Calculation of the Average and Its Standard Deviation

Sometimes an investigator has to determine a quantity that is a function of more than one random variable. In such cases, it is very important to know how to calculate the error of the complex quantity in terms of the errors of the individual random variables. This procedure is generally known as propagation of errors and is described in this section. 

It has already been mentioned that the x, s are determined experimentally, which means that average values Xj, X2, X3, , Xj are determined along with their standard errors o-j, r2. Two questions arise  

It is assumed that the function fix, . x ) can be expanded in a Taylor series around the averages X , =i.m - 

The term 0(x, — x,) includes all the terms of order higher than first, and it will be ignored. Thus, the function is written 

Equations 2.81 and 2.83 are the answers to questions 1 and 2 stated previously. They indicate, first, that the average of the function is calculated using the average values of the random variables and, second, that its standard error is given by Eq. 2.83. Equation 2.83 looks complicated, but fortunately, in most practical cases, the random variables are uncorrelated—i.e., p, = 0, and Eq. 2.83 reduces to 

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