Medians and Resampling

Abstract

Classical statistical theory provides powerful tools for estimating population means from samples and for establishing error ranges for desired confidence levels, and these were presented in Chapters 8 and 9. If outliers in a sample interfere with using the mean, the same tools can be applied to estimate the trimmed mean. If the asymmetrical shape of a sample (skewness) interferes with using the mean, transformations can be applied as a correction, and this makes it possible to proceed with significance testing, as we will see in Chapters 11–15. While this works fine for significance testing, it puts the measurements on a scale that is not intuitive and makes it difficult to talk about them straightforwardly. It would just not be at all easy to talk meaningfully about estimates of, say, the mean logarithm of site area in two periods. The median may be a more useful index of center in such a case, and the best estimate of the median in a population is the median in the sample. There is, however, no abstract theoretical basis for establishing error ranges for this estimated median at any particular confidence level because there is no theoretical way to determine the center, spread, or shape of the special batch (or sampling distribution) of the median as there is for the mean. The contribution of exploratory data analysis to this difficulty was to recognize that the special batch can be approximated by resampling, or repeatedly selecting samples from the sample itself.