Statistics · Averages & range

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Mean, median, mode & range

The three averages and the range from a raw list — what each one is actually for, why the median needs the data sorted first, why the range is a measure of spread and not an average, and why every reverse-mean question is really a question about the total.

Statistics · Averages & range

Mean, median, mode & range

Why it works

A set of data is unwieldy: fifty numbers say fifty things at once. An average is the bargain you strike with that mess — one number you are willing to send out into the world in place of all of them. "The average rainfall in June is

52

mm" is a promise that if you had to describe June with a single figure,

52

is the figure you would stand behind.

Nothing decides which single number that should be, which is why there are three of them. Each answers "what is typical here?" honestly, but in a different way, and each is the right answer to a different question.

The mean is the share-it-out-equally number. Pool everything and redistribute it fairly. If four people have

\pounds 3

\pounds 5

\pounds 10

and

\pounds 2

, they have

\pounds 20

between them; share that equally and everyone gets

\pounds 5

. So

\text{mean} = \frac{\text{total of the values}}{\text{how many values there are}}.

Read that the other way round and you have the single most useful fact in the whole topic:

\text{total} = \text{mean} \times \text{number of values}.

A mean never really knows the individual values at all — it only knows their total. Any two lists of the same length with the same total have exactly the same mean. That is why every "reverse" mean question is solved by going to the total: a mean is a disguised total, and undoing the disguise is one multiplication. Told that the mean of

6

numbers is

14

, you know at once that they add to

6 \times 14 = 84

— even though you know nothing whatever about the six numbers. Told that after one is removed the remaining five have mean

15

, you know those add to

5 \times 15 = 75

. The number that left carried away

84 - 75 = 9

. No algebra, no guessing: totals in, totals out. (Sense-check it: taking a value away made the mean rise, so the value removed must have been below the old mean — and

9 < 14

. ✓)

The median is the middle value — the halfway line, with as many values below it as above. Its position is

\frac{n+1}{2}

: with

n = 9

that is the

5

th value (four below, four above), and with

n = 8

it is position

4.5

, meaning halfway between the

4

th and the

5

th, so you take those two and average them. Notice that

\frac{n+1}{2}

gives a position, not a value — position

5

, not the number

5

.

And here is why sorting first is not a formality. "Middle" is a claim about rank — about how many values are smaller and how many are larger. The order the data happens to be written in is an accident of collection: the order the buses arrived, the order the students queued. The middle of an unsorted list tells you which value was recorded sixth, which is a fact about the clipboard, not about the data. Sort the list and the position finally means something. Forgetting to sort is the single most common lost mark on this topic.

The mode is the value that occurs most often, and it is the only one of the three that survives when the data are not numbers. You cannot add up "red, red, blue, silver" and divide by four, and you cannot line colours up in size order, so the mean and median are simply undefined — but "red is commonest" is still true and still useful. A shoe shop restocks by mode, not by mean: a size

7.4

shoe does not exist.

The range is not an average. It is

\text{largest} - \text{smallest}

: a single number measuring spread — how far apart the data are, not where they sit. Two teams can both average

2

goals a match while one scores

2, 2, 2, 2

and the other scores

0, 0, 6, 2

. The means are identical; the ranges (

0

and

6

) capture the thing that actually separates them, consistency. Quoting a range as an average is a category error — it answers a different question. (Test yourself: add

100

to every value in a list. Every average goes up by

100

; the range does not move at all.)

So which one is honest? The mean is the only measure that uses every value, which is usually its virtue and is exactly its weakness when one value is wild. Eight salaries clustered near

\pounds 22\,000

, plus one director on

\pounds 120\,000

, make nine people with a mean of nearly

\pounds 33\,000

— a figure that eight of the nine earn less than. One extreme value dragged the total, so it dragged the mean. The median barely notices: push that

\pounds 120\,000

up to

\pounds 1\,200\,000

and the middle value does not shift by a penny, because the median only cares about rank.

Median — skewed data, or data with an outlier.
Mode — categories (colours, sizes, brands), or when you need a value that

actually occurs.

Mean — reasonably symmetric data, when you want every value to count.
Range — never an average; use it to say how spread out the data are.