AND UNEQUAL VARIANCE
Serious loss of power may result with nonnormal distributions. If all distributions were normal, the mean and standard deviation of Chapter 2 would be ideal statistics. This point may be emphasized by comparing mean and median. Both are equal in a normal population, and either could be used to estimate central tendency from a sample. But statistical theory shows that the median is more variable from sample to sample. The sample mean has a shorter confidence interval, in other words, one reason for preferring it over the median.
With a nonnormal population, in contrast, the sample mean may be more variable than the median. Even one or two extreme scores can have crippling effects on the confidence interval that represents the mean.
The first line of protection against extreme scores lies in experimental procedure. This extrastatistical issue deserves emphasis before we proceed to statistical techniques. Your expectation about the shape of the data is important in planning your experiment. Small changes in experimental procedure sometimes markedly reduce the likelihood of extreme scores. And it goes without saying that the data should always be inspected for extreme scores.
Of the statistical procedures for dealing with extreme scores, the trimming method of Section 12.1 may be far and away the best. Although as yet little used, it appears to be safe and effective.
Three other ways to deal with nonnormality have been developed by statisticians. Outlier rejection techniques have specialized applicability and do not seem robust (Section 12.3). Transformation to make the data more normal is sometimes helpful, although not a general purpose tool (Section 12.4).
A different tack is taken with distribution-free tests, which handle extreme scores by reducing the data to ranks. Hence no assumption is needed about the specific shape of the population (Section 12.2). Distribution-free tests are largely limited to one-way designs, however, for which it may be preferable to apply regular Anova to the ranks themselves.