100 BIOMETRY 
of values lying one above and one below the true value 
required—e.g., the average stature of the whole of a 
race—such that it is an even chance that the value 
actually found will lie between them. Or the same 
thing may be expressed in another way. If we plot in 
the form of a curve a long series of actual determina- 
tions of a particular value, the probable error of a 
single determination will be nearly equal to the 
quartile of the curve so obtained. We may illustrate 
this state of things from our example of tossing coins, 
or still better by the essentially similar case of draw- 
ing balls out of a bag which contains a very large 
number of balls—black and white in equal numbers. 
Here the value to be determined experimentally is 
the relative number of black balls to white, which we 
know as a matter of fact to be equality; and our 
single determination may consist in drawing out a 
hundred balls, which are afterwards returned to the 
bag. If we do this 1,000 times, and plot the number 
of black balls drawn each time, we shall arrive approxi- 
mately at a curve having its mode at 50, and possessing 
a standard deviation which it is possible to determine 
from the instructions given in the footnote to p. 99. 
Multiplying o by 0°6745 gives us the quartile, which 
represents the probable error of a single determination. 
That is to say, it is an even chance whether any single 
determination differs from 50 by more or less than g. 
In this particular example the quartiles would be found 
to lie very nearly at 46°6 and 53°4,so that the value of 
the probable error is 3:4. 
The properties of the normal curve tell us a number 
