2 THE MATHEMATICAL TREATMENT OF DATA 



cluster around the "true" value, measurements will avail us nothing. 

 Thus we are led to the assumption that the value around which the results 

 cluster is near to the "true" value; indeed, if we think about the problem 

 carefully, we come to the realization that we have no way of ever know- 

 ing the "true" value. So, we define the "true" value as the value around 

 which measurements do, in fact, cluster. Of course, we now need to 

 specify a means of extracting the true value (we henceforth omit the 

 quotation marks, which, nevertheless, are still there so far as knowing 

 the "truth" is concerned). If we could make an infinite number of 

 measurements, the value which recurred most frequently would be taken 

 as the true value. What do we do about the fact that we normally take 

 only a few measurements and essentially never take a number even 

 remotely resembling a very large number? How do we extract an esti- 

 mate of the true value from a small, finite set of n measurements? The 

 answer involves an analysis of kinds of errors. 



In principle there are two kinds of errors in measurements. First, there 

 are the so-called systematic errors — such as those resulting from the use 

 of an inaccurate scale. This kind of error source is eliminated only 

 through painstaking examination of the measurements themselves and 

 analysis of the machines and procedures involved. We have no way of 

 being sure that all such errors have been eliminated ; we hope for the best. 



The second kind is the chance error with which biometry is concerned. 

 We assume that these chance errors are just as likely to yield measure- 

 ments that are higher than the true value, as they are to yield measure- 

 ments that are lower than the true value. On this assumption, the devia- 

 tions, d i (=x i —a), of the individual measurements, x h from the true 

 value, a, should total zero. This condition can be stated algebraically 

 and we now show how this enables us to determine the value of a from 

 a given set of measurements. 



J] di = d x + d 2 + d 3 + • • ' + d n = Yl ( x i ~ a ) 



i i 



= xi + -r 2 + x 3 + • • • + x n — na. 

 Setting the sum of the deviations equal to zero gives 



= x-i + x 2 + x 3 + • • • + x n — na 

 or 



Xi + X 2 + X 3 + • • • -f X n Hi Xi 



a = 



n n 



This condition leads directly to the choice of the arithmetic mean as the 

 desired average. With this choice, the average deviation is zero, since 

 there must be as much positive deviation as negative deviation. To get 



