In the region close to the minimum of the objective function, , the system approximates to a linear least-squares system, for which
Therefore, the parameter values are (approximately) linear combinations of the observed data values and the errors on the parameters, , can be obtained by error propagation from the observations, , using the linear formula. Let the variance-covariance matrix for the observations be denoted by  and that of the parameters by . Then,
When , this simplifies to
In most cases the errors on the observations are un-correlated, so that  is diagonal.
If so, each weight should be the reciprocal of the variance of the corresponding observation. For example, in a potentiometric titration, the weight at a titration point, , can be given by
where  is the error in electrode potential or pH,  is the slope of the titration curve and  is the error on added volume.
When unit weights are used (, ) it is implied that the experimental errors are uncorrelated and all equal: , where  is known as the variance of an observation of unit weight, and  is an identity matrix. In this case  is approximated by
where  is the minimum value of the objective function and  and  are the number of data and parameters, respectively.
In all cases, the variance of the parameter  is given by  and the covariance between parameters  and  is given by . Standard deviation is the square root of variance. These error estimates reflect only random errors in the measurements. The true uncertainty in the parameters is larger due to the presence of systematic errors—which, by definition, cannot be quantified.
Note that even though the observations may be uncorrelated, the parameters are always correlated.