Science Puzzle
The Billionaire in the Café
Nine people are having coffee in a café. Each of them earns £30,000 a year, so the average income in the room is £30,000. Then a billionaire with an income of £1 billion a year walks in and orders a flat white.
What happens to the average income in the café?
The Answer
The average leaps to just over £100 million, yet it describes nobody in the room.
The everyday average is the mean: add up everyone's income and divide by the number of people. Nine incomes of £30,000 come to £270,000. Add the billion and the total is £1,000,270,000, and divided between ten people that's £100,027,000 each. On paper, the typical customer is now a multimillionaire. In reality nine out of ten earn exactly what they did a minute ago.
One huge value has dragged the mean a long way from everyone else. A better guide here is the median: line all ten incomes up from smallest to largest and take the one in the middle. The fifth and sixth incomes are both £30,000, so the median is £30,000, and the billionaire hasn't moved it at all.
Neither number is wrong; they answer different questions. The mean tells you what everyone would get if the money were shared out equally, while the median tells you about a typical person. That's why official figures for things like household income and house prices usually quote the median. A handful of very rich households, or a few mansions, would push the mean far above what most people actually have.
The principle: Mean versus median. A mean can be dragged a long way by a single extreme value, while the median, the middle value, describes a typical case far better when the data are lopsided.