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• #### Appendix K: Statistical Analysis of Bullet Lead Data by Karen Kafadar and Clifford Spiegelman 169-214

ments with constant variance. That is, an estimate of log(µxk) is the logarithm of the sample average of the three measurements on bullet k, and a plot of these log(averages) shows more normally distributed values than a plot of the averages alone. We denote the variances of and as and and the variances of the error terms and as and respectively. It is likely that the between-bullet variation is the same for the populations of both the CS and the PS bullets; therefore, since should be the same as we will denote the between-bullet variances as Similarly, if the measurements on both the CS and PS bullets were taken at the same time, their errors should also have the same variances; we will denote this within-bullet variance as or σ2 when we are concentrating on just the within-bullet (measurement) variability.

Thus, for three reasons—the nature of the error in chemical measurements, the approximate normality of the distributions, and the more constant variance (that is, the variance is not a function of the magnitude of the measurement itself)—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, µx denotes the mean of these log(measurement) values, and εi denotes the error in this ith measurement. Similarly, let yi denote the logarithm of the ith measurement on a given PS bullet and the same element, µy denote the mean of these log(measurement) values, and ηi denote the error in this ith measurement.

### NORMAL (GAUSSIAN) MODEL FOR MEASUREMENT ERROR

All measurements are subject to measurement error:

Ideally, εi and πi are small, but in all instances they are unknown from measured replicate to replicate. If the measurement technique is unbiased, we expect the mean of the measurement errors to be zero. Let and denote the measurement errors’ variances. Because µx and µy are assumed to be constant, and hence have variance 0, and The distribution of measurement errors is often (not always) assumed to be normal (Gaussian). That assumption is often the basis of a convenient model for the measurements and implies that

(E.1)

if µx and σx are known (and likewise for yi, using µy and σy). (The value 1.96 is often conveniently rounded to 2.) Moreover, will also be normally

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