The median

The median is the value in the middle once the data are sorted from smallest to largest: half the data are smaller than it or equal to it, and half are larger than it or equal to it.

How many minutes Ido waited for the bus each day of a week:
$\displaystyle 12, 5, 8, 20, 6, 9, 7$

• Sorted from smallest to largest: $\displaystyle 5, 6, 7, 8, 9, 12, 20$.
• There are 7 data. The middle one is the fourth: three before it and three after it.
• The median: 8 minutes.

When the number of data is even, two values sit in the middle, and the median is their mean.

The goals of 8 players in a season: $\displaystyle 10, 4, 12, 3, 8, 4, 9, 6$.

• Sorted: $\displaystyle 3, 4, 4, 6, 8, 9, 10, 12$.
• In the middle: the fourth, 6, and the fifth, 8.
• The median: $\displaystyle {\frac{6+8}{2}}={7}$ – a number that is not among the data.

In a sorted list of $\displaystyle n$ data the median’s place is $\displaystyle \frac{n+1}{2}$: with 7 data the fourth place, with 8 data place “4.5” – between the fourth and the fifth.

In a frequency table the values are sorted already, and the median’s place is found by adding up the frequencies.

The number of brothers and sisters of 24 pupils:

• 24 data: the median is the mean of the 12th and the 13th value.
• Counting: values 1–3 are 0; values 4–11 are 1 ($\displaystyle {3+8}={11}$); values 12–18 are 2 ($\displaystyle {11+7}={18}$).
• The 12th and the 13th value are both 2, so the median: 2.

Note: the mode here is 1, and the median 2.

An outlier is a value very far from the rest of the data. An outlier pulls the mean towards it, and hardly moves the median.

The monthly pay of six workers in a small business, in shekels:
$\displaystyle 7000, 8000, 8000, 9000, 10000, 12000$

• The mean: $\displaystyle {\frac{54000}{6}}={9000}$ shekels. The median: $\displaystyle {\frac{8000+9000}{2}}={8500}$ shekels.

Now we add the pay of the business’s owner, 51,000 shekels:

• The mean: $\displaystyle {\frac{105000}{7}}={15000}$ shekels – up by 6,000 shekels.
• The median: the 4th value of 7, $\displaystyle 9000$ shekels – up by only 500 shekels.

No worker earns 15,000 shekels: the mean here is higher than every worker’s pay. The median describes the typical pay better.

Each measure of centre fits a different question.

• The mode – to know what is most common: which shoe size a shop should order most of. It is also the only measure for categorical data, such as a favourite colour.
• The median – when there are outliers: pay, flat prices.
• The mean – when the total matters, and the data are all close to each other: how many pizzas to order for a class, from the number of slices a pupil eats on average.

Find the median.

The ages of the players of a youth team: $\displaystyle 15, 13, 16, 14, 13, 17, 15, 16, 14$.

How many hours six pupils slept the night before the class trip: $\displaystyle 8, 7, 9, 6, 10, 7$.

The number of goals in each game, in 21 games of a schools’ league:

Find the median. Write which place in the list it is in.

The marks of 9 pupils in a test:
$\displaystyle 85, 90, 78, 88, 92, 19, 80, 86, 84$

Work out the mean.

Find the median.

Which measure – the mean or the median – better describes the mark of a typical pupil in the group? Explain.

Which measure of centre – the mode, the mean or the median – fits each question? Explain.

1. A clothes shop is ordering shirts: in which size should it order the most shirts?
2. How much does a typical worker earn in a company whose manager earns 20 times as much as each worker?
3. How many bottles of water to buy for a trip of 30 pupils, when you know how many bottles a pupil drinks on a trip?
4. What is the class’s favourite colour?

Write five numbers whose median is 10 and whose mean is 12.