Exploring data: Graphs and numerical summaries

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# 1.4.6: Measures of dispersion

During the above discussion of suitable numerical summaries for a typical value (measures of location), you may have noticed that it was not possible to make any kind of decision about the relative merits of the sample mean and median without introducing the notion of the extent of variation of the data. In practice, this means that the amount of information contained in these measures, when taken in isolation, is not sufficient to describe the appearance of the data. A more informative numerical summary is needed. In other words, if we are to be happy about replacing a full data set by a few summary numbers, we need some measure of the dispersion, sometimes called the spread, of observations.

The range is the difference between the smallest and largest data values. It is certainly the simplest measure of dispersion, but it can be misleading. The range of β endorphin concentrations for collapsed runners is 414−66=348, suggesting a fairly wide spread. However, omitting the value 414 reduces the range to 169−66=103. This sensitivity to a single data value suggests that the range is not a very reliable measure; a much more modest assessment of dispersion may be more appropriate. By its very nature, the range is always going to give prominence to outliers and therefore cannot sensibly be used in this way.

This example indicates the need for an alternative to the range as a measure of dispersion, and one which is not over-influenced by the presence of a few extreme values. In fact, we shall discuss in turn two different measures of dispersion: the interquartile range and the standard deviation.

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