Geometric means are widely used in a variety of scientific disciplines. They are the natural parameter of interest for a lognormal random variable because a ratio of lognormal random variables has a known lognormal distribution, and the geometric mean of a lognormal ratio is equal to the ratio of the individual geometric means. Some common uses for the geometric mean with survey data include estimating average population growth rates, bacterial contamination rates, and chemical concentration rates. At present, none of the SAS/STAT ® survey procedures directly computes geometric means. However, with SAS/STAT software and a little programming, you can easily estimate a geometric mean and its variance from sample survey data. The SAS source code for this example is available as an attachment in a text file. In Adobe Acrobat, right-click the icon in the margin and select Save Embedded File to Disk. You can also doubleclick to open the file immediately. Analysis Following Wolter (1985), suppose NY denotes the population mean of a characteristic y and Ny denotes an estimator of NY based on a sample of fixed size n. The natural estimator for the exponential function D e NY is O D e Ny Suppose v. Ny / denotes an estimator of the variance of Ny that is appropriate to the particular sampling design. Then, the Taylor series estimator of variance is v. O / D e 2 Ny v. Ny/ These results can be applied directly to the problem of estimating the geometric mean of a finite population characteristic x because the geometric mean is the exponentiation of the mean of the natural logarithm. That is, the geometric mean can be expressed as a finite population quantity by12
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