Sec. 2.1 ARITHMETIC MEAN AND STANDARD DEVIATION 15 



The truth of the general theorem that the sum of the algebraic values 

 of the Xi always is zero is established as follows. By definition and 

 simple algebra, 2x = %(X - /*) = SX - SO*) ; but SX = Nfi, and 

 S(/u.) also = Nfi because this symbol requires that we add N terms 

 obtained by letting i have values from 1 to N, inclusive. The fi stays 

 constant for each i; therefore, SO*) = Nfi. Since Sz = A/"/* — Nfi, it 

 is always equal to zero, as was to be shown. 



As a consequence of the truth of the above theorem, a measure 

 of the variation about the arithmetic mean cannot be based upon the 

 algebraic sum of the Xi. Therefore, one of two actions should be 

 taken: (a) Ignore the signs of the x t and obtain their mean there- 

 after. Or (6), find some other relatively simple function of the Xi 

 which has more of the desirable properties (2.11) to (2.15) than are 

 obtained by method a. The latter procedure has proved to be the 

 more successful and therefore will be considered first. As a matter 

 of fact, it involves a function of the squared deviations, X?. 



The quantity 



(2.12) a = Vx(xi 2 )/N , 



where <r, the Greek letter sigma, has been found by statisticians to be 

 a good measure of the variability of a set of numerical measurements 

 about their arithmetic mean. Just why it should be so useful cannot 

 be shown to the student at this time, but it does have more of the 

 desirable properties of measures of variation than any other such 

 measure which has yet been devised. The quantity defined by 

 formula 2.12 is called the standard deviation of the Xi about their 

 mean, fi. It would be zero if all the Xi were equal; the more dis- 

 persed they are about the mean, the larger the standard deviation 

 tends to be. For example, consider the weights of problem 2.11 and 

 of Set 1. The former obviously are more dispersed and generally 

 more variable than the latter. The two standard deviations are 18.0 

 and 2.4, respectively, which certainly is a concise way to point out 

 that, although the mean weights of the two squads are the same, 

 their dispersions about that mean are far from the same. 



The square of the standard deviation, a 2 , is called the variance 

 of the Xi about fi. There are some relatively advanced statistical 

 procedures in which it is preferable to work with the variance in- 

 stead of the standard deviation, but the latter will be used most of 

 the time in this book. 



From the definition of o- contained in formula 2.12 it appears that 

 each Xi must be calculated and squared, but such is not the case. If 



