18 



THE MATHEMATICAL TREATMENT OF DATA 



Fig. 7. A plot of some experimental data and a line which is to be tested by 

 the chi-squared criterion for its goodness of fit to the data. 



The plot of the data and the theoretical curve are shown in Fig. 7. 

 The theoretical curve was derived by methods to be discussed in a later 

 section. The equation is y th = —0.1 + 1.3.r. The values of y exp and y th 

 at the experimental values of x are 



x = 1 2 3 4 5 

 2/th = 1.2 2.5 3.8 5.1 6.4 

 z/ exp =21547 



The deviations are, for the straight line, 



(2 - 1.2), (1 - 2.5), (5 - - 3.8), (4 - - 5.1), and (7 - - 6.4). 

 Their chi squared is then given by 



X' 



0.64 . 2.25 . 1.44 



1.21 0.36 



1.2 



= 0.533 



= 2.613 



2.5 3.8 ' 5.1 6.4 



0.900 + 0.379 + 0.237 + 0.564 



There are five measurements, and we have chosen the two parameters 

 of the straight line (the intercept on the y-axis and the slope of the line) 

 to make a best fit, so there arc only three independent measurements. 



Looking at the chi-squared table, we see that for three independent 

 measurements we could get a chi squared of 3.3 or more in one of each 



