4 THE MATHEMATICAL TREATMENT OF DATA 



This average variation is called the variance, and we shall use the 

 term, even though it seems unnecessary to lose the directly given de- 

 scription inherent in the longer name. 



Now, we could specify the results of our set of n measurements by 

 saying they yield the estimate of the true value to be a ± V. This seems 

 somewhat silly, for it would mean, for instance, that we would give the 

 length of something as 5.7 cm ± 2.3 cm 2 , and it is silly to talk of squared 

 centimeters as expressing the variation in the measurement of centi- 

 meters. It would appear more reasonable to take the square root of the 

 variance, so that the average and its variation would then have the same 

 dimensions. Thus, in the illustration above, we would say 5.7 cm ±1.5 

 cm, and this statement has a sensible ring to it. 



Adopting this change, we need to name the new quantity — the square 

 root of the variance — and it is called the standard deviation. It is usually 

 denoted by s, or by the equivalent Greek letter cr (sigma). So the results 

 of a set of measurements are summarized as a ± s, where 



a = > 



n 



and 



s = 



\ Li U'j — a )' 



n 



Now that we have this formidable looking apparatus, we have to ask 

 what it means. Of what use is the averaging procedure? What signifi- 

 cance attaches to this standard deviation? The procedures of statistics 

 supply answers to these questions, and the answer is based upon finding 

 that sets of n measurements do not exhibit many different kinds of 

 clustering around the average a but that, surprisingly, only two kinds of 

 clustering are actually obtained. To be concrete, if we plot n it the num- 

 ber of times we get the measurement x i} we obtain only two different 

 curves, as sketched in Fig. 1. In practice, we divide n t by the total 

 number of measurements to obtain the fraction of measurements which 

 yield the result x t . This fraction is called the frequency, f i} of a par- 

 ticular measurement, and it is / { which is indicated in the figure. 



Note that these curves differ in that one is symmetrical, the other 

 asymmetrical. Now, it has been possible to derive mathematical ex- 

 pressions for frequency curves under assumptions to be set forth later. 

 It turns out that there are basically two types of curves, and the two 

 types, when graphed, look very much like the two empirically obtained 

 curves. There is a famous quotation to the effect that mathematicians 

 believe their theoretical curves are relevant because they have been 

 shown to exist in nature, while experimental scientists believe the experi- 



