# Statistics for Environmental Engineers

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Figure 3.8 summarizes and compares the trickling filter data of Figure 3.7 by showing the average with error bars that are plus and minus two standard errors (the standard error is an estimate of the standard deviation of the average). This has some weaknesses. The standard error bars are symmetrical about the average, which may lead the viewer to assume that the data are also distributed symmetrically about the mean. Figure 3.7 showed that this is not the case. Also, Figure 3.8 makes the 12 trickling filters appear more different than Figure 3.7 does. This happens because in a few cases the averages are

FIGURE 3.7 Box-and-whisker plots to compare the performance of 12 identical trickling filters operating in parallel. Each panel summarizes 35 measurements.

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FIGURE 3.8 The trickling filter data of Figure 3.7 plotted to show the average, and plus and minus two standard errors.

strongly influenced by the few extreme values. If the purpose of using error bars is to show the empirical distributions of the data, consider using box plots. That is, Figure 3.8 is better for showing the precision with which the mean is estimated, but Figure 3.7 reveals more about the data.

Often, repeated observations of the dependent variable are made at the settings of the independent variable. In this case it is desirable that the plot show the average value of the replicate measured values and some indication of their precision or variation. This is done by plotting a symbol to locate the sample average and adding to it error bars to show statistical variation.

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