Mean Median Mode and Range –

Mean, median, mode and range are the four key statistical measures you must know for GCSE Maths. They appear on every exam board — AQA, Edexcel and OCR — and crop up in both Foundation and Higher papers. Whether the data is presented as a list, a table or a grouped frequency chart, you need to be confident selecting the right average and interpreting what it tells you about the data set. This guide takes you through each measure, gives clear worked examples at both tiers, flags the mistakes examiners see every year, and finishes with practice questions. For a full overview of every topic, see our complete GCSE Maths topics list.

What Is an Average?

An average is a single value that represents a data set. In GCSE Maths you work with three types of average — the mean, the median and the mode — plus the range, which measures spread.

Mean

The mean is the most commonly used average. You calculate it by adding all the values together and dividing by how many values there are.

The mean uses every piece of data, which makes it representative, but it can be affected by extreme values (outliers).

Median

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

• If there is an odd number of values, the median is the exact middle value.
• If there is an even number of values, the median is the midpoint of the two central values — add them together and divide by two.

The median is not affected by outliers, so it is often a better average when data contains extreme values.

Mode

The mode is the value that appears most often. A data set can have no mode, one mode, or more than one mode (bimodal or multimodal). The mode is the only average that can be used with non-numerical (qualitative) data, such as favourite colours.

Range

The range measures the spread of the data.

A large range means the data is widely spread; a small range means the data is consistent. The range is heavily affected by outliers.

Step-by-Step Method

Finding the Mean

1. Add every value in the data set together to find the total.
2. Count how many values there are.
3. Divide the total by the number of values.
4. Give your answer to a suitable degree of accuracy (the question will tell you if rounding is needed).

Finding the Median

1. Arrange the values in ascending order.
2. Count the number of values, n.
3. Use the formula (n + 1) ÷ 2 to find the position of the median.
4. If n is odd, the median is the value at that position. If n is even, the position will be X.5 — find the mean of the values either side.

Finding the Mode

1. Look at the data set and identify which value appears most frequently.
2. If two or more values share the highest frequency, they are all modes.

Finding the Range

1. Identify the highest and lowest values.
2. Subtract the lowest from the highest.

Worked Example 1 — Foundation Level

Question: A pupil scored the following marks in seven spelling tests: 6, 9, 3, 7, 9, 5, 8. Find the mean, median, mode and range.

Working:

Mean: Total = 6 + 9 + 3 + 7 + 9 + 5 + 8 = 47. Number of values = 7. Mean = 47 ÷ 7 = 6.71 (to 2 d.p.).

Median: Arrange in order: 3, 5, 6, 7, 8, 9, 9. Position = (7 + 1) ÷ 2 = 4th value. The 4th value is 7.

Mode: The value 9 appears twice; all others appear once. Mode = 9.

Range: 9 − 3 = 6.

Worked Example 2 — Higher Level

Question: The table below shows the number of goals scored by a football team in 20 matches.

Calculate the mean number of goals per match and find the median.

Working:

Mean: Multiply each value by its frequency, then add the products.

• 0 × 4 = 0
• 1 × 6 = 6
• 2 × 5 = 10
• 3 × 3 = 9
• 4 × 2 = 8

Total = 0 + 6 + 10 + 9 + 8 = 33. Total frequency = 4 + 6 + 5 + 3 + 2 = 20.

Mean = 33 ÷ 20 = 1.65 goals.

Median: There are 20 values, so the median lies between the 10th and 11th values. Using cumulative frequency: values 1–4 are 0, values 5–10 are 1, values 11–15 are 2. The 10th value is 1 and the 11th value is 2, so the median = (1 + 2) ÷ 2 = 1.5 goals.

Common Mistakes

• Forgetting to order the data before finding the median. The median only works when values are in ascending order.
• Confusing the position with the value for the median. If the formula gives position 4, the median is the value at position 4, not the number 4.
• Dividing by the wrong number when calculating the mean from a frequency table — you must divide by the total frequency, not the number of rows.
• Saying there is no mode when two values share the highest frequency — the data set is bimodal and both values are modes.
• Including outliers carelessly when asked which average best represents the data — mention that the mean is affected by outliers while the median is not.

Exam Tips

1. Show every step — write down the sum and the count separately before dividing for the mean. This earns method marks even if your final answer is wrong.
2. Use the (n + 1) ÷ 2 rule to find the median position. Examiners expect to see this working.
3. Read the question carefully — if it says "estimate the mean", you are likely dealing with grouped data and should use midpoints. See our guide on frequency tables and grouped data.
4. Check units and rounding — the question may specify "to 1 decimal place" or "to 2 significant figures".
5. For formula reminders, see our GCSE Maths formulas list.

Practice Questions

Question 1 (Foundation): The ages of five friends are 12, 14, 14, 15 and 17. Find the mean, median, mode and range.

Question 2 (Foundation): A shop records how many umbrellas it sells each day for a week: 3, 0, 5, 2, 8, 1, 2. Find the median and range.

Question 3 (Higher): The table shows the number of pets owned by 30 students.

Calculate the mean number of pets.

Question 4 (Higher): Two data sets have the following means: Set A mean = 12, n = 5; Set B mean = 18, n = 3. Find the overall mean of both sets combined.

Ready to practise these skills with instant, personalised feedback? Try our AI-powered GCSE Maths tutor at gcsemathsai.co.uk — it adapts to your level and helps you build confidence before exam day.

Related Topics

• Frequency Tables and Grouped Data
• Bar Charts, Pie Charts and Pictograms
• Cumulative Frequency and Box Plots
• Sampling Methods

Summary

The mean, median, mode and range are essential statistical tools in GCSE Maths. The mean adds all values and divides by the count — it uses every data point but is sensitive to outliers. The median is the middle value when data is ordered — it resists extreme values. The mode is the most frequent value and can apply to non-numerical data. The range measures spread by subtracting the smallest value from the largest. At Foundation level, you will work mostly with listed data; at Higher level, expect frequency tables and questions asking you to compare data sets using averages and range. Always show your working clearly, order data before finding the median, and check whether the question asks for an exact answer or a rounded one.