MISCELLANEA. 
I. The Distribution of the Means of Samples which are not 
drawn at Random. 
By student. 
It is one of the advantages of the noi'mal curve that if samples are drawn at random from 
any population, no naatter how distributed, the distributions of the statistical constants of the 
samples rapidly approach the Gaussian as the samples grow large. 
This being so, the result of grouping 2000 in samples of 25 given in Drs Greenwood and 
White's very interesting paper in Biometrika is surprising. 
For it is easy to show that if Bi , B-i be the constants of the distribution of the means of samples 
of n drawn at random, corresponding to /3i , ^2 in the original frequency distribution, then* 
A=^' and 5.-3 = ^. 
11 " n 
But in this case ft = 1-7977 and ifi = -4756 : ^^= 0719, 
/32-3 = 2-5790, ^2 -3= -3185: ^'^ = •1032. 
n 
Now neither of these can be considered significant with a sample of 80 means but at the same 
time they are both sufficiently different to suggest that the conditions which led to the theoretical 
result have not been fulfilled. 
The first thing which occurred to me was that as Sheppard's corrections had been used for 
the means but not for the original distribution it might be well to try applying them to both. 
This however makes but little difference, for we get 
ft = 1 •9898: || = -0792, 
ft -3 = 2-7725: ^|^=-1109. 
I next considered the possibility that the samples were not strictly random but that there 
was some slight correlation between successive observations. 
I therefore assumed that the individuals composing the sample were more like each other 
than to the rest of the population, that in fact there was homotyposis, and working from this 
* Henderson, R. : Journal Inst, of Actuaries, Vol. xli. pp. 429 — 442. 
