Writer gets answer

Tom Barringer, Stockton
    A letter to the editor — “Report Card data ‘misleading’” — appeared in Feb 24 issue of The Beacon. In the letter, the writer makes comparisons between median and average values of teachers’ salaries. In general, however, the average and the median of a set of data are not the same value. Most people know how to compute the average: Add ‘em up and divide by the number of observations. The median may be a little less familiar: Sort the data from smallest to largest then find the middle value. That’s the median. (If there are an even number of observations, the median is the average of the two middle values).
   How different can the average and median be? Consider two example data sets: (1, 2, 2, 2, 2, 3) and (1, 1, 1, 1, 1, 100). For the first one the average is 1 + 2 + 2 + 2 + 2 + 3 = 12 divided by 6 = 2, and the median is 2 + 2 = 4 divided by 2 = 2. So the average and median are the same. For the second data set the average is: 1 + 1 + 1 + 1 + 1 + 100 = 105 divided by 6 = 17.5 and the median is 1 + 1 = 2 divided by 2 = 1. In the second case the average and median are quite different. The reason that they were the same in the first case is the data are ‘symmetrical’ (The so-called ‘bell curve’ is another example). They differed in the second case because the data are ‘asymmetrical’ or ‘skewed’. As it turns out, the median tends to do better than the average when data are skewed. In the first example, most of the observations are 2s, and both the average and the median reflect that. But in the second example, where most of the observations are 1s, the median tells us that, but the average does not.
   The take-away here is: (1) The average and the median do not always have the same value, and (2) The more skewed the data are, the more the average and median differ from one another. Best to compare averages to averages and medians to medians.
   Further down in the article the writer refers to the “average median.” This differs from both the average and the median. You would have to ask a statistician what properties the average median has and how it differs from the average and the median of a set of data.
   By the way, next week there may be a short quiz.