Average vs. average

I have run across people who, when calculating a mean, will discard their two or three highest and two or three lowest pieces of data and calculate the mean for the rest of their data. What they want to do is protect themselves against the effects of skew, specifically the distortion of a mean by a few extreme scores.

That probably doesn't hurt, but there is a simpler and much more effective way of dealing with this problem – use the median. The median is the score that is midway between the highest and the lowest. In other words it is the true average of your set of data (the mean is an estimate of the median). So use the MEDIAN function in your spreadsheet rather than the MEAN function.

There are some exceptions to this rule, though. If you're using your data to estimate a total – the total value of donations to an organization, for example – you'd use the mean. If you want to compare two sets of data with a statistical test you would usually be better off to use the mean.

And if the SKEWNESS function in your spreadsheet provides a skewness coefficient for your set of data that is higher than -1.00 and less than 1.00 you normally don't worry about this at all.

The TRUTH about means and medians

One of the many bees I have in my bonnet is buzzing about how people talk about the mean and the median. I just read a research report in which the mean was described as the average and the median was just called the median. It was a pretty good report, so I suspect the author was trying to help her non-statistically-trained readers. I still think this can be misleading, though.

The mean of a set of scores is simply the result of adding them up and dividing by the number of scores. The median, on the other hand, is the score which has equal numbers of other scores above and below it.

The average of a set of scores is its midpoint. That is, the median is the average. The mean is an estimate of the median. We use it because it can be manipulated more effectively and profitably than the median can, usually without affecting the validity of conclusions.

Most people know that skew may make the mean inaccurate. However, most do not know that there are criteria for deciding if use of the mean should be reconsidered. I reconsider if the skewness coefficient (which is provided by spreadsheets as well as statistical software) is greater than 1 or less than -1.

Above all, do not do what some people do and throw out your highest and lowest scores, or the two highest and two lowest etc., as a protection against skew. While that probably does little harm, it doesn't help either. If you're using your data for descriptive purposes, the best solution is to use the median. That way you get the benefit of all your data.

If you're using the data for inferential purposes, you should of course be using statistical tests. I suggest you compare the result of a test of differences between means with the result of a test of differences between medians (this is often a good idea even with unskewed data). If you're comparing either means or medians without a test, you are wasting your time. You need to know how likely a difference is to happen by accident before you can decide how important it is.

I should note that there are occasions when discarding data before calculating the mean can be useful; however, these are occasions that are best handled by people trained to deal with them. If you ever find yourself having to estimate the location parameter of a Cauchy distribution, discarding data from the tails before calculating a mean can be helpful, but even more helpful is having someone do who's been trained to do it and does it a lot.

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The Truth about Means and Medians © 2011, John FitzGerald