About the z-multiplier in total error calculations.

BACKGROUND

Total error (TE) in analytical measurement is calculated as systematic error (SE) plus z-times random error (RE). The z-multiplier is typically chosen at the 95% probability level, being 1.96 in the absence of SE is of considerable magnitude (one-sided 95% probability). Up to now, no SE/RE ratio dependent z-values have been considered. Here, we present z-values for SE/RE ratios ranging from 0 to 1.

METHODS

The z-multiplier (95% probability level) was empirically obtained by modulation of the standard normal distribution with the EXCEL NORMDIST function (Microsoft EXCEL 2002). In total, five distributions representing SE/RE ratios between 0 and 1 were calculated, so that the total probability outside a TE boundary +/- 1.96 sigma was in the order of 5.013%.

RESULTS

For distributions with SE/RE ratios ranging from 0 to 1 that satisfy the aforementioned total probability outside a TE boundary +/- 1.96 sigma, we found that the associated z-multipliers exhibit values from 1.96 (SE/RE = 0), 1.769 (SE/RE = 0.25), 1.68 (SE/RE = 0.5), 1.651 (SE/RE = 0.75) TO 1.645 (SE/RE = 1). The respective probability values beyond +/- 1.96 sigma were 2.5%/2.5%, 1.165%/3.849%, 0.368%/4.645%, 0.081%/4.934%, and 0.013%/5% for the SE/RE ratios of 0, 0.25, 0.5, 0.75, and 1.

CONCLUSIONS

The results show that at SE/RE ratios > 0.75 the one-sided 95% probability level is practically reached. The results allow a refined calculation of TE at specified SE/RE ratios and a general understanding of the relevance of two- and one-sided probabilities at different SE/RE ratios.