Mean, median, mode and range, explained

By The STEM-Tutor team · Updated October 2026

An average is a single value that represents a whole set of data. GCSE Maths uses three averages, the mean, median and mode, plus the range to measure spread. Each has its own strengths, and knowing which to use is part of the skill.

The three averages and the range

• Mean: add up all the values and divide by how many there are.
• Median: put the values in order and take the middle one. With an even number of values, take the mean of the middle two.
• Mode: the value that appears most often. There can be more than one mode, or none.
• Range: the largest value minus the smallest. It measures spread, not average.

A memory aid: mode is the most common; the median is in the middle (like the central reservation, or median strip, of a road); the mean is the one you have to work out.

Worked examples

Example 1: all four

Find the mean, median, mode and range of 4, 7, 2, 9, 7, 5, 8.

• Order: 2, 4, 5, 7, 7, 8, 9.
• Mean = 42/7 = 6. Median (4th value) = 7. Mode = 7. Range = 9 − 2 = 7.

Example 2: an even number of values

Find the median of 12, 3, 8, 15, 6, 10.

• Order: 3, 6, 8, 10, 12, 15. The middle two are 8 and 10.
• Median = (8 + 10)/2 = 9. The median does not have to be one of the data values.

Example 3: a missing value

The mean of five numbers is 12. Four of them are 10, 15, 9 and 14. Find the fifth.

• The total must be 5 × 12 = 60.
• The four known numbers add to 48, so the fifth is 12. Working with totals is the key to every missing-value question.

Example 4: a changed mean

The mean height of 10 students is 152 cm. A new student of height 163 cm joins. Find the new mean.

• Old total = 10 × 152 = 1520 cm. New total = 1683 cm.
• New mean = 1683/11 = 153 cm.

Example 5: combining two groups

Class A has 20 students with a mean score of 70. Class B has 30 students with a mean score of 60. Find the mean score of all 50 students.

• Totals: 20 × 70 = 1400 and 30 × 60 = 1800, so 3200 altogether.
• Combined mean = 3200/50 = 64. It is not 65, the average of 70 and 60, because class B has more students and so pulls the mean towards its value.

Averages from a frequency table

When data comes in a frequency table, each value appears as many times as its frequency.

Example 6: a frequency table

The number of goals scored in 20 matches: 0 goals, 4 matches; 1 goal, 7; 2 goals, 5; 3 goals, 3; 4 goals, 1. Find the mean, median and mode.

• Mean: total goals = 0 + 7 + 10 + 9 + 4 = 30, so mean = 30/20 = 1.5 goals.
• Median: with 20 values, it lies between the 10th and 11th. Running totals: 4, 11, …. Both the 10th and 11th values are 1, so median = 1 goal.
• Mode: the highest frequency is 7, for 1 goal. Note the mode is the value, not its frequency.

Estimating the mean from grouped data

With grouped data, such as heights in ranges, you do not know each exact value. Use the midpoint of each class as an estimate, multiply by the frequency, add up, and divide by the total frequency. The answer is an estimate, and exam questions will say so.

Example 7: grouped data

Times to run 100 m: 12–14 s, 3 students; 14–16 s, 8; 16–18 s, 6; 18–20 s, 3. Estimate the mean.

• Midpoints: 13, 15, 17, 19. Products: 39, 120, 102, 57. Total = 318.
• Estimated mean = 318/20 = 15.9 s.
• The modal class is 14–16 s, and the median is in the 14–16 class too, since the 10th and 11th values lie there.

Averages in everyday life

News reports, sports statistics and school reports use averages all the time, and the choice matters. Average house prices are usually given as medians, because a few very expensive homes would push the mean up. A shoe shop orders the modal size most often. A teacher might report a class's mean test score. When you read a statistic, ask which average was used and whether it gives a fair picture.

Which average should you use?

• Mean: uses every value, so it is the most representative when the data has no extreme values. But a single outlier can drag it a long way.
• Median: not affected by extreme values, so it is better for skewed data such as house prices or salaries.
• Mode: the only average for non-numerical data, such as favourite colours or shoe styles in a shop.

Example 8: choosing an average

Five salaries at a small company are £22,000, £24,000, £25,000, £26,000 and £150,000. Which average best represents a typical salary?

• Mean = £49,400, which is higher than four of the five salaries.
• Median = £25,000, which is much more typical. The one very high salary distorts the mean, so the median is the better choice.

Comparing two sets of data

To compare two groups, make two comparisons: one using an average and one using a measure of spread such as the range. Always explain in context. For example: “Class A had a higher mean score (68 compared with 61), so they did better on average. Class A's range was smaller (20 compared with 35), so their scores were more consistent.”

Common mistakes

• Forgetting to order the data before finding the median.
• Giving the frequency instead of the value as the mode.
• Dividing by the number of rows instead of the total frequency in a table.
• Using class boundaries instead of midpoints for grouped data.
• Comparing data without context or without a measure of spread.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are averages on the Foundation paper?

Yes. Mean, median, mode and range, including from frequency tables, are on both Foundation and Higher papers. Estimating the mean from grouped data appears on both too.

What is the difference between range and interquartile range?

The range uses the largest and smallest values, so it is affected by outliers. The interquartile range measures the spread of the middle half and is a Higher tier topic.

Key takeaways

• Mean: total ÷ number of values.
• Median: the middle value of ordered data.
• Mode: the most common value; range: largest minus smallest.
• Use totals for missing-value questions.
• Use midpoints to estimate the mean of grouped data.

Related guides

• Box plots
• Histograms
• Cumulative frequency