Averages and spread: mean, median, mode, range, IQR and standard deviation

Imagine two classes sit the same test. Class A has 30 students and Class B has 28. You are given all 58 marks and asked, "Which class did better?"

Staring at 58 numbers will not help. You need to squash the list down to a few numbers that capture what it is like. Two questions matter:

• Where is the middle? What is a typical mark? Numbers that answer this are called averages, or measures of central tendency ("central tendency" just means "where the data tends to sit").
• How spread out is it? Are the marks all bunched together, or scattered from very low to very high? Numbers that answer this are called measures of variation (or spread).

You need both. Two classes can have the same typical mark but one can be very consistent while the other has some top scores and some very low ones. This lesson gives you the standard tools for both questions.

Here is the notation we will use. Each one is also explained again where it first appears.

The mean is what most people mean by "the average". Add up all the values, then divide by how many there are:

$\bar{x} = \frac{\Sigma x}{n}$

Why does this make sense? Imagine five friends with £2, £4, £6, £8 and £10 in their pockets. If they pooled all the money (£30) and shared it out equally, each would get £6. The mean is that "fair share" value. It is also the balance point of the data: if each value were a weight on a ruler, the ruler would balance on the mean.

Because the mean uses every value, it reacts to every value. That is a strength (nothing is ignored) and a weakness (one extreme value can drag it a long way). We will see that in a moment.

The median is the middle value once the data is in order. Half the values are below it and half are above it.

Step 1: put the values in order, smallest to largest. (Skip this and the answer is wrong.)

Step 2: find the middle.

• If $n$ is odd there is exactly one middle value. It is in position $\dfrac{n+1}{2}$.
• If $n$ is even there are two middle values. The median is halfway between them, which means you add them and divide by 2.

Why bother with the median when we have the mean? Because the median ignores how extreme the biggest and smallest values are. Think of the wages in a small shop: four staff earn about £20 000 and the owner takes £220 000. The mean wage is £60 000, which describes nobody. The median is £20 000, which describes a typical worker. When a few values are unusually large or small (outliers), the median is the fairer "typical" value.