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The mean, the median and the mode are three different ways of describing the middle of a set of numbers. The mean is the total of the values 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.
When a test question simply says “average” with no further detail, it means the mean. When it wants one of the other two, it will normally name it.
The three averages side by side
| Measure | How to work it out | Best used when |
|---|---|---|
| Mean | Add every value, then divide by how many values there are | The values are fairly evenly spread, with no extremes |
| Median | Put the values in order and take the middle one | A few unusually high or low values would drag the mean away from typical |
| Mode | Count how often each value occurs and take the most frequent | The data is categories or repeated whole values, such as shoe sizes or ratings |
The mean is the only one of the three that uses every value, so changing any single number always changes the mean. The median and the mode can stay exactly where they are. That is the main reason a test asks about more than one of them.
How to calculate the mean
- Add all the numbers together.
- Count how many numbers there are.
- Divide the total by the count.
Example. Your test scores are 85, 90, 80 and 95.
- Add them: 85 + 90 + 80 + 95 = 350.
- Count them: there are 4 scores.
- Divide: 350 ÷ 4 = 87.5.
The mean is 87.5.
Example with an extreme value. A small team logs these numbers of calls: 18, 21, 19, 22 and 120.
The total is 18 + 21 + 19 + 22 + 120 = 200, and 200 ÷ 5 = 40. The mean is 40, which is higher than four of the five values. One unusual number has pulled the mean away from anything typical, and this is exactly the situation in which a test will ask for the median instead.
How to calculate the median
- Put the values in order, smallest to largest.
- If there is an odd count of values, the median is the middle one.
- If there is an even count of values, the median is the mean of the two middle ones.
Odd count. Take 12, 7, 3, 9 and 15. In order they are 3, 7, 9, 12, 15. There are 5 values, so the median is the 3rd, which is 9. The shortcut for the position is (n + 1) ÷ 2, and (5 + 1) ÷ 2 = 3.
Median of an even count of values
This is the case people get wrong. With an even count there is no single middle value, so you take the two middle values and average them.
Median of 4 numbers. Take the scores 85, 90, 80 and 95. In order they are 80, 85, 90, 95. With 4 values the middle two are the 2nd and the 3rd, which are 85 and 90. Their mean is (85 + 90) ÷ 2 = 175 ÷ 2 = 87.5, so the median is 87.5. Here the mean and the median happen to be the same number, which is common when the values are evenly spread.
Median of 6 numbers. Take 6, 9, 14, 21, 22 and 30, which are already in order. With 6 values the middle two are the 3rd and the 4th, which are 14 and 21. Their mean is (14 + 21) ÷ 2 = 35 ÷ 2 = 17.5, so the median is 17.5. Note that 17.5 is not one of the values in the set: the median does not have to be a number that actually appears.
The general rule for an even count of n values is that the two middle positions are n ÷ 2 and (n ÷ 2) + 1.
How to calculate the mode
- Put the values in order, which makes repeats easy to see.
- Count how many times each value occurs.
- The mode is the value with the highest count.
Example. The scores 85, 90, 85 and 95, in order, are 85, 85, 90, 95. The value 85 appears twice and everything else appears once, so the mode is 85.
When there is no mode
Take 3, 8, 11 and 14. Every value appears exactly once, so no value is more frequent than any other and the set has no mode. That is a valid answer, and in a multiple choice question “no mode” is often one of the options.
When there are two modes
Take 2, 4, 4, 7, 9, 9 and 11. The value 4 appears twice and the value 9 appears twice, and everything else appears once. Two values tie for most frequent, so the set has two modes, 4 and 9, and is described as bimodal. Give both. A set can have more than two modes for the same reason.
Which average a question is asking for
- The word “average” on its own means the mean.
- Wording about a typical value in data with extremes, such as salaries or house prices, points to the median.
- “Most common”, “most popular”, or anything about categories, points to the mode.
- In a symmetrical distribution, the familiar bell curve, all three land on the same number. The further the data is from symmetrical, the further apart they sit.
Numerical test items often pair these with the range, which is the largest value minus the smallest. For 6, 9, 14, 21, 22 and 30 the range is 30 - 6 = 24.
Practice questions
1. Find the mean of 14, 19, 11 and 16.
2. Find the median of 31, 24, 40 and 27.
3. A team logs these numbers of calls per day: 12, 15, 12, 19, 15 and 21. Give the mode, the median and the mean.
Answers
1. 14 + 19 + 11 + 16 = 60, and 60 ÷ 4 = 15. The mean is 15.
2. In order: 24, 27, 31, 40. With 4 values the middle two are 27 and 31, so the median is (27 + 31) ÷ 2 = 58 ÷ 2 = 29.
3. In order: 12, 12, 15, 15, 19, 21. Both 12 and 15 appear twice, so the set is bimodal and the modes are 12 and 15. With 6 values the middle two are the 3rd and the 4th, both of which are 15, so the median is 15. The total is 12 + 15 + 12 + 19 + 15 + 21 = 94, and 94 ÷ 6 = 15.67 to two decimal places.
Practice these under test conditions
Mean, median and mode turn up throughout numerical reasoning tests, usually inside a table or chart question rather than as a bare list of numbers, so the calculation is only half the work. For the underlying arithmetic against the clock, work through basic numeracy tests, and to see how these questions sit within a full paper, try our free aptitude tests.