PROBABLE ERROR 93 



mentally 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. 92. 

 Multiplying <r 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 q. 

 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 

 of useful things about the probable error. In the first 

 place its value varies inversely as the square root of 

 the number of variates that is to say, that in such a 

 case as we have just described the probable error varies 

 inversely as the square root of the number of balls 

 drawn each time. We can realize this point more 

 clearly when we remember that the linear dimensions 

 of a curve vary with the square root of its area (the 

 number of variates) ; the accuracy of our determination 

 varies in fact with the quartile, which is the linear 

 distance from the mode of a certain perpendicular. 



We have seen that it is an even chance whether a 

 single determination differs from the proper value by 

 more or less than the amount of the probable error, 



