APPLICATIONS OF STATISTICAL ANALYSIS 



17 



The chief use of chi squared is in determining the ratio for a whole set 

 of different points with different theoretical values. For instance, if we 

 wish to compare an experimental set of values with any theoretical 

 curve, we use chi squared. In this instance, the variance in the denom- 

 inator is replaced by the mean value predicted by the curve, since the 

 Poisson Distribution afforded us the insight that the mean is equal to 

 the variance. The numerator remains n times the experimental variance 

 even though in application of chi squared there is usually only one 

 measurement made at each point, so that, usually, n = 1. Next, we 

 sum the expression for all the points of the curve at which experimental 

 values were taken. This sum over the various points of the curve is 

 indicated by a subscript p to distinguish it from the sum of deviations at 

 individual points. This sum of chi squareds is then written as 



= E 



(x p — a p y 



Since, as we have remarked, the means are good estimates of the theo- 

 retical values at each point, the expression is more usually written as 



= E 



(* 



exp 



•r t h) : 



Zth 



This is the formula given in most textbooks. 



Since the value of chi squared at each point should be unity, on the 

 average, the sum should total simply to the number of independent 

 measurements. Indeed, if we look at tables for chi squared, we find that 

 the values are very closely equal to the number of independent measure- 

 ments at a probability of occurrence equal to Va- The following table 

 will assist us in demonstrating the use of chi squared. 



As an illustration, we compare some experimental data with a straight- 

 line formula to find whether it can represent the data adequately. 



