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118 Analytical Chemistry 2.0dIf s mb is not known, we can replace itwith s mb ; equation 4.24 then becomes( S )tsmb= S +nA DL mbYou can make similar adjustments to otherequations in this section.See, for example, Kirchner, C. J. “Estimationof Detection Limits for EnvironmentalAnalytical Procedures,” in Currie, L.A. (ed) Detection in Analytical Chemistry:Importance, Theory, and Practice; AmericanChemical Society: Washington, D.C., 1988.tive hypothesis is that the analyte is present in the sample. To detect theanalyte, its signal must exceed S mb by a suitable amount; thus,( S ) = S + zσ 4.24A DL mb mbwhere (S A ) DL is the analyte’s detection limit .The value we choose for z depends on our tolerance for reporting theanalyte’s concentration even though it is absent from the sample (a type1 error). Typically, z is set to three, which, from Appendix 3, correspondsto a probability, a, of 0.00135. As shown in Figure 4.16a, there is only a0.135% probability of detecting the analyte in a sample that actually isanalyte-free.A detection limit also is subject to a type 2 error in which we fail to findevidence for the analyte even though it is present in the sample. Consider,for example, the situation shown in Figure 4.16b where the signal for asample containing the analyte is exactly equal to (S A ) DL . In this case theprobability of a type 2 error is 50% because half of the signals arising fromsuch samples are below the detection limit. We will correctly detect the analyteat the IUPAC detection limit only half the time. The IUPAC definitionfor the detection limit indicates the smallest signal for which we can say, ata significance level of a, that an analyte is present in the sample. Failing todetect the analyte does not imply that it is not present in the sample.The detection limit is often represented, particularly when discussingpublic policy issues, as a distinct line separating detectable concentrationsType 2 Error = 50.0%Type 1 Error = 0.135%Type 1 Error = 0.135%(a)S mb(S A) DL= S mb+ 3σ mb(b)S mb(S A) DL= S mb+ 3σ mbFigure 4.16 Normal distribution curves showing the probability of type 1 and type 2 errors for the IUPACdetection limit. (a) The normal distribution curve for the method blank, with S mb = 0 and s mb = 1. Theminimum detectable signal for the analyte, (S A ) DL , has a type 1 error of 0.135%. (b) The normal distributioncurve for the analyte at its detection limit, (S A ) DL = 3, is superimposed on the normal distribution curvefor the method blank. The standard deviation for the analyte’s signal, s A , is 0.8, The area in green representsthe probability of a type 2 error, which is 50%. The inset shows, in blue, the probability of a type 1 error,which is 0.135%.