Suppose someone asks how tall the people in your class are. You could read out twenty-eight separate heights — but nobody would take that in. So instead you give one number that stands in for the whole list: a typical height, a number that speaks for the group. That single stand-in number is what we mean by an average.
An average is a summary. It throws away detail on purpose, so that a long list becomes one figure you can hold in your head and compare. "The average rainfall in April." "The average score on the test." Each one folds many numbers down into a single one.
Here is the part most people miss: there is more than one way to choose that stand-in number. The two you'll meet most are the mean and the median. They usually land close together — but when they don't, the gap between them tells you something real about the data. Today is about understanding what each one hides and what each one reveals.
The mean, the median and the mode are all called averages — three different answers to the same question, "what is the typical value here?" They can disagree, and that disagreement is information, not a mistake.
The word average comes from an old shipping term, avaria — the cost of damage to goods at sea. When cargo was lost in a storm, the loss was shared out evenly between everyone who owned part of the shipment. "Sharing the loss equally" is exactly what the mean still does today.
Take a small set of test scores out of ten: 3, 7, 7, 8, 9. Watch how each kind of average is built from exactly the same five numbers.
With an even number of values there is no single middle one, so you take the two middle values and find the mean of just those two. For 4, 6, 9, 11 the two in the middle are 6 and 9, so the median is (6 + 9) ÷ 2 = 7.5. Always order the list first — an unordered median is meaningless.
Two more words finish the family. The mode is the value that appears most often — in 3, 7, 7, 8, 9 the mode is 7, because it shows up twice and nothing else repeats. The range is not an average at all; it measures spread — the largest value minus the smallest. Here the range is 9 − 3 = 6. Mode tells you the commonest; range tells you how stretched out the data is.
Tap each card — the four words, quickly:
A set of numbers can have two modes, or none at all. If every value is different, there is no mode. If two values tie for "most common", the data is called bimodal — and that often means you've accidentally mixed two different groups into one list.
Imagine a small company of seven people. Six of them earn fairly ordinary salaries. The seventh is the owner, who pays herself a great deal more. Here is the whole payroll, already in order.
| Person | Salary (£ per year) |
|---|---|
| Worker 1 | 18,000 |
| Worker 2 | 20,000 |
| Worker 3 | 21,000 |
| Worker 4 | 22,000 |
| Worker 5 | 24,000 |
| Worker 6 | 25,000 |
| The owner | 150,000 |
| Total | 280,000 |
That single large salary is called an outlier — a value far from the rest. The mean feels every value, so the outlier hauls it upward to £40,000. The median only cares about position, not size, so the owner's huge figure shifts nothing: it's still just "the highest one", and the middle stays put at £22,000.
When data has a few extreme values — salaries, house prices, the wealth of a town — the median usually tells the truer story of "typical". When data is fairly even with no wild outliers — heights, exam marks, daily temperatures — the mean works well and uses every value. Whoever picks the average is, quietly, choosing which story to tell.
In 1998, the small town of Lake Wobegon joke became a real statistic: when the basketball player Shaquille O'Neal walked into an ordinary bar, the mean wealth of everyone inside leapt into the millions — while the median barely moved. Statisticians call it the "Bill Gates walks into a café" problem: one outlier, and the mean stops describing anybody real.
A bar chart turns numbers into heights so your eye can compare them at a glance. To read one properly you do three things, in order: look at the axis labels to see what's being measured, check the scale up the side, then read each bar across to that scale. Here is a tidy one — the number of books a reading group finished in each of five months.
Read April across to the scale and you land on the tallest bar of the five; read March and you land on the shortest. Because the scale begins at zero and rises in even steps of one, a bar that looks twice as tall as another really does stand for twice as many books. That is the quiet promise an honest chart makes — and the next page is about charts that break it.
The bar chart was invented by a Scotsman, William Playfair, in 1786 — astonishingly late, given how obvious it now seems. Before him, data lived only in tables of numbers. Playfair also gave us the line graph and the pie chart. Critics of his day called the pictures "frivolous"; today they run the world.
Here are the results of a survey: three brands of phone, and how many people out of a hundred preferred each. The numbers are identical in both charts below — only the scale changes. Watch what that does to your eye.
In the left-hand chart the bars are nearly level — which is the truth, because 94, 92 and 90 out of a hundred are barely different. In the right-hand chart Brand A's bar towers three times higher than Brand C's, and the eye screams "huge difference!" Nothing in the data changed. Only the axis was cut — it begins at 88, not zero — so a four-point gap got stretched into a cliff.
1 · A cut axis — a vertical scale that doesn't start at zero makes small gaps look enormous. 2 · An uneven scale — steps that aren't equal (0, 10, 50, 100) bend the shape of the data. 3 · A missing label — no numbers up the side at all, so you can't check anything. Always read the axis before you trust the bars.
You've built each average by hand. Now watch Khan Academy work through the same three from a single list of numbers. As it goes, say each answer to yourself before he does — and notice that "average" never means just one thing.
Fresh one. Five spelling-test scores: 6, 8, 7, 9, 10. Find the mean.
Fresh one. Five reaction times in order: 11, 12, 14, 15, 18. Find the median.
This chart shows how many goals a netball team scored in each of five matches. The scale starts at zero and rises in steps of one. Read each bar across to the scale to answer the three questions below — the numbers are in the chart, not in the words.
A newspaper runs the headline: "Average pay at TechCo soars to £85,000!" You later learn that TechCo has 200 staff. Nearly all of them earn around £30,000 — but the three founders each pay themselves several million pounds a year.
Write a short paragraph explaining what's going on. Use the words mean, median and outlier. Say which average the headline has almost certainly used, why it gives a misleading picture of a "typical" worker, and which average would be more honest. Three or four sentences is plenty.
strong You spotted the move at the heart of it — that the £85,000 is a mean, and that three multi-million salaries are outliers dragging it far above what almost everyone actually earns. Naming the founders as outliers is exactly the word the examiner is listening for.
try this You mention the median would be "better" — take it one step further and say what the median would roughly be here. With nearly everyone on about £30,000, the person in the middle earns about £30,000, so that's the median. Putting a number to it shows you understand why it's the honest one.
to add One sentence on why the newspaper chose the mean — it makes a far more dramatic headline — turns a correct answer into a sharp one. That's the difference between describing the maths and reading the data honestly, which is the whole point of today.
You learned what an average is for, and that there's more than one. You can find a mean and a median, and you know why a single outlier drags one but not the other. And you've seen how a chart can tell the truth or bend it, all depending on where the axis starts. From here on, when a headline waves a number at you, you'll know to ask: which average, and what's the scale? That question, Florence, is most of statistics.