Averages and measures of spread

Mean, median, mode and range

An average is a single value that represents a whole set of data. There are three, and each one answers a slightly different question.

• The mean is the total of all the values divided by the number of values. It answers "if the total were shared out equally, how much would each get?"
• The median is the middle value once the data is in order. With $n$ values, it is the $\frac{n+1}{2}$th value. When $n$ is even there are two middle values, and the median is halfway between them.
• The mode is the value that occurs most often. A set of data can have two modes, or no mode at all if every value occurs once. The mode is the only average that works for qualitative data: you can say "the modal colour is blue", but not "the mean colour".

The range is not an average. It measures spread: range $=$ largest value $-$ smallest value. A small range means the data is consistent; a large range means it is spread out.

For example, for $2,3,3,7,10$: the mean is $\frac{25}{5}=5$, the median is 3 (the 3rd value), the mode is 3 and the range is $10-2=8$.

Why order matters for the median. The median is the value with as many values below it as above it. In an unordered list, the value in the middle position can be anything, so always sort the list first, and count to check you have every value.

Averages from a frequency table

Data is often given in a frequency table rather than a list. The table is just a short way of writing the list: a row "value 2, frequency 8" means the value 2 appears 8 times.

Mean. Every value must be counted as many times as it occurs, so multiply each value by its frequency.

1. Add a column for (value $\times$ frequency), often written $fx$.
2. Add up this column: the total of all the data.
3. Add up the frequency column: the number of values.
4. Mean $=\frac{\text{total of}fx}{\text{total frequency}}$.

The most common mistake is to divide by the number of rows instead of the total frequency.

Median. Find its position, $\frac{n+1}{2}$, where $n$ is the total frequency. Then add up the frequencies down the table (a running total) until you reach that position.

Mode. The value with the highest frequency. Give the value, not the frequency.

Range. The largest value minus the smallest value that actually occur (ignore any value with frequency 0).

Working backwards from the mean

Since mean $=\frac{\text{total}}{\text{number of values}}$, it follows that

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

This one rearrangement answers most "missing value" questions.

• A missing value. If the mean of 5 numbers is 8, their total is 40. Subtract the four numbers you know.
• A value added or removed. Work out the total before and after; the difference is the value added or removed.
• Combining groups. To find the mean of two groups together, add the two totals and divide by the total number of values. Do not average the two means unless the groups are the same size: a group of 30 should count three times as much as a group of 10.

For an unknown frequency in a table, write the total of $fx$ and the total frequency in terms of the unknown, set mean $\times$ total frequency $=$ total of $fx$, and solve.

Choosing the best average

Each average has a job it does best.

• Mean. It uses every value, so it is the most representative when the data has no extreme values. It is the one to use when totals matter (sharing costs, fuel per journey). But one very large or very small value (an outlier) drags it towards that value.
• Median. It is not affected by extreme values, because it only depends on the middle of the ordered data. Use it for data with outliers or a long tail, such as salaries or house prices.
• Mode. It is the only average for qualitative data, and the right choice when the most common value matters: a shoe shop wants to know which size sells most, not the mean size. It can be unhelpful if the data has several modes or none.

For example, the annual salaries (in thousands of dollars) at a small firm are 18, 20, 21, 21, 23, 25 and 89. The mean is $\frac{217}{7}=31$, which is more than six of the seven salaries, because of the manager's 89. The median, 21, describes a typical salary much better.

The same thinking applies to spread: the range uses only the two most extreme values, so a single outlier can make it misleadingly large. (Extended candidates use the interquartile range instead, below.)