Averages!

An "average" is a means of representing a group of values by a single figure. In doing so it will give some sort of central tendency, but say nothing about the broad range values, that is, about its distribution. So, for instance, a group where all the value are 50 (very equal), will have the same average as a group where half are zero and half are 100 in value (- unequal).

There is a further complication, because there is no definitive idea of what figure should represent "central tendency". We normally use the word "average" to indicate the result when all values are added together and divided equally among the number of cases which produce them. This is the Mean. Otherwise the central value could be the middle one - when all are lined up in order and the central one in ranking is taken. This is the Median. Alternatively, we could take the most popular value encountered. This is called the Mode.

In most large data sets, which tend to have a "bell" shaped pattern and the highest point in frequency corresponds broadly to the middle value, there may be fairly small differences between mean, mode and median. but in distributions of values which are not symmetrical or equal about the central value, the three measures may give different results.

Today, on the Institute of Economic Affairs website, Len Shackleton illustrates the problem over the "gender pay gap". How does male pay, in general, compare with female pay? We leave aside the obvious Harman ploy, to try to include all workers, part time and full time, as there would obviously be a larger gap between male and female worker's pay, as women tend to dominate part time jobs and lower pay.

Comparing full time workers only, we find that using the Mean, the normal understanding of average, the pay gap between men and women fell by 1% to 16.4% in the year to April 2009. Using the Median, because the Mean value is distorted by a few very high earning most male workers, the fall was from 12.6% to 12.4%.

(If all workers, part time and full, were lumped together, the gender difference fell from 22.5% to 22.0%. Clearly for polemical or political reasons Harriet Harman would prefer to lump workers together, as this (falsely) shows the largest gender gap.

Which measure do you quote? It depends on your political purpose.

Shackleton ends with amusing results. If you consider merely those working in the public sector, surely those jobs most within the government's control, then the median measure of the gender gap has widened in each of the last three years, and the mean measure widened in each of the last two years.

This suggests that Harman, in attacking the private sector which has reduced the gender gap to very small proportions, is attacking the wrong target!