Average traps

The average salary at the company is 4800 euros a month, the average waiting time at the checkout is 12 minutes, the average household owns 1.4 cars. Numbers like these are everywhere, and they sound objective. But an average can paint a very distorted picture if you do not know how it was made. Once you know the tricks, you are much harder to impress.

What the mean really does

The classic average is the mean: add up all the values and divide by how many there are. If five friends get 10, 12, 14, 16 and 18 euros of pocket money, the mean is $\frac{70}{5} = 14$ euros. Here it works beautifully, because the values sit close together. The mean spreads the total evenly across everyone in your head, as if each person had exactly the same amount.

The millionaire in the pub

Now for the famous trick. Nine people sit in a pub, each earning 2000 euros a month. Average income: 2000 euros, obviously. Then a millionaire with a monthly income of 102000 euros walks in. New mean: $\frac{9 \cdot 2000 + 102000}{10} = 12000$ euros. Statistically everyone now earns 12000 euros on average, even though nothing changed for nine out of ten people. A single outlier drags the mean way up.

The median keeps its cool

There is a second kind of average that refuses to be impressed by outliers: the median. Sort all values by size and take the one exactly in the middle. With an even number of values, take the mean of the two middle ones. In the pub, the median is 2000 euros both before and after the millionaire shows up: with ten people the fifth and sixth of the sorted line count, and both of them still earn 2000 euros. The median describes the typical person, not the grand total.

When which value is honest

Both values are useful, they just answer different questions. The mean is honest when a total is to be shared out evenly: rainfall per month, revenue per opening day, goals per match. The median is honest when you want to know how the typical person is doing: incomes, rents, house prices, waiting times. That is exactly why statistics offices put the median first when they report incomes. If someone quotes only the mean when typical cases are the point, it is worth asking: where is the median, and who might be the millionaire in the pub here?