MEASUREMENTS AND THEIR VARIATION 3 



information about the deviations, we cannot use the average deviation, 

 since a set of measurements with large deviations from the average will 

 appear equivalent to a set with small deviations, in that both have an 

 average deviation of zero. It is plausible that we are really interested in 

 the absolute values of the deviations as a measure of the accuracy and 

 reproducibility of our measurements. Now, it is not simple mathemati- 

 cally to work with absolute values. Therefore statisticians have used the 

 least complicated way which is almost equivalent to using the absolute 

 values: they use the squares of the values of the deviations of the indi- 

 vidual measurements from the average. Since these are always positive, 

 the sum of these squared deviations is also always positive, and the 

 quality of the set of measurements can be inferred from the smallness of 

 the sum of the squared deviations. 



Now, however, we must ask what average must be taken to minimize 

 the sum of the squared deviations; this value of a will be the one around 

 which the individual measurements cluster according to our proposed 

 criterion of least-squared deviations. 



For those who cannot use the differential calculus, the result is given 

 a few lines below. For those who can use the calculus, the derivation 

 follows. 



We wish to minimize the expression for the total variation: 



2 d 2 i = J2 ( x i ~ «) 2 = (*i - «) 2 + ( x 2 - a) 2 + • • • + (x» - a) 2 . 



i i 



To minimize this expression we differentiate with respect to a, keeping 

 everything else constant, and set the result equal to zero: 



2(.n - o)(-l) + 2(x 2 - o)(-l) + • • • + 2{x n - o)(-l) 



= 2£(s< - o)(-l) = 0. 



i 



Dividing by 2 and collecting terms, we find 



— Oi + x 2 H + x n ) + na = 0. 



Thus, finally, solving for a, we obtain 



•>'l + X 2 + • • • + Xn 2~2iX{ 



a = 



n n 



We have thereby shown that the value of a which minimizes the sum 

 of the squared deviations is again just the arithmetic mean. 



We need to introduce a few new terms at this point. First, V, the 

 average variation is simply the total variation divided by the number of 

 independent measurements taken: 



v _ . E i di 2 _ £ f (xi — a) 2 

 n n 



