Reading Data: Mean, Median, Spread — and the Lie of the Average

The average is a compression, and compression loses things

Every summary of data throws information away on purpose. That's the point of a summary — you can't hold a thousand numbers in your head, so you replace them with one or two that stand in for the whole. The question is never "is the summary lossy?" It always is. The question is what did this particular summary throw away, and does it matter here?

Most people know one summary: the average. And the average is the single most confidently misused number in public life — quoted about salaries, house prices, exam results, waiting times — usually by someone who has no idea what it's hiding. By the end of this you'll read any "the average X is Y" and immediately ask the two questions that expose what it left out.

There are only three things you ever need to know about a set of numbers: where the middle is, how spread out they are, and what shape they make. Get those three and you've understood the data. Miss the second and third — as almost everyone does — and the first one lies to you.

Part one — where's the middle?

There are three different "middles", and they answer different questions.

The mean — add them up, divide by how many

The everyday average. Add every value, divide by the count. Its defining property: every single value pulls on it. One huge number drags the mean toward itself, no matter how many small numbers sit on the other side.

That's its strength (it uses all the data) and its fatal weakness (a single outlier can move it to a place where no actual data point lives).

The median — line them up, take the one in the middle

Sort the values from smallest to largest and read off the middle one (with an even count, average the middle two). Its defining property: it doesn't care how extreme the extremes are. Replace the richest person in the room with someone a thousand times richer and the median doesn't move at all — the person in the middle is still in the middle.

The mode — the most common value

The value that appears most often. Usually the least useful of the three for continuous numbers, but the only one that makes sense for categories ("the most common grade was a B") — you can't take the mean of a grade.

The one rule to carry: when the mean and the median are far apart, the data is skewed, and the median is almost always the more honest "typical" value. When a report gives you the mean and not the median, ask why.

The lie of the average, made concrete

A bar has nine people in it earning £25,000 each. The average salary in the bar is £25,000 — and it's also the typical salary, because everyone earns roughly that. Fine.

Now the tenth person walks in earning £2,000,000.

• The mean salary is now about £222,500.
• The median salary is still £25,000.

The mean just produced a number that is false about nine of the ten people in the room. Nobody earns £222,500. It is a real calculation and a useless summary. "The average person in this bar earns £222,500" is technically true and completely misleading — and this is exactly the move behind most "average salary / average wealth" headlines, because income and wealth are always dragged upward by a long tail of very large values.

The tell: whenever the thing being measured has no upper limit and a floor at zero — money, house prices, view counts, wait times — the mean runs high and the median is what you actually want. Someone choosing to report the mean anyway is usually choosing the bigger, more flattering, or more alarming number.

Part two — how spread out is it? (the part everyone skips)

Two classes both average 60% on an exam.

• Class A: everyone scored between 57 and 63.
• Class B: half the class scored 90, the other half scored 30.

Same mean. Completely different reality. In Class A the average describes everyone. In Class B the average describes no one — it's the midpoint of two groups, and treating "60" as the typical student misses the entire story, which is that the class split in two.

The centre tells you where; the spread tells you whether the centre means anything. A mean without a spread is half a summary, and it's the missing half that hides the interesting part.

Two ways to measure spread:

• Range — biggest minus smallest. Simple, but one freak value blows it up.
• Standard deviation — roughly, the typical distance of a value from the mean. Small standard deviation = everyone clustered near the middle (Class A). Large = values scattered far from it (Class B). You don't need to compute it by hand — every spreadsheet has STDEV — you need to ask for it.

A more outlier-resistant cousin is the interquartile range (IQR): the range of the middle 50% of the data, ignoring the extreme quarters at each end. It pairs naturally with the median.

Part three — what shape does it make?

Sketch the values as a histogram in your head and one of three shapes usually appears:

• Symmetric (a hump in the middle). Mean ≈ median. The average is trustworthy. Heights, exam marks in a big cohort.
• Skewed (a long tail one side). Mean pulled toward the tail, median stays put. Income, wealth, house prices, response times. Use the median.
• Bimodal (two humps). The average lands in the empty valley between them, describing nobody — the 60% class, or "average daily screen time" across people who barely use their phone and people glued to it. The average is a lie here; the story is that there are two groups.

You can't know which shape you've got from a mean alone. That's precisely why a single number is never enough.

Percentiles — the summary that actually respects the spread

A percentile tells you the value below which a given share of the data falls. "You're at the 90th percentile" means 90% scored at or below you. Exam results, growth charts, response-time targets and income brackets are all reported this way for a reason: percentiles carry the shape and the spread with them, where a mean throws both away. "Median (50th), 25th and 75th" tells you more in three numbers than any average tells you in one.

What this means for you

• A number alone is not information. "The average is X" is half a fact. Ask for the spread and the shape before you believe the middle.
• Mean far from median ⇒ skew ⇒ trust the median. This one habit defuses most misleading money and time statistics you'll ever read.
• Ask "spread?" every single time. Same average, wildly different reality — the spread is where that difference lives.
• Watch for the empty valley. If the data is really two groups, the average describes neither. "The typical user" can be a person who doesn't exist.
• Whoever chose the summary chose what you'd see. Reporting the mean over the median, or hiding the spread, is a choice — and usually not an innocent one.

Try it: summarise three real datasets honestly (30 min)

Reading data is a habit; build it on real numbers.

1. Find three real datasets with at least ~15 values each — anything numeric: your class's test scores, the prices of every listing for one used phone model, daily steps for a month, house prices on one street, follower counts of accounts you follow.
2. For each, in a spreadsheet, compute the mean (AVERAGE), the median (MEDIAN) and the standard deviation (STDEV), and eyeball the shape.
3. Write:

DATASET: (what it is, how many values) ........................
MEAN: ......  MEDIAN: ......  STD DEV: ......
SHAPE: symmetric / skewed / bimodal
THE MISLEADING SUMMARY: (which single number would fool someone,
   and what it hides) .........................................
THE HONEST ONE-LINE SUMMARY: (the number(s) you'd actually quote)

✅ Finish check: three datasets, each with mean/median/spread computed, the shape named, and — the whole point — the specific single number that would mislead someone identified in each.

Summary card

• Every summary is lossy. Ask what this one threw away.
• Mean — uses every value, so outliers drag it. Median — the middle one, ignores extremes. Mode — most common; the only one that works for categories.
• Mean far from median ⇒ skewed data ⇒ report the median. Money, prices and times are almost always skewed.
• The lie of the average: one big value makes the mean describe nobody. "£222,500" in a room of £25k earners.
• Spread is the missing half. Same mean, opposite realities. Ask for range / standard deviation every time.
• Shape: symmetric (average is fine), skewed (use median), bimodal (average describes no one).
• Percentiles carry shape and spread; a mean throws both away.

Sources

1. Huff, D. — How to Lie with Statistics, 1954
2. Spiegelhalter, D. — The Art of Statistics, 2019
3. Rosling, H. — Factfulness, 2018
4. Wheeler, D. — Understanding Variation, 1993

Next lesson: DT-02 — How Charts Lie: Eight Manipulation Techniques (L1) Related: DT-03 Correlation Is Not Causation · DT-09 Verifying a Source · PS-04 Presenting Data · MN-06 Reading Financials Path: Hard to Fool — 1/6