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The mean is the total divided by the number of values, the mode is the most common value, and the range is the largest minus the smallest. Free notes.
Measures of Central Tendency and Spread is part of Averages and Spread in GCSE AQA Higher Maths.
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The mean is the total divided by the number of values, the mode is the most common value, and the range is the largest minus the smallest. Free notes.
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Use this lesson to stay inside Averages and Spread while connecting the detail back to the wider Statistics topic.
Read the core explanations from Measures of Central Tendency and Spread before testing yourself from memory.
Two football teams both score an average of 2 goals per game. Does that mean they perform identically? Not necessarily — one team might score exactly 2 every match, while the other swings between 0 and 6. The average tells you the centre; the spread tells you the story.
This is exactly what the examiner expects you to do: not just calculate averages, but use them alongside spread to compare and interpret distributions. Both skills appear together on exam papers, so you need both in your toolkit.
There are three measures of central tendency you need to know: the mean, the median, and the mode. Each one describes the centre of a data set differently, and each has situations where it works best.
The median is the middle value when all data is arranged in order. If there is an even number of values, the median is the mean of the two middle values. Because it sits in the middle of ordered data, it is not affected by extreme values the way the mean is.
The mode is the value that appears most often. A data set can have more than one mode, or no mode at all if every value appears the same number of times. For grouped data, you identify the modal class — the class interval with the highest frequency.
Theory is straightforward; the exam tests whether you can apply these methods accurately under pressure. Work through this example step by step, paying attention to the ordering step for the median — it is where most marks are dropped.
Worked example: Finding Mean, Median and Mode from Raw Data
The ages of seven students in a revision group are: 15, 16, 14, 16, 15, 17, 16. Find the mean, median and mode.
Answer: Mode = 16, Median = 16, Mean ≈ 15.6
Notice that when data is fairly symmetric and has no extreme values, the mean, median and mode are close together — as they are here. When they differ significantly, that is a signal worth investigating.
Exam questions often present data in a frequency table rather than as a raw list. The method adapts slightly: you multiply each value by its frequency, sum those products, then divide by the total frequency.
Worked example: Calculating the Mean from a Frequency Table
The table shows the number of siblings reported by 20 students. Calculate the mean number of siblings. Siblings (x): 0, 1, 2, 3, 4 Frequency (f): 4, 7, 5, 3, 1
Answer: Mean = 1.5 siblings
Now that you have the centre of a data set, you need to describe how spread out the values are. The range is the measure of spread at GCSE level: it is the difference between the largest and smallest values.
The range is quick to calculate, but it has a weakness: it depends entirely on the two extreme values. This is where outliers become important. An outlier is a value that sits far away from the rest of the data — and a single outlier can dramatically inflate the range, making the data look far more spread out than it really is.
For example, if nine students score between 55 and 72 on a test, but one student scores 12, the range becomes $72 - 12 = 60$. Remove that outlier and the range drops to $72 - 55 = 17$. The outlier has completely changed the picture. This is why GCSE Higher exam questions often ask you to consider the effect of outliers on your measures.
Knowing how to calculate each average is only half the skill. The other half is knowing which one to use — and why. This comes up regularly in GCSE Higher exam questions that ask you to 'explain which average is most appropriate'.
| Measure | Best used when... | Weakness |
|---|---|---|
| Mean | Data has no extreme outliers and you want to use all values | Distorted by outliers — one extreme value pulls it away from the centre |
| Median | Data has outliers or is skewed — it is resistant to extreme values | Does not use all the data values in its calculation |
| Mode | Data is categorical (e.g. shoe sizes, favourite colours) or you need the most common value | May not exist, or there may be multiple modes; not useful for numerical analysis |
| Modal class | Data is grouped in a frequency table and individual values are unknown | Only gives the class interval, not a precise value |
A classic exam scenario: a company reports the 'average' salary of its employees as £45,000. If the CEO earns £500,000 and everyone else earns around £25,000, the mean is misleading. The median would give a far more representative picture of what a typical employee earns.
