# Statistics for Environmental Engineers

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Biological counts usually need to be transformed to make the variance uniform over the observed range of values. The paired analysis will be done on the differences between inlet and outlet, so it is the variance of these differences that should be examined. The differences are plotted in Figure 17.3. Clearly, the differences are larger when the counts are larger, which means that the variance is not constant over the range of population counts observed. Constant variance is one condition of the f-test because we want each observation to contribute in equal weight to the analysis. Any statistics computed from these data would be dominated by the large differences of the high population counts and it would be misleading to construct a confidence interval or test a null hypothesis using the data in their original form.

A transformation is needed to make the variance constant over the ten-fold range of the counts in the sample. A square-root transformation is often used on biological counts (Sokal and Rohlf, 1969), but for these data a log transformation seemed to be better. The bottom section of Figure 17.3 shows that the differences of the log-transformed data are reasonably uniform over the range of the transformed values.

Table 17.2 shows the data, the transformed data [z = ln(y)], and the paired differences. The average difference of ln(in) — ln(out) is d = X din /17 = -0.051. The variance of the differences is s2 = X(d;d )2/ 16 = 0.014 and the standard error of average difference ss = s /JY7 = 0.029.

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