# Statistics for Environmental Engineers

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Authors often fail to tell the reader what the error bars represent. Error bars can convey several possibilities: (1) sample standard deviation, (2) an estimate of the standard deviation (standard error) of the statistical quantity, or (3) a confidence interval. Whichever is used, the meaning of the error bars must be clear or the author will introduce confusion when the intent is to clarify. The text and the label of the graph should state clearly what the error bars mean; for example,

• The error bars show plus and minus one sample standard deviation.

• The error bars show plus and minus an estimate of the standard deviation (or one standard error) of the statistic that is graphed.

•    The error bars show a confidence interval for the parameter that is graphed.

If the error bars are intended to show the precision of the average of replicate values, one can plot the standard error or a confidence interval. This has weaknesses as well. Bars marking the sample standard deviation are symmetrical above and below the average, which tends to imply that the data are also distributed symmetrically about the mean. This is somewhat less a problem if the errors bars represent standard errors because averages of replicates do tend to be normally distributed (and symmetrical). Nevertheless, it is better to show confidence intervals. If all plotted averages were based on the same number of observations, one-standard-error bars would convey an approximate 68% confidence interval. This is not a particularly interesting interval. If the averages are calculated from different numbers of values, the confidence intervals would be different multiples of the standard error bars (according to the appropriate degrees of freedom of the f-distribution). Cleveland (1994) suggests two-tiered error bars. The inner error bars would show the 50% confidence interval, a middle range analogous to the box of a box plot. The outer of the two-tiered error bars would reflect the 95% confidence interval.

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