Standard deviation estimate

Method Standard deviation, % Estimated sample error, %... 

If optimization is aehieved using the proeesses ehosen, then the standard deviation estimated for eaeh eomponent toleranee ean be eompared to the required assembly standard deviation to see if overall eapability on the assembly toleranee has been aehieved. If exeessive variability is estimated at this stage on one or two eharaeteristies, then redesign will need to be performed. Guidanee for redesign ean be simplified by using sensitivity analysis, used to estimate the pereentage eontribution of the varianee of eaeh eomponent toleranee to the assembly varianee, where the varianee equals cr. It follows that from equation 3.3 ... 

The measures of dimensional variability from Conformability Analysis (CA) (as described in Chapters 2 and 3), specifically the Component Manufacturing Variability Risk, q, is useful in the allocation of tolerances and subsequent analysis of their distributions in probabilistic design. The value is determined from process capability maps for the manufacturing process and knowledge of the component s material and geometry compatibility with the process. In the specific case to the th component bilateral tolerance, it was shown in Chapter 3 that the standard deviation estimates were ... 

In Case a, only the standard deviation estimated from the experimental data for the test sample is needed, which is then used to normalize the difference Xmean f - The quotient difference/.5 c, the so-called Student s f, is compared with the critical tc for a chosen confidence level and f n - 1. (Use Eq. 1.13.)... 

Data for the carbon dioxide-methanol binary are available from Hong and Kobayashi (1988). The parameter values and their standard deviations estimated from the regression of these data are shown in Table 14.2. 

Standard Deviation estimate for the distribution of systematic errors (Sb2) ... 

FIG. 11.11 Electron-density difference maps on Li2BeF4 calculated with all reflections < sin 6/1 = 0.9 A"1 (81 K). (a) Based on the neutral atom procrystal model, (b) based on the ionic model. Contour levels are drawn at intervals of 0.045 eA"3.1 Full lines for positive density, dashed lines for negative and zero density. The standard deviation, estimated from [2Lff2(F0)]1/2N, is 0.015 eA-3. Source Seiler and Dunitz (1986). 

For the time being, let us use the standard deviation, estimated by any of the above techniques, as our measure of experimental error. We all know intuitively that as we run more tests, we get a better knowledge of the property we are measuring. We also know that the improvement in knowledge depends to some extent on the number of tests that we run What is the relationship involved ... 

The lower (LL) and upper (UL) acceptance limits for the sample mean are given for various values of the standard deviation of location means and the within-location standard deviation (SE). For our example, after rounding the standard deviation estimates up to the more restrictive values, the combination of 3.5 for the standard deviation of location means and SE of 3.9 is off the table, so this combination has too large a combination of standard deviations to pass the criterion. Therefore, the criterion fails. If the USP 25 tablet criterion were evaluated instead of the capsule criterion, this would be even more restrictive and would also fail the criterion. 

Here and throughout this paper, all uncertainties are standard uncertainties, i.e., one standard deviation estimates.) Assuming CPT invariance holds, we take... 

Fontenot and Schork [ 140] studied the miniemulsion polymerization of methyl methacrylate using hexadecane as the costabihzer. A portion of their results are shown in Table 4. Polydispersities are listed for macroemulsion, and miniemulsions subjected to varying durations of sonication, at two levels of initiator. In this and all cases following, the macroemulsions and miniemulsions were made from the same recipe, but with the costabihzer left out of the macroemulsion. The miniemulsions and macroemulsions were polymerized by the same procedure except that the sonication was ehminated for the macroemulsions. It may be seen that at both initiator levels, the macroemulsion is slightly more narrow than some of the miniemulsions, but broader than others. An estimate of the standard deviation of the polydispersity measurement is given as 0.01-0.02, and may be applied to all of the polydispersity data reported. With this standard deviation estimate, it may be seen that the differences in polydispersity between the macro- and miniemulsions are not hkely to be significant. 

The program automatically creates the list of peaks where the intensity is 3 times greater than the standard deviation estimated on the background noise and, for each peak, proposes one or more possible identifications. The analyst determines the most probable identification. The presence of lines of the same element, corresponding to different series, is one criteria for ascertaining the presence of this clement. 

To the statistician the process of sampling consists of drawing from a population a finite number of units to be examined. From sample statistics, such as mean and standard deviation, estimates are made of the population parameters. By appropriate tests of significance, confidence limits are placed on the estimates. Sampling for chemical analysis is an example of statistical sampling in that conclusions are drawn about the composition of a much larger bulk of material from an analysis of a limited sample. 

In a critical examination of pharmaceutical mixing, Train showed that samples of a random mixture of equal parts A and B must contain at least 800 particles if 997 out of every 1000 samples were to lie between 10%i of the stated composition, that is, the proportion, p, of A = 0.5 0.05, where a = 0.05/3. If limits of l%i were substituted, 90,000 particles must be present in each sample. The true standard deviation is given by a. The standard deviation estimated by the withdrawal of a number of samples is denoted as s. 

Fig. 4. Temperature dependence rfthe C spin-lattice relaxation time Tj in KjCao, plotted as T]T versus T. Error bars are standard deviations estimated from five Tj measurements at 298 K.

The resulting histogram for the diffusivity, shown in Figure 3.10, shows that the resulting distribution is not normal. The mean value for the calculated dijfusivity is Doj = —1.03 X 10 1.1 X 10 cmh, but neither the mean nor the standard deviation estimated for this distribution has statistical meaning. 

Figure 3.10 Distribution for the oxygen diffusion coefficient obtained by use of Monte Carlo simulations to assess propagation of errors for Example 3.1. The distribution is not normal, and the standard deviation estimated for such a distribution has no statistical meaning as a confidence interval.

Standard deviation and relative standard deviation estimated from ANOVA of 3 sets of 16 replicates (see text) b Standard deviation and relative standard deviation estimated from duplicate results (15 sets) for a range of concentrations and matrices (see text)... 

The precision of measurement is defined as the standard deviation of measurement (o). If o (for each experiment) is known or can be estimated then the standard deviation of the estimate of each coefficient ft, in the model can also be calculated or estimated. It can be determined whether the coefficient is statistically significant, that is whether there is a 95% probability that the absolute value of the effect is greater than zero. We will call the standard deviation estimation of the coefficient parameter ft, its standard error. 

Where t = the Student s t value appropriate for a 99% confidence level (one-sided) and a standard deviation estimate with n-1 degrees of freedom and s = the standard deviation of the replicate analyses of standards or samples with low concentration of analyte. Figure 5 depicts the MDL as an error distribution. As originally proposed, the procedure did not allow for blank correction. The latest version of the procedure does allow for blank correction, although the standard deviation of analyte response rather than the standard deviation of the blank Is used to calculate the detection limit. The distribution shown In Figure 5 Is that of the result (presumably after blank correction If necessary). 

Several problems arrise when this procedure is followed. Because the centroid of the calibration data is far removed from zero, the CL s diverge greatly near zero due to the long extrapolation associated with uncertainty in the slope. Furthermore, if the variance tends to Increase with concentration, the pooled standard deviation estimate is Inflated relative to the actual standard deviation near the DL. As a consequence, the estimated DL is frequently above the lowest standard. Bailey et al. (2i) reported recoveries up to 97 for spiked concentrations of... 

---