l 5 
Recent Advances in the Study of Heredity. 
would be bound to maintain that such diminution took place. Now, 
in both these cases it is, of course, evident to the reader that the 
larger the number of individuals composing a given generation, 
the less will the interpretation based upon it be open to the objection 
that the difference between observation and expectation is due to 
chance ; that is to say, to the small number of individuals recorded. 
What we want to know, therefore, is how much the ratio determined 
from a generation of a given size may be expected to deviate, on the 
average , from expectation. Here statisticians come to our aid, and 
have provided us with a formula which enables us to estimate this. 
Suppose that we have a hundred F a families, each one of which 
consists of a thousand individuals, the theory of probability does 
not lead us to expect that the determination of the proportion of 
recessives will be exactly 25% in each of these hundred families. 
On the contrary, basing our expectation on the theory of probability 
we should anticipate that if the frequency of the hundred ratios 
were plotted in the form of a frequency curve, such a curve 
would be of the normal type. The mean of this curve would 
probably near at 25% and the deviations would become less numerous 
as they increased in magnitude in the plus or minus direction ; and 
there would be a point on the base-line of the curve in the plus 
direction dividing the number of deviations in the plus direction 
into half, and a similar point, on the minus side of the mean, dividing 
the deviations in the minus direction into half. The value on either 
side of the mean, which divides the number of deviations on one 
side of the mean into two, that is to say the value corresponding to 
the point we have just been considering, is called the Probable 
Error. 
But if each of the hundred F 2 families had consisted of 10,000 
instead of 1,000 individuals, we should expect this point to be nearer 
the mean, that is to say, we should expect the deviations in this 
case to be smaller and to cluster more closely round the mean. In 
other words we should expect the Probable Error to he smaller. 
If, therefore, we can calculate this value beforehand for a generation 
of given size, we should expect half the ratios to fall inside the 
deviation indicated by this point, and half of them to fall outside. 
Now the only question of real interest is, how far outside the 
Probable Error deviations may he expected to fall, for as anyone 
who has dealt with the determination of such ratios will know, 
deviations can fall much further outside this point than they can on 
the inside, for on the inside they are limited by the mean. The 
