20 THE MATHEMATICAL TREATMENT OF DATA 



TREATMENT OF EXPERIMENTAL DATA 



1 . Data-fitting using the least-squares principle 



The least-squares principle was used to obtain an index of the varia- 

 tion in data. It has other uses, one of which is the subject of this sec- 

 tion. The problem of finding a theoretical expression to fit data utilizes 

 this principle to choose the parameters of the curve so as to minimize 

 the variation around the curve. We illustrate the approach by finding the 

 expressions for the parameters of a straight line which is to be fitted 

 to the data. 



The equation of a straight line is 



y = a + bx. 



Consider the difference between the theoretically and the experimentally 

 determined y: 



2/th — */ex P = a + bx — y QXp . 



These differences are sketched in Fig. 8. 



Our principle of least squares tells us to square this difference, add all 

 the squared differences for all the points, and then to choose a and b so 

 as to minimize the sum. For those with mathematical facility, the deri- 

 vation of a and 6 will be presented. For other readers, the results are 

 given below. 



!lth = (i+bx 



Fig. 8. A plot of some experimental data relating y and x and a theoretical 

 line whose slope and intercept are to be chosen to minimize the total deviation 

 of the theoretical values of y from the experimental values of y. The short 

 vertical lines are the deviations themselves: ?/ th — ?/ exp . 



