474 Theory of the Average. [CHAP. xix. 



a height as just to include one half of the men above (or 

 below) the mean. (In practice this would be found to re 

 quire about 171 inches: that is, one quarter of any large 

 group of such men will fall between 69 and 7071 inches.) 

 Divide this number by 4769 and we have the modulus. In 

 the case in question it would be equal to about 3 6 inches. 



Under the assumption with which we start, viz. that the 

 law of error displays itself in the familiar binomial form, or 

 in some form approximating to this, the three methods indi 

 cated above will coincide in their result. Where there is 

 any doubt on this head, or where we do not feel able to cal 

 culate beforehand what will be the rate of dispersion, we 

 must adopt the second plan of determining the modulus. 

 This is the only universally applicable mode of calculation: 

 in fact that it should yield the modulus is a truth of defini 

 tion; for in determining the error of mean square we are 

 really doing nothing else than determining the modulus, as 

 was pointed out in the last chapter. 



10. The position then which we have now reached is 

 this. Taking it for granted that the Law of Error will fall 



into the symbolic form expressed by the equation y = -^~e~ n2x \ 



X/TT 



we have rules at hand by which h may be determined. We 

 therefore, for the purposes in question, know all about the 

 curve of frequency: we can trace it out on. paper: given 

 one value, say the central one, we can determine any 

 other value at any distance from this. That is, knowing 

 how many men in a million, say, are 69 inches high, we can 

 determine without direct observation how many will be 67, 

 68, 70, 71, and so on. 



We can now adequately discuss the principal question of 

 logical interest before us; viz. why do we take averages or 

 means? What is the exact nature and amount of the ad- 



