Karl Pearson 
243 
Interchanging p and 'p we obtain precisely the same result for all the four terms 
in the product of the square terms in 8w(,/ with the Unear terms in 8»/p. Thus 
the cubic terms in the values of o-,^^ o-^^ , , are also zero. We have thus 
established to a high order of approximation that (i) Wi^ = mean of array in sampled 
population, (ii) that there is no correlation between the means in any two different 
arrays, while (iii) to a lower order of approximation only a^^^ = '^jij^^'p- 
Now we know that the distribution of means of samples of const ont si:e taken 
from a population following any law of frequency approximates very rapidly as 
the size of the sample increases to the normal law. How far may we extend this 
result to the present case of the means of arrays the total frequency of which 
varies from sample to sample? 
With the view of considering the approach to a normal distribution, let us 
investigate the third moment coefficient of for the samples of the ^^th array. 
From (iii) keeping only lowest order terms we have 
8 {Sngj,a\,) 
and accordingly 
8 m, 
+ 3S 
6,S 
or, using (a), (h) and (c) on p. 244, 
S (8»,p8«,'j,8v;, 'j,) w/Cg'X^ 
1 
\ N J N ' 
"qp ''q J) "q p ' 
S 
S (v„ 
The last two terms vanish with the factor S (n^p*,^) 
2 
Thus 
iV2 
S (Smj,) 
where ^1^^ is the third moment coefficient of the array about its own mean in the 
population sampled. We have seen also that to the same degree of approximation 
— . — - = "iP^ , where .a, 
A vp 
Accordingly if j,B^ be the value of the first 
j8-coefficient for the distribution of the means of the pt\\ array in samples: 
I (8m,) 
A 
A 
