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Figure 4. When measurements are made for a sample, the χ values ob tained should follow normal distribu tion around a mean value. If a sample were measured to have a mean at the XL value, the distribution of these χ values around XL would resemble the blue line in Figure 4. One-half of the time the measurement would fall below the XL value and could not be considered a true signal according to the IUPAC definition. To avoid this high probability, the limit of xi is set at 3si away from XL- This level is cho sen so the areas a and β are equal. If si = SB then this level is 3SB away from XL or 6SB away from XB. By using k = 3, the area of the χι distribution curve below XL, β, is no less than 0.0013. Thus, there is a 0.13% chance that an χ value measured at χι would fall below the XL limit and not be con sidered as a true signal. This idea of further statistically separating the blank measurements distributions and true signal distribu tions has been proposed by the ACS Subcommittee on Environmental An alytical Chemistry and has been termed the limit of quantitation (2). Since the numerical significance of the analyte concentration increases as the analyte signal increases above XL, a minimum criterion, representing the ability to quantify the sample, can be established reasonably far way from XB· This criterion, called the limit of quantification (LOQ), is 10σ away from XB. For limit of detection work, σ = SB- Samples that are measured as having a signal, x, where χ > IOSB are termed to be in the region of quantita tion while samples where 3SB ί χ ί IOSB are termed to be in the region of
detection. By setting the quantitation level as IOSB or the identification limit as 6SB,
a much higher probability is afforded that the sample signal is not just a random fluctuation of the blank. However, when making comparisons using LOQ or ci to IUPAC CL values, the analyst must bear in mind the dif ference in the k factors for each limit.
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Analytical Sensitivity Error The previous models for calculating detection limits consider the error in the blank measurements. These mod els also consider the analytical sensi tivity, m, as a well-defined value. In practice, however, m may have signifi cant error due to nonlinearity in the calibration curve, or measurement er rors. The following proposed detection limit approaches include errors associ ated with measurements of the analyt ical sensitivity. The first method, a graphical approach, includes the stan-
718 A · ANALYTICAL CHEMISTRY, VOL. 55, NO. 7, JUNE 1983
Figure 5. Analytical calibration curve of signal, x, vs. concentration, c, show ing graphical approach to limit of detec tion calculation White dashed lines are the limit of error bars. c L is obtained from x L if no error occurs in slope. CR is obtained from x L if slope error is significant
dard deviation of the slope, s m , in the CL expression. The second method, a propagation of errors approach, con siders the standard deviation of the concentration, sc. This value is calcu lated by including the standard devia tions of the blank, slope, and intercept in the equation. The statistical expres sions for these values are listed in Table I. Although these models require ad ditional calculations, most linear re gression analyses are performed using calculators or microcomputers. With additional programming, these calcu lations can be easily performed, allow ing more accurate determinations of CL values to be made. These values may also be used for a truer compara tive look at the ability of an analytical method or instrument to quantify trace elements (or compounds) in a sample.
Graphical Approach
To obtain a more reliable CL value, the m value should be expressed as a confidence interval m ± t„s m , where s m is the standard deviation of the slope and t„ is a t distribution value chosen for the desired confidence level, a, and the degrees of freedom, υ. The insertion of this interval into Equation 7 produces ks c L = —— B m ± t„s m
ηηΛ (I")
The effect of the inclusion of the con fidence interval can best be seen by referring to Figure 5. The error bars (confidence interval) generated around the regression line are indicat ed as white dashed lines. Because of error in the slope, three concentration
