As it stands, the above expression takes into account the effect of the algebraic sign of the changes . Such a relation can be used to correct R for the effects of known systematic errors, but is not practical when the errors are random in nature. This is because the standard deviation calculated in the case of random errors has no sign associated with it. For the case of random errors an expression for the error associated with R can be derived as follows. To simplify the algebra, only two independent variables x and y will be considered.
All summations are from for the case of n measurements.
First calculate the variance of R . This means that on the average the product is just as likely to be positive as negative, and the sum on the extreme right will be zero. The first two summations are in fact the variances in x and y respectively. In the general case of more than two variables similar terms may be added for the other variables. Here the partial derivatives are to be evaluated using the mean values of the independent variables.