# There are an infinite number of ways to be average

Over the last few days, I have been struck by the number of uses of the statistics term ‘median’ by a number of politicians and commentators. This is a relatively technical term and I rarely hear it from politicians, or indeed anyone else on TV or in newspapers. The terms has been used in the context of public sector pay. The statement is that median public sector pay is higher than median private sector pay. And it is, it is £22,902 in the public sector and £20,575 in the private sector. However, the mean public sector pay is lower than the mean pay in the private sector, £25,892 versus £27,195 in the private sector.

I would say that this is just shameless cherry picking of stats, but there you go. But from a stats/data analysis point of view it is does bring out the fact that in general there are an infinite number of ways of measuring the mean of a distribution, such as the distribution of incomes here. Also, that when the distributions are broad, as they are here, that different measures of this average can given significantly different answers.

The mean and media of a distribution are defined as follows. The mean pay, in say the public sector, is the total pay of all the public sector workers, divided by the total number of public sector workers. The median pay is the pay such that half of people in, for example the private sector, earn morethan this pay, and half earn less.

In other words, if you earn the mean pay you get the pay that everyone would get if the pay were shared equally and everyone got the same (this is unlikely to happen of course, this is what theoretical physicists call a thought experiment). If you earn the median pay, as many people earn more than you as earn less, you are in the middle in that sense. The mean and median are usually different, they are only the same in special cases.

Here I assume what is happening is that the mean of the private sector is dragged up by a few very high earners in the private sector. If you add even a few people earning £1 million + a year that significantly increases the mean because the mean is proportional to the total which increases a lot if you add very high earners. But these people do not increase the median by much. As they are very few in number, they do not change the point at which half earn more and half earn less, which is what the median is.

A BBC article has a nice plot of the distribution of incomes from 2006/7 (it is an old article). You can see a tail in the distribution going out to large values where the high earners are – the plot is truncated, the really high earners would not fit onto it.

So, there are many ways to measure an average, and no one way is right or wrong. They are all just ways to capture some information about the typical values of what are in this case, millions of salaries. (I cannot resist pointing out the total cost to the taxpayer depends on the mean, not median.) In any case, if someone uses a less common method of assessing an average, you should probably ask why.