Thursday, 26 February 2004

Central tendency

Vance of Begging to Differ takes issue with’s claim that the Bush administration’s claim that the average tax cut is $1,586 is “misleading,” because using the mean instead of the median† is improper. Vance writes:

I can think of a valid justification for either measure. If you’re trying to understand the overall economic effects of the tax cuts, for example, an average is entirely applicable.

In the case where data is “normally distributed”—following the “bell curve” known to statisticians—the mean and the median are essentially the same.* When they differ, the data is said to be skewed, and measures of central tendency and dispersion that assume a normal distribution (like the mean) are generally misleading, as they don’t properly describe the distribution. The income distribution, for example, is skewed right.‡

To cast things in non-mathematical terms, when people think about averages they are thinking in terms of things that are most typical, rather than in terms of distributions. And, in general, the median better reflects this perception of average than the mean. While there may be technical value to the mean for specialists and those who want to engage in further analysis, I think the median does a better job of reflecting the “most typical” observation in most data patterns.

† For non-math geeks: the “mean” is the most commonly used “average”; normally, it refers to the arithmetic mean, which is simply the sum of all the observations divided by the number of observations. The “median” is the number you’d find if you sorted all of the observations and took the middle one; if there are an even number of observations, you take the two observations closest to the middle and take the mean of them. Statisticians call averages “measures of central tendency,” because they like to use complex terms for simple concepts. (Another measure of central tendency is the “mode”; it is the observation that is most frequently observed.)
* This also applies to some other symmetric distributions, like the logit and t distributions; it does not (generally) apply to distributions like the Poisson, F, or χ² (chi-square).
‡ Skewness is described by the side with the longer tail.