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Appendix E: Basic Principles of Statistics
Pages 133-141

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From page 133...
... In addition, as discussed below, the logarithmic transformation yields more normally distributed data as well as transformed measureiNote that the notation used in this Appendix differs from that used in the body of the report.
From page 134...
... logarithmic transformation of the measurements is advisable. In what follows, we will assume that xi denotes the logarithm of the ith measurement on a given CS bullet and one particular element, ,UX denotes the mean of these log~measurement)
From page 135...
... Under those conditions, x-, y and sp are highly efficient estimates of fix, ,uy, and c,, respectively, where sp is a pooled estimate of the standard deviation that is based on both samples: sp = :[nX - list + (ny - 1)
From page 136...
... Replacing 1.334 with the quartiles 1.198, 1.288, 1.403, and 1.413 yields values of 1.892, 1.760, 1.616, and 1.604, respectively all smaller than the FBI value of 2. The FBI value of 2 would correspond to an approximate error of 0.03.
From page 137...
... Only a few of the standard deviations in the datasets were greater than 0.2 (see the section titled "Description of Data Sets" in Chapter 3~. The case of CABL differs from the classical situation primarily in the reversal of the null and alternative hypotheses of interest.
From page 138...
... DIGRESSION: LOGNORMAL DISTRIBUTIONS This section explains two benefits of transforming measurements via logarithms for the statistical analysis. The standard deviations of measurements made with inductively coupled plasma-optical emission spectroscopy are generally proportional to their means; hence, one typically refers to relative error, or coefficient of variation, sometimes expressed as a percentage, (sxlx)
From page 139...
... l I_ _ N em/ ,0 O I' .
From page 140...
... Thus, to obtain more-normally distributed data and as a by-product a simple calculation of the COV, the data should first be transformed via logarithms. Approximate confidence intervals are calculated in the log scale and then can be transformed back to the original scale via the antilogarithm, (eX-2SD ex-+2SD' DIGRESSION: ESTIMATING c,2 WITH POOLED VARIANCES The FBI protocol for statistical analysis estimates the variances of the triplicate measurements in each bullet with only three observations, which leads to highly variable estimates a range of a factor of 10, 20, or even more (%20902, %20 to 2~- Assuming that the measurement variation is the same for both the PS and CS bullets, the classical two-sample t statistic pools the variances into sp (Equation E.2)
From page 141...
... Thus, if a given standard deviation is <3 = 1.732 times larger than the pooled standard deviation for that element, one should consider remeasuring that element, in that the precision may be larger than expected by chance alone (5% of the time)


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