The mean is the total divided by how many values there are. The median is the middle value once the numbers are in order. The mode is the value that appears most often. The range is the largest value minus the smallest. For 2, 4, 4, 5 and 10: the mean is 25 ÷ 5 = 5, the median is 4, the mode is 4 and the range is 10 − 2 = 8.
Mean, median and mode are three kinds of average: each one gives a single typical value for a set of data. The range is different: it measures how spread out the data is. All four appear from Year 6 and KS3 through to GCSE Foundation and Higher.
| Meaning | How to find it | 2, 4, 4, 5, 10 | |
|---|---|---|---|
| Mean | The “fair share” average | Add the values, then divide by how many there are | 25 ÷ 5 = 5 |
| Median | The middle value | Put the values in order and find the middle one | 4 |
| Mode | The most common value | Find the value that appears most often | 4 |
| Range | How spread out the data is | Largest value − smallest value | 10 − 2 = 8 |
A way to remember them: mode is the one there is most of; median sounds like medium, the one in the middle; and the mean is the “mean” one, because it takes the most work.
When there is an even number of values there are two middle values. The median is halfway between them: add the two and divide by 2. For 3, 5, 8 and 10 the middle values are 5 and 8, so the median is (5 + 8) ÷ 2 = 6.5. The median does not have to be one of the values in the list.
For a long list, the median is the (n + 1) ÷ 2th value, where n is the number of values. With 15 values it is the 8th value. That gives the position of the median, not the median itself.
If two values are tied for most common, both are modes: 1, 3, 3, 6, 6, 9 has modes 3 and 6. If every value appears the same number of times, there is no mode. The mode is the only average you can use for non-numerical data, such as the most popular colour or shoe brand. In a frequency table it is called the modal value or modal class.
The mean uses every value, so it is the most representative when the data has no extreme values. One very large or very small value (an outlier) pulls it a long way. The median ignores extreme values, which is why house prices and salaries are usually reported as medians. The mode suits data where you want the most popular choice. The range is not an average at all: compare ranges to say which set of data is more consistent.
Finding the median without putting the values in order first.
Writing the range as “2 to 10” instead of the single number 10 − 2 = 8.
In a frequency table, dividing by the number of rows instead of the total frequency Σf.
Giving the position (n + 1) ÷ 2 as the median, instead of the value in that position.
Untimed practice with worked answers. Every answer is a whole number. This extra activity does not count towards your daily progress.
MathsDaily sets ten fresh questions a day across the whole curriculum, with new numbers every morning. Averages and range come up regularly in the Year 6, KS3 and GCSE sets, with a tip and a worked solution if you get stuck.
Start Year 7 MathsDaily →The mode is the value that appears most often in a set of data. In 2, 4, 4, 5, 10 the mode is 4. A set can have more than one mode, or none if every value appears equally often.
Mean, median and mode are all types of average. When people say “the average” without saying which, they usually mean the mean: the total divided by the number of values.
Put the numbers in order and find the two middle values. The median is halfway between them: add them and divide by 2. For 3, 5, 8, 10 the median is (5 + 8) ÷ 2 = 6.5.
No. The range measures spread: how far apart the largest and smallest values are. A small range means the data is consistent.
Yes. The averages and range are not on the GCSE Maths formula sheet, so you need to know how to find them, including the mean from a frequency table. See the GCSE Maths formula sheet for the full list of what is given.
Free A4 PDFs with worked answers. See every printable, Year 1 to A-level.