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About the z-multiplier in total error calculations
-
Dietmar Stöckl
Published/Copyright:
November 1, 2008
Abstract
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 (two-sided 95% probability) and 1.645 when 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 σ 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 σ, 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 σ 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.
Clin Chem Lab Med 2008;46:1648–9.
Keywords: one-sided 95% probability; ratio between systematic and random error; two-sided 95% probability
Received: 2008-5-28
Accepted: 2008-7-15
Published Online: 2008-11-01
Published in Print: 2008-11-01
©2008 by Walter de Gruyter Berlin New York
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Keywords for this article
one-sided 95% probability;
ratio between systematic and random error;
two-sided 95% probability
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