Mean vs Median: Which Average Should You Trust?

Nine people in a bar each earn around $40,000 a year. Then a billionaire walks in. The average income in the bar is now over $100 million. Nobody’s richer except in the statistics, and if anyone told you “the average person in this bar earns $100 million”, you’d know you were being misled. You just might not be able to say how.

The trick is that “average” can mean two quite different things, and they only agree when the data is well behaved.

Two kinds of average

The mean is what most people call the average: add everything up and divide by how many there are. For the bar, that’s the nine salaries plus a billion, divided by ten.

The median is the middle value: line everything up from smallest to largest and take the one in the middle. Half are below it and half above. In the bar, the median barely moves when the billionaire arrives. It’s still about $40,000.

For many kinds of data, like heights or the weights of apples, the two are nearly the same, because values are spread fairly evenly either side of the middle. They split apart when the data is skewed: when a few values are far bigger (or smaller) than the rest. Move the top earner in the toy above and watch it happen.

Why the mean runs away

The mean uses the size of every value, so one huge value pulls it a long way. The median only cares about order: the billionaire counts as one person above the middle, however rich, exactly as a person earning $41,000 would.

This is why incomes, house prices and wealth are usually reported as medians. In the US in 2023, the median household income was about $80,000 a year, while the mean was about $115,000. Most households earn less than the mean. The few very high incomes pull it upwards and nothing equivalent pulls it down, because nobody earns less than zero.

Wealth is even more lopsided than income. In most rich countries the mean wealth of a household is far above the median, because a small share of households owns a large share of the total. A headline about “average wealth” can describe almost nobody.

The shape of skewed data

It helps to picture it. For heights, a chart of how many people are each height is a symmetrical hump, with the mean and median together in the middle. For incomes, the hump is bunched at the low end with a long tail stretching off to the right. The median sits in the hump, where most people are. The mean is pulled out along the tail.

Lots of real data is shaped like this:

• Incomes and wealth, with a long tail of high earners.
• House prices, with a tail of mansions.
• Waiting times, where most waits are short and a few are very long.
• Sizes of cities, companies and online followings, where a few are enormous.
• Insurance claims and losses, where most are small and a rare few are catastrophic.

For these, if you want to know about a typical case, the median is the better guide. And the gap between mean and median is itself a clue: a large gap tells you the data has a long tail.

Which one should you use?

• Use the median for the typical case when the data is skewed: incomes, house prices, wealth, waiting times, city sizes.
• Use the mean when the total matters, or the data is roughly symmetrical. If you want to know how much a company pays its staff in total, or how much fuel a fleet uses, the mean is what you need, because mean times count equals the total.
• Look at both. If someone gives you only one, ask for the other. A big gap is useful information in itself.

There’s also a halfway house: the trimmed mean, which drops the most extreme values at each end before averaging. Judges in diving and gymnastics do this when they throw out the highest and lowest score, so that one generous or harsh judge can’t swing the result.

The mode

There’s a third average, the mode: the most common value. It’s useful for categories, like the most popular shoe size or the most common answer to a survey. For measurements that rarely repeat exactly, such as salaries to the dollar, it’s not much use. But it can reveal something the others hide: data with two separate humps, like a class where half the students found a test easy and half found it hard, has a mean and median somewhere in the gap between them, where almost nobody actually scored.

When the mean misleads about lives

One of the most persistent historical myths comes from reading a mean as if it were typical. You may have heard that people in the Middle Ages died at around 30. That figure is a mean life expectancy at birth, and it’s dragged down by the huge number of children who died in their first five years. People who survived childhood often lived into their 50s, 60s and beyond. There’s a toy that shows it in our article on medieval life expectancy.

The same reasoning applies today. “The average person who gets this illness survives five years” might be a median, a mean or something else, and the shape behind it matters. In a famous essay, The Median Isn’t the Message, the palaeontologist Stephen Jay Gould described being told in 1982 that the median survival for his cancer was eight months. He realised that a median says half live longer, that the curve had a long tail to the right, and that he had reasons to think he’d be in it. He lived another twenty years.

Three averages from everyday life, decoded

“The average house price rose 8%.” If it’s a mean, a handful of expensive sales in one smart neighbourhood can move it, even if a typical home barely changed. Official house price indices usually report a median, or a carefully adjusted index, for exactly this reason. If a story gives you only a mean, ask what the median did.

“The average wait in A&E is two hours.” Waiting times have a long tail: most people are seen in an hour or so, and a few wait all night. A mean of two hours might hide a median of 70 minutes and a nasty tail of 12-hour waits. For waiting times, the most useful figures are often the median and a high percentile, such as how long the slowest one in ten waited.

“The average family has 2.4 children.” No family has 2.4 children. This one’s a mean of whole numbers, and it’s useful for planning schools, but it tells you little about any family. The mode, two, is more like what you’d meet on a street.

In each case the number isn’t wrong. It answers one question, and the headline treats it as if it answered another.

A trick to watch for

Because the mean and median can differ so much, whoever chooses which to report can shape the story. A company might quote the mean salary to look generous; a union might quote the median to show that most staff earn less. Both can be accurate. A government might quote mean household income growth when a few high earners have done very well, or median growth when most households have.

So when you see “the average”, ask three things:

1. Which average? Mean or median?
2. How lopsided is the data? Are there a few huge values?
3. What’s the question? The typical case, or the total?

The answers usually tell you whether the number in front of you describes most people or almost nobody.

If you want to test your instincts, the statistics test has questions on averages, skewed data and the traps around them.