KiRSTiNE Smith 
3 
(a) The Mean Value. 
For the sample in hand wc have 
The mean value of y for a great number of samples coincides, according to the 
suppositions, with the mean of the population and this we choose for the zei'o point 
of y. The squared standard deviation of v/ is therefore found simply by squaring 
the expression above, suniming for all the samples imagined and taking the mean 
value of the result. We thus find 
where a bar above a summation indicates that the mean value has to be taken 
of the sums for all samples, i.e. for the population. Let the standard deviation 
of the population be s and the correlation coefficient for individuals of the same 
class r, we then have 
S (2/1^) = nqs' 
and t (2/12/2) = ^nq (q-l) rs-. 
As individuals of different classes are uncorrected S (2/12/2) is equal to 0, and 
accordingly we find 
crjf = ~{l+{q-l)r}* (1). 
This contains « and r for the population, which are, as a rule, only known from 
the sample in hand. It will be seen in the following, what is the approximation 
obtained by putting s and r equal to the values found from the sample. 
(b) The Mean Value of cr- — the presiniiptive Standard Deviation. 
For our sample we find 
'^-'=^^^(:y^)-Tr (2). 
By taking the mean of a- for a great number of samples we find from this, 
remembering that the mean of 2/^ equals o--", 
.J. = 4l_l±<£^-^, 
\ nq J ^ ' 
When we take the value found foro-- as an approximation to cr-, we find accord- 
ingly the. presumptive value of the standard deviation of the population by the 
formula 
„o-^ = s- = a'^ 
nq-\l+(q-l)r} ' 
which for ?^ = 0 or g = 1 takes the form known for uncorrelated observations. 
* Vide Comptes-Rendiis des Trav. du Lab. Carhberg, Vol. xiv. No. 11, 1921, Copenhagen, p. .32. 
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