median

The middle value once the numbers are put in order, half above and half below. Papers report it rather than an average when extreme values would drag an average away from the typical case.

The NIST/SEMATECH e-Handbook of Statistical Methods, the reference work maintained by the United States National Institute of Standards and Technology with the SEMATECH consortium, defines the median as the value of the point which has half the data smaller than that point and half the data larger than that point. Computing one means sorting the values and reading off the middle: with an odd number of observations that is the single value at position n plus 1, divided by two; with an even number it is the average of the two values lying either side of the middle. The same handbook's treatment of percentiles says it a second way. The pth percentile is a value such that at most 100p percent of the measurements are less than it, and the 50th percentile is called the median.

What separates a median from an average is what each does with extreme values. The handbook states that extreme values in the tails distort the mean, and that these extreme values do not distort the median since the median is based on ranks. A median asks only which value sits in the middle of the order; how far away the largest one lies does not enter the arithmetic. Where a distribution is lopsided the two measures part company, and the handbook says which way: the mean and median are not the same, and the mean will be pulled in the direction of the skewness.

Two things follow for reading a study. A median says where the middle of the order is and nothing about the spread around it, so a median of half an hour a night is equally consistent with a group in which almost everyone logged about half an hour and with one in which some logged two minutes and others two hours. The spread has to be stated separately, which is what an interquartile range or a standard deviation is for. And a median follow-up of 7.9 years does not mean the study watched anyone for 7.9 years in particular: it means half the participants were followed for longer than that and half for less. Where a paper reports a median in place of a mean, the shape of the data is usually the reason, and by the handbook's account the figure it prints is one that a handful of unusual participants cannot move.

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