Mean, Median, Mode and Range: KS3 Maths Guide

Mean, median, and mode are three measures of average that describe the typical value in a data set. Range measures how spread out the data is. Together, these four statistics are the foundation of all data handling at KS3 and GCSE.

What is the mean?

The mean is the most common type of average. It is calculated by adding all the values and dividing by the number of values.

Formula: Mean = (sum of all values) ÷ (number of values)

Worked example 1

Find the mean of: 5, 8, 3, 12, 7.

Sum: 5 + 8 + 3 + 12 + 7 = 35
Number of values: 5
Mean: 35 ÷ 5 = 7

Answer: mean = 7

Worked example 2

A student scores 62, 75, 58, 81, and 74 in five tests. Find the mean score.

Sum: 62 + 75 + 58 + 81 + 74 = 350
Mean: 350 ÷ 5 = 70

Answer: mean = 70

When to use the mean

The mean uses every value in the data set, which makes it the most representative average when the data has no extreme outliers. However, a single very large or very small value can pull the mean away from the typical value — in those cases, the median is often more informative.

What is the median?

The median is the middle value when data is arranged in order from smallest to largest.

If there is an odd number of values, the median is the middle one.

If there is an even number of values, the median is the mean of the two middle values.

Position of the median: (n + 1) ÷ 2, where n is the number of values.

Worked example 3: odd number of values

Find the median of: 3, 8, 1, 5, 9, 2, 7.

Arrange in order: 1, 2, 3, 5, 7, 8, 9
Number of values: 7. Middle position: (7 + 1) ÷ 2 = 4th value.
4th value = 5

Answer: median = 5

Worked example 4: even number of values

Find the median of: 12, 4, 19, 7, 15, 6.

Arrange in order: 4, 6, 7, 12, 15, 19
Number of values: 6. Middle positions: 3rd and 4th values.
3rd value = 7, 4th value = 12. Median = (7 + 12) ÷ 2 = 19 ÷ 2 = 9.5

Answer: median = 9.5

When to use the median

The median is not affected by outliers, which makes it a better average for skewed data sets. For example, the median house price is more useful than the mean house price in an area where a few very expensive properties would distort the mean.

What is the mode?

The mode is the value that appears most often in a data set.

A data set can have one mode, more than one mode (bimodal or multimodal), or no mode (if all values appear equally often).

The mode is the only average that can be used with non-numerical (categorical) data.

Worked example 5

Find the mode of: 4, 7, 2, 7, 3, 9, 7, 1.

Count the frequencies:
1 appears once, 2 once, 3 once, 4 once, 7 three times, 9 once.

Answer: mode = 7

Worked example 6: two modes

Find the mode(s) of: 5, 3, 8, 3, 5, 1, 6.

3 appears twice, 5 appears twice — all other values once.

Answer: modes = 3 and 5 (bimodal)

When to use the mode

The mode is most useful when you want to know the most popular or most common item — for example, the most frequently ordered shoe size, or the most common response to a survey question.

What is the range?

The range is not a measure of average — it is a measure of spread. It shows how far apart the smallest and largest values are.

Formula: Range = highest value − lowest value

A small range means the data is clustered together; a large range means the data is spread out.

Worked example 7

Find the range of: 14, 3, 22, 9, 17.

Highest: 22. Lowest: 3.
Range: 22 − 3 = 19

Answer: range = 19

Comparison table

Measure | What it tells you | Affected by outliers? | Can be used with non-numerical data?
Mean | Arithmetic average of all values | Yes | No
Median | Middle value when ordered | No | No
Mode | Most frequent value | No | Yes
Range | Spread of the data | Yes | No

Using all four measures together

Worked example 8

A Year 8 class records how many books they read over the summer: 0, 1, 1, 2, 2, 2, 3, 4, 5, 20.

Mean: (0 + 1 + 1 + 2 + 2 + 2 + 3 + 4 + 5 + 20) ÷ 10 = 40 ÷ 10 = 4
Median: 10 values; middle two are the 5th and 6th: both are 2. Median = (2 + 2) ÷ 2 = 2
Mode: 2 (appears three times)
Range: 20 − 0 = 20

Interpretation: The mean of 4 is pulled up by the outlier 20. The median and mode of 2 better represent the typical student. The large range (20) shows how spread out the data is.

Common mistakes to avoid

Mistake 1 — Not arranging data in order before finding the median.
The median is not the value in the middle of the list as written — it is the middle value after sorting.

Mistake 2 — Finding the mean of a list with different frequencies.
If a value appears multiple times, multiply it by its frequency before adding. Do not count it just once.

Mistake 3 — Thinking range is an average.
Range measures spread, not average. It is reported alongside the averages, not instead of them.

Mistake 4 — Calculating median wrongly for an even-count data set.
Find the mean of the two middle values; do not simply pick one of them.

How averages fit the KS3 national curriculum

The Department for Education's KS3 mathematics programme of study requires pupils to "describe, interpret, and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous, and grouped data; and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)." The AQA GCSE Mathematics specification positions averages and range as core statistical skills that underpin data interpretation across science, geography, and everyday life.

Frequently asked questions

Which average is the best one to use?

There is no single best average — it depends on the data. Use the mean when data is roughly symmetrical and has no extreme outliers. Use the median when data is skewed or contains outliers. Use the mode when you want the most popular value, or when data is categorical (such as favourite subjects). In KS3 and GCSE questions, you may be asked to justify your choice, so always explain why.

Can the mean, median, and mode all be the same?

Yes. In a perfectly symmetrical, unimodal distribution — such as 2, 4, 6, 8, 10 — the mean, median, and mode can all coincide. In this example, mean = (2+4+6+8+10)/5 = 6, median = 6, and there is no mode. The three measures are equal more commonly in idealised distributions than in real-world data.

What happens to the mean if I add the same number to every value?

The mean increases by that same number. For example, if the mean of a data set is 12 and you add 5 to every value, the new mean is 17. This is because adding 5 to each of n values increases the total sum by 5n, and 5n ÷ n = 5. The median and mode also increase by 5; the range does not change.

How do I find the mean from a frequency table?

Multiply each value by its frequency, sum the results, then divide by the total frequency. For example: value 3 appears 4 times (contributing 12), value 5 appears 2 times (contributing 10), value 7 appears 1 time (contributing 7). Total = 29, total frequency = 7. Mean = 29 ÷ 7 ≈ 4.14.

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