TREATMENT OF EXPERIMENTAL DATA 



23 



Fig. 9. A plot of some experimental data (o) ; their deviations (---) from 

 the mean, y; and their deviations (— -) from a line proposed to fit the data. 

 The extent to which x and y are said to be correlated is given by the extent 

 to which the line is a better representation of the data than is the value y. The 

 criterion for better representation is the ratio of the sum of the squares of the 

 deviation from the line to the sum of the squares of the deviations from the 

 mean, y. The quantitative use of the criterion is given in the text. 



Consider the ratio of these two variations: 



E(y — Vth) 2 



Z(y - y) 2 



If there were a perfect correlation between y and x — that is to say, if 

 the line fit the data perfectly — the numerator would be zero. If the line 

 is less than a perfect fit, the numerator becomes increasingly large until, 

 finally, if there is absolutely no connection between y and x, there will 

 be as much variation around the line as around the mean value of y. 

 Then the ratio will be unity. Thus the values of this ratio vary from 

 zero for a perfect correlation to unity for completely unrelated things. 

 Since we wish to speak of the extent to which things are "co-related," the 

 statisticians have chosen to use one minus this ratio as the index of 

 correlation, R. Then perfect relation is unity and complete lack of 

 relation is zero. 



R 



1 



E(y - V) 2 



There is one more complication to be taken care of. The way we 

 have chosen the relationship index, it is always positive. This presents 

 a problem, because it is possible for things to be negatively connected, in 

 that an increase in one variable correlates with a decrease in the other. 



