Standard deviation of slope

Standard deviation of slope Sj, Standard deviation of intercept ... 

Fig. 2. Compensation plot for cracking and related (see text) reactions on nickel (Table I, A). The error in the individual points is indicated by the size of each cross the line was calculated by the least squares method (Appendix II),and standard deviations of slope (erj and intercept (nB) are indicated,...

Standard deviation of slope Sb Standard deviation of intercept sa... 

Analytical ligures-of-merit are shown in Table 1. Calibration curves obtained over a 3-week period had only a 5% relative standard deviation of slope. This system served as an amperometric detector for HPLC with various citrate, phosphate buffers in the pH 2.1-3.0 range and with 0.1 M monochloroacetic acid in 1% acetonitrile at pH 3.0 as mobile phases. 

Standard deviation of slope 90.8772 Standard deviation of intercept 0.2927... 

The reader is asked to find the standard deviations of the slopes matrix in Problem 9 below. 

Eind the standard deviations of the slopes in matrix (3-78) for row 2, which refers to absorbances measured at 525 nm. 

A more useful representation of uncertainty is to consider the effect of indeterminate errors on the predicted slope and intercept. The standard deviation of the slope and intercept are given as... 

The test for the significance of a slope b is formally the same as a t-test (Section 1.5.2) if the confidence interval CI( ) includes zero, b cannot significantly differ from zero, thus ( = 0. If a horizontal line can be fitted between the plotted CL, the same interpretation applies, cf. Figures 2.6a-c. Note that si, corresponds to fx ean). that is, the standard deviation of a mean. In the above example the confidence interval clearly does not include zero this remains so even if a higher confidence level with t(f = 3, p = 0.001) = 12.92 is used. 

Results The uncertainties associated with the slopes are very different and n = H2, so that the pooled variance is roughly estimated as (V + V2)/2, see case c in Table 1.10 this gives a pooled standard deviation of 0.020 a simple r-test is performed to determine whether the slopes can be distinguished. (0.831 - 0.673)/0.020 = 7.9 is definitely larger than the critical /-value for p - 0.05 and / = 3 (3.182). Only a test for H[ t > tc makes sense, so a one-sided test must be used to estimate the probability of error, most likely of the order p = 0.001 or smaller. 

Data from several laboratories within the Interregional Research Project No. 4 (IR-4) in the USA have been evaluated for determining the values of MDL and MQL. These data have been presented in Table 1. The two-step procedure described in the EPA guideline was used to calculate the values of MDL and MQL. For the first step, the slope, intercept and RMSE values for the first three calibration curves of each study were separately calculated, then the IDL and IQL values calculated and the value of LQQ estimated for the method. These values were compared with the actual values of LLMV. The standard deviation of the spike recoveries at the LLMV (xllmv) was used to calculate the MDL and MQL. The values of LLMV were separately determined by the laboratory not using any of the methods described in this article. 

To construct the Hill plot (Figure 5.10E), it was assumed that fimax was 0.654 fmol/mg dry wt., the Scatchard value. The slope of the plot is 1.138 with a standard deviation of 0.12, so it would not be unreasonable to suppose % was indeed 1 and so consistent with a simple bimolecular interaction. Figure 5.10B shows a nonlinear least-squares fit of Eq. (5.3) to the specific binding data (giving all points equal weight). The least-squares estimates are 0.676 fmol/mg dry wt. for fimax and... 

The DL and QL for chromatographic analytical methods can be defined in terms of the signal-to-noise ratio, with values of 2 1-3 1 defining the DL and a value of 10 1 defining the QL. Alternatively, in terms of the ratio of the standard deviation of the blank response, the residual standard deviation of the calibration line, or the standard deviation of intercept (s) and slope (5) can be used [40, 42], where ... 

Figure 54-1, however, still shows a number of characteristics that reveal the behavior of derivatives. First of all, we note that the first derivative crosses the X-axis at the wavelength where the absorbance peak has a maximum, and has maximum values (both positive and negative) at the point of maximum slope of the absorbance bands. These characteristics, of course, reflect the definition of the derivative as a measure of the slope of the underlying curve. For Gaussian bands, the maxima of the first derivatives also correspond to the standard deviation of the underlying spectral curve. 

Phosphoric Acid. The 2nd-order rate method for analyzing the TGA data was statistically best (Table IV) for the cellulose/H PO samples. This suggests that the conclusions from a prior study which assumed a lst-order reaction (29) may need to be reexamined. While Wilkinson s approximation method gave high r values, the rate constant is determined by the intercept rather than the slope in this method. Thus, the standard deviation of the rates determined by Wilkinson s approximation method is still relatively high when compared to the other methods. In addition, the reaction order as determined by the Wilkinson approximation method was unrealistically high, ranging from 2.6 to 5.8. 

Several other useful values can be obtained with a complementary procedure, similar to MAXSLOPE except it calculates variances instead of regression slopes (see Grove Meehl, 1993). An investigator can use these values to calculate parameters of latent distributions (e.g., mean and standard deviation of the taxon). These estimates are not biased by nuisance correlations, unlike MAXCOV estimates. Unfortunately, these calculations are fairly arduous, so we will not describe them here (for more details see Grove Meehl, 1993). 

From this the standard deviations of the slope, sm, and intercept, Sb, are... 

Also determine the slope and intercept of the least-squares line for this set of data. Determine the concentration and standard deviation of an analyte that has an absorbance of 0.335. 

The beauty of this completely random approach to the analyte detection limit is the direct applicability of the statistical hypothesis testing formalism. Also, long-term trends in calibration slope or backgrounds have little influence. One important assumption is made that the form of the calibration curve [Equation 2c] is fixed. Also, a subtle change has occurred, the operation is no longer linear, with A in the denominator. Thus, the distribution of x is only asymptotically normal, as the relative standard deviation of becomes smaller. 

The simplified term (standard deviation of the residuals divided by the estimated slope of the calibration line), the procedural standard deviation, will often be accepted as a measure of precision. 

Depending on method type or intent, the LOD or LOQ may need to be determined. ICH guidelines describe several approaches and allow alternative approaches, if scientifically justifiable. Suggested approaches include calculation based on signal-to-noise ratio (typically set at 3 1 for LOD and 10 1 for LOQ) standard deviation of the response and slope standard deviation of the response of a blank, or a calibration curve. 

The LOQ can be expressed as lOtr/S, where tr is the standard deviation of the response and S is the slope of the calibration curve. The LOQ and the method used for determining the LOQ should be presented. The limit should subsequently be validated by the analysis of a suitable number of samples known to be near or prepared at the LOQ. 

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