402 
R. A. FISHER ON THE CORRELATION BETWEEN 
1. Let us suppose that the difference caused by a single Mendelian factor is 
represented in its three phases by the difference of the quantities a, d, — a, and 
that these phases exist in any population with relative frequency P, 2Q, It, where 
P + 2Q + ft = 1. 
Then a population in which this factor is the only cause of variability has its 
mean at 
to = P a + 2QcZ - Ra, 
so that 
P('a - m ) + 2Q (d - m) - R(a + to) = 0. 
Let now 
P(a -■ to ) 2 + 2QG - to ) 2 + R(a + to ) 2 = <i 2 ..... (I) 
a 2 then is the variance due. to this factor, for it is easily seen that when two such 
factors are combined at random, the mean square deviation from the new mean is 
equal to the sum of the values of a 2 for the two factors separately. In general the 
mean square deviation due to a number of such factors associated at random will be 
written 
a 2 = 2a 2 . (ii) 
. To justify our statement that a 2 is the contribution which a single factor makes 
to the total variance, it is only necessary to show that when the number of such 
factors is large the distributions will take the normal form. 
If now we write 
/x 3 = P(a - m) 3 + 2Q (d — to) 3 - R(ct + to) 3 
/x 4 = P(a - to) 4 + 2Q(cZ - to) 4 + R(a + to) 4 , 
and if M 3 and M 4 are the third and fourth moments of the population, the variance 
of which is due solely to the random combination of such factors, it is easy 
to see that 
M 3 = 2 /x 3 
M 4 -3(r 4 = 2(/x 4 -3a 4 ). 
Now the departure from normality of the population may be measured by means of 
the two ratios 
ft = — §? and 
B - M4 
P2~— T 
The first of these is 
(2/, 3 ) 2 /(Sa 2 ) 3 , 
and is of the order where n is the number of factors concerned, while the second 
n 
differs from its Gaussian value 3 also by a quantity of the order - . 
2. If there are a great number of different factors, so that a- is large compared to 
every separate a, we may investigate the proportions in which the different phases 
occur in a selected array of individuals. Since the deviation of an individual is 
simply due to a random combination of the deviations of separate factors, we must 
expect a given array of deviation, let us say x, to contain the phases of each factor 
