Mean, Median and Mode: The Three Averages

Three ways to describe the middle

All three are called averages. Each summarises a whole data set with one number, and each does it differently.

Measure	How to find it
mean	add everything, divide by how many
median	put in order, take the middle
mode	the value that appears most

Take the test scores $6, 7, 7, 9, 11$.

$$\text{mean} = \frac{6 + 7 + 7 + 9 + 11}{5} = \frac{40}{5} = 8$$ $$\text{median} = 6, 7, \mathbf{7}, 9, 11 \;\Rightarrow\; 7$$ $$\text{mode} = 7 \quad (\text{it appears twice})$$

Three different answers from one data set, and none of them is wrong.

The mean shares the total out evenly

$$\text{mean} = \frac{\text{sum of the values}}{\text{number of values}}$$

The mean answers: if everyone had the same amount, how much would that be? Five students scoring $40$ marks between them average $8$ each.

That also explains why the mean need not be a value anyone actually got. The mean of $4$ and $7$ is $5.5$, and the mean number of children per family is famously not a whole number.

The median needs the data in order

Sorting first is not optional. The median of $9, 2, 5$ is $5$, and reading the middle of the unsorted list would give $2$.

With an odd count there is one middle value:

$$3, 5, \mathbf{8}, 9, 12 \;\Rightarrow\; \text{median} = 8$$

With an even count there are two, so take their mean:

$$3, 5, \mathbf{8}, \mathbf{9}, 12, 15 \;\Rightarrow\; \frac{8 + 9}{2} = 8.5$$

Why the choice between them matters

Here are the salaries at a small company, in thousands:

$$28, \; 30, \; 31, \; 32, \; 34, \; 35, \; 350$$

$$\text{mean} = \frac{540}{7} \approx 77 \qquad \text{median} = 32$$

The mean salary is $77$ thousand. Nobody earns anywhere near that. Six of the seven people earn under $36$, and the seventh earns $350$.

That single value is an outlier, and it drags the mean up by more than $45$ thousand while moving the median not at all. The mean uses every value, so an extreme one has extreme influence. The median only cares about position, so a value ten times too big counts exactly as much as a value slightly too big — it is one entry above the middle either way.

This is why house prices, salaries and household incomes are reported as medians. A few very large values are normal in that data, and the mean would describe none of the people in it.

Data	Better measure	Because
test scores, similar range	mean	uses all the information
salaries, house prices	median	resists outliers
shoe sizes, favourite colour	mode	the most common value is the useful one

The mode is the only one that must be a real value

The mode is the value appearing most often, and a data set can have:

• one mode: $2, 3, 3, 5$ has mode $3$
• two modes: $2, 2, 5, 5, 8$ has modes $2$ and $5$
• no mode: $1, 4, 7, 9$ — every value appears once

Mode is also the only average that works for data that is not numbers. There is no mean favourite colour, but there is a most common one.

Worked examples

Common mistakes

Practice problems