Karl Pearson 
149 
The hypothesis is justified therefore if no cell be taken so small that its contents 
are very small compared with the size of the sample. 
(ii) That the sampling takes place out of a population indefinitely greater 
than the sample. If this be not true, then the distribution of frequency of any 
given cell for a series of samples follows not a binomial but a hypergeometrical 
series. The necessary modifications in the formulae are not very substantial and 
have been discussed elsewhere*. 
Supposing the above two conditions to be fulfilled, then in true random 
samples the mean of a cell frequency will be 
m,= M'^ (xii), 
where M is size of sample, the contents of the sth cell in the sampled population 
N, and condition (i) amounts to saying that no cell is to be chosen so that nJN 
is indefinitely small. 
Further, the frequency of the sth cell will follow the binomial 
ns 
N ^ 
(-1)}"' 
and thus have a standard deviation given by 
^) (xiii). 
Lastly the correlation Vss' between deviations in the sth and s'th cells is given by 
CT«.CT„,'r,,. = - (xiv) 
(5) The following deduction of the value of is a variant from my Phil. Mag. 
proof. I owe the suggestion of it to Mr H. E. Soper, although I have deviated some- 
what from his track. Let an indefinitely large population N consist of the classes 
Co, Ci, ... Ci in the quantities Wq. '^i. ••• "i^i respectively. Then = nJN = 
chance of drawing a member of the class C,, and the standard deviation of 'the 
distribution of frequency in samples of M drawn from the popvdation will in this 
class Cs be as above 
a, = VMps {l-lh) (xiii)'^K 
Further, the mean of samples for this class will be Mp^ by (xii). 
In the next place the correlation between deviations from the means in classes 
Cs and Cf will be in our present notation 
CTsa^r,,. = - Mn,7i,./N^ (xiv)'^^ 
or by (xiii)'^ie r,, = - V^^^s ■ ^ 
* Phil. Mag. p. 239, 1899 and Biomelrika, Vol. v. p. 174 
