Mean vs Median: Which One to Use When?

Full transcript

Word for word what is said in the video — the same text as the subtitles. Times are minutes:seconds into the video.

0:00Twenty incomes — and a promise ▶

Here are twenty people and their twenty incomes — most of them somewhere around forty thousand. Ask for the average income here, and you get a perfectly sensible answer. By the end of this video, one person will walk through that door and make that average almost meaningless — without making it wrong.

0:22Two kinds of middle: balance point & middle person ▶

There are two honest ways to find the middle of these numbers. The mean is the balance point: add up all twenty incomes and share them out equally — the seesaw balances at 41,000. If everyone here pooled their income and split it evenly, each would walk away with exactly that. The median is the middle person: line everyone up and point at the seat halfway down — here, that's 40,000. Two different questions: where does the money balance, and where do the people sit.

0:54On a bell, they agree ▶

So why did we never need this distinction before? Because on a bell curve, it doesn't exist. Video one showed the mean, the median and the mode all landing on the same spot. The boxplot from last time was perfectly symmetric around it — even the quartiles sit at fixed spots on the bell. When data is symmetric like this, the two middles agree. And then the mean is the usual choice — it simply carries the most statistical tools for later. The trouble starts when the data leans.

1:33The billionaire walks in ▶

Now a billionaire walks in, with an income of forty million. Watch the balance point: to keep forty million in balance, the triangle races to the right — the mean rockets from 41,000 to almost two million. Nobody in this room earns anything like two million. And did anyone get richer when he walked in? Nobody did — the mean rose because the pot rose: it is a property of the pot, not of the people in the room. And now watch the median: one extra person means the middle seat shifts by one — from 40,000 to 41,000. Both numbers are correct — they just answer different questions, and one of those questions stopped being yours.

2:16Which one to use when ▶

So which one should you use? Use the mean when totals matter. The mean times the headcount returns the whole pot: 41,000 times twenty is the entire payroll of 820,000 — budgets, costs and totals live here. Use the median when the typical member matters: half earn less, half earn more, and no billionaire can drag it away. Salaries, house prices, waiting times — anything you would summarise as what people usually get wants the median. And it explains a familiar news puzzle: a country's average income can climb while most pay cheques stand still — one very good year at the very top is enough. You meet this choice every day in headlines. When a report quotes the average income, the average house price, the average delay — your questions are now automatic. Is the data symmetric enough for a mean, and did they use the balance point or the middle person? And how do you check that symmetry? There are rules of thumb and formal tests for exactly that — a story for later in this series.

3:35Two faces you've met: the typo and the box ▶

You have already seen both faces of this. In video three, one typo dragged the mean and sigma into nonsense, while the median never flinched — same mechanism, because the mean chases extremes. Back then it was a typo; today's billionaire is real — and the mean cannot tell the difference. It gets dragged by errors and honest extremes alike. And last time you met the boxplot: the median is drawn right on it, the mean is not. For our heights the two would sit almost on top of each other, because heights are symmetric. Incomes are not, and that difference is about to become the whole story.

4:18In one picture ▶

So, in one picture: the mean is the balance point, the median is the middle person. On symmetric data they agree, and then the mean is the usual choice. One extreme value drags the mean, but only slides the median one seat. Use the mean for totals and budgets; use the median for the typical member. And when someone quotes you an average, ask which one they mean.

4:46What's next: why data leans ▶

The billionaire pulled our data into a long right tail — and that shape is everywhere: incomes, waiting times, house prices. Why does so much data lean right? That is next time: skewed distributions, and why your data leans. Tell me below — where has an average misled you? And if you want to keep learning, subscribe — the next video will find you. Until then: happy learning.