Median and spread

The median is the number in the middle once you have lined them all up by size.

3, 5, 7, 9, 11
      ↑
  the median is 7

If there is an even number of them there is no single middle number. Then you take the mean of the two middle ones:

3, 5, 7, 9
   ↑  ↑
(5 + 7) / 2 = 6

The range is how spread out the numbers are: the largest minus the smallest.

3, 5, 7, 9, 11   →   11 − 3 = 8

Why two different measures of the middle? Look at this:

Salaries at a small company
25 000, 26 000, 27 000, 28 000, 900 000

Measure	Value	What it says
Mean	201 200	misleading — nobody earns that
Median	27 000	what a typical person earns

A single very large number pulls the mean up enormously. The median does not care.

When do you use which?

Situation	Choose	Why
Salaries, house prices, wealth	median	a few extremely high values
Test results in a class	mean	usually fairly even
Waiting times	median	a few very long ones
Number of siblings	median	a few people have a great many
Temperature over a week	mean	no extreme values

Rule of thumb: if there are a few values that stick out a long way — use the median.

The range has a problem: it is built on only two numbers, the largest and the smallest. A single odd value ruins it.

5, 6, 6, 7, 7, 8, 100
range = 95

But almost all of them lie between 5 and 8! That is why the interquartile range is often used instead: divide into four equally large parts and look at the distance between the first and the third boundary. The outermost values are then ignored.

A good habit: report both — a measure of the middle and a measure of the spread. «The median is 27 000 and most people lie between 25 000 and 28 000» says far more than one number alone.

Collect numbers of your own and do the arithmetic.

Task 1 — shoe sizes. Ask ten people for their shoe size.

Write the numbers: ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
Sort them:         ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
Median (the two middle ones, divided by 2): ___
Largest − smallest: ___
Mean: ___

Did the median and the mean come out roughly the same? They usually do for shoe sizes.

Task 2 — add an extreme value.

Imagine one more person turns up, and that person has shoe size 60. Redo the arithmetic:

New median: ___     (did it change much?)
New mean: ___       (did that change much?)
New range: ___

What you should see: the median barely moves. The mean jumps. The range explodes.

Task 3 — find a real example.

Look for a news story where a mean is used. Would the median have given a different picture? It is often about salaries, prices or waiting times — and the median would often have been more honest.

Something to think about: why do you think the writer sometimes picks the mean precisely because it looks better?