Error Analysis Using the Variance-Covariance Matrix

The variance-covariance matrix for a linear fit to y = b + mx is defined, and its use in computing the errors for derived quantities from the fit is illustrated. The matrix formalism is used to derive analytical expressions for the vertical and horizontal error bands, calibration curve error bands, and the error in the area under the regression line. Although most spreadsheet regression routines do not make the complete variance-covariance matrix available, the missing covariance matrix element can be computed easily. An Excel spreadsheet is presented that uses the matrix formalism to compute the error in analyte concentration from a fit of standard additions data.