(This equation assumes the data are uncorrelated.) Since the dependence of on is not linear as Equation (13) suggests, due to the dependence of and on , evaluation of this expression is very complicated. The original version of this paper contained an error in the result of this calculation, and a corrected calculation has not yet been done. To first order, however, ignoring the nonlinearity one obtains the approximation

that is, the variances of the parameters are given simply by the diagonal elements of the inverse of the normal matrix defined in Equation (13). (The off-diagonal elements of this matrix are the covariances of the parameters.) For well-behaved data such as those used for illustration by York (1966), this approximation is good to within a few percent.

If the experimenter does not have standard errors for the measured quantities , but only relative uncertainties, the resulting fit is the same using these relative uncertainties, but the variances in the fitted parameters are given by expression (20) multiplied by , where is the number of degrees of freedom of the problem. If the errors are known a priori, then the goodness of fit can be inferred from the value of , which should be close to unity for normally distributed errors. This constitutes a test of the -component hypothesis as set forth in the introduction.