - 6^1 - 9) 
we find f - r = -050,094, 
r = -64192. 
We shall see later that the actual value is 
7^= -64194. 
or the approximation is very close. 
The skewness is given by 
Sh. = (r - f)/ar = -36847, 
thus indicating that there is but little approach to normahty. 
Fig. 1, p. 338, shows the excellent fit of a Pearson curve of Type II to the dis- 
tribution. The equation is 
5-7536 / T \ 178-5135 
^ = •31004 1 -;^) 1 + 
31075/ V 9-64157 
We see that when n = 25, Pearson's curves — fitted by moments not by range — 
adequately describe the frequencies, but there is still no real approach to a 
Gaussian distribution. 
The series-expansions which have been given for the determination of the 
moments are of very little service when n is less than 25. We have therefore to 
consider formulae for deducing in succession the moments about r = 0 for w = 5 
to w = 25. 
* The values were in every case worked out to nine places of decimals. 
I Pearson : Mathematical Contributions to the Theory of Evolution, xii, p. 7. Drapers' Company 
Research Memoirs, Biovietric Series, Cambridge University Press. 