This is where the specification really tests your understanding. Comparing two data sets means using both a measure of central tendency and a measure of spread — and linking your comments back to the context of the question.
A comparison that only mentions averages is incomplete. A comparison that only mentions spread is also incomplete. mark schemes award marks for both, so you need both in your answer.
Worked example: Comparing Two Data Sets — Exam-Style Question
Two athletes record their 100 m sprint times (in seconds) over eight races. Aisha: 12.1, 12.4, 12.0, 12.3, 12.2, 12.5, 12.1, 12.4 Ben: 11.8, 13.1, 12.0, 11.9, 13.4, 12.2, 11.7, 13.1 Compare the performance of the two athletes. [4 marks]
Answer: Aisha is faster on average (mean 12.25 s vs 12.4 s) and more consistent (range 0.5 s vs 1.7 s).
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-21.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
The mode is the value that appears most often in a data set. A data set can have more than one mode, or no mode at all if every value appears the same number of times. In the ages 14, 15, 15, 16, 16, 16, 17, the mode is 16 because it appears three times.
The mean is calculated by adding all the values together and dividing by how many values there are. For the ages 14, 15, 15, 16, 16, 16 and 17, the total is 109, and 109 ÷ 7 ≈ 15.6, so the mean age is about 15.6.
The range is the difference between the largest and smallest values in a data set, so range = largest value − smallest value. It is the measure of spread used at GCSE. For sprint times from 11.7 to 13.4 seconds, the range is 13.4 − 11.7 = 1.7 seconds.
To find the mean from a frequency table, multiply each value by its frequency to make an fx column, add up the fx column, then divide by the total frequency. For example, 20 students with a total fx of 30 siblings give a mean of 30 ÷ 20 = 1.5.
The three measures of central tendency are the mean, the median and the mode. Each one describes the centre of a data set differently: the mean adds all values and divides by how many there are, the median is the middle value in order, and the mode is the most frequent value.
An outlier is a value that sits far away from the rest of the data. A single outlier can dramatically inflate the range, making the data look far more spread out than it really is, and it also pulls the mean away from the centre.
A data set can have more than one mode, because the mode is simply the value that appears most often and several values can tie. A data set can also have no mode at all if every value appears the same number of times, which is one weakness of the mode.
The mean is affected by outliers because it uses every data point, so one extreme value pulls it away from the centre. If a CEO earns £500,000 and everyone else earns around £25,000, the mean salary is misleading and the median gives a more representative picture of a typical employee.
The mean is all the values added together and divided by how many values there are, the median is the middle value when the data is in order, and the mode is the value that appears most often. The mean uses every data point, so it is sensitive to extreme values; the median is not.
An outlier can dramatically inflate the range, because the range depends entirely on the largest and smallest values. If nine students score between 55 and 72 but one scores 12, the range is 72 − 12 = 60; without the outlier it drops to 72 − 55 = 17.
Use the median when the data has outliers or is skewed, because it is resistant to extreme values, while the mean is distorted by them. If a CEO earns £500,000 and everyone else earns around £25,000, the median gives a far more representative picture of a typical salary than the mean.
The modal class is the class interval with the highest frequency in grouped data. It is used when data is grouped in a frequency table and the individual values are unknown. Its weakness is that it only gives the class interval, not a precise value.
Two data sets are compared using both a measure of central tendency, such as the mean or median, and a measure of spread, such as the range, and each comment is linked back to the context. A comparison that only mentions averages, or only mentions spread, is incomplete.
The position of the median is (n + 1) ÷ 2, where n is the number of values, counted along the data once it is in order. For 7 values, (7 + 1) ÷ 2 = 4, so the median is the 4th value. With an even number of values, the median is the mean of the two middle values.
The mode is the best average when data is categorical, such as shoe sizes or favourite colours, or when the most common value is needed. Its weaknesses are that it may not exist, there may be multiple modes, and it is not useful for numerical analysis.