Averages and spread

Calculate and compare mean, median, mode and range, and reason about how data changes affect them.

Also known as: mean median mode, range, standard deviation, statistics.

SAT · Problem-Solving and Data Analysis GCSE Maths · Statistics IGCSE Maths · Statistics

Key ideas

Mean = total ÷ number of values; total = mean × number of values.
Median is the middle value of ordered data (average the two middle values if there is an even count).
Range = largest − smallest; standard deviation measures typical distance from the mean.
Outliers affect the mean far more than the median.

Measures of average

Mean: add the values and divide by how many there are. The mean of $4, 7, 9, 12, 8$ is $\frac{40}{5} = 8$ .
Median: order the data and take the middle. For $3, 4, 5, 7, 8, 9$ the median is $\frac{5 + 7}{2} = 6$ .
Mode: the most common value.

Working backwards with the mean

Because total $=$ mean $\times$ count, missing values are easy to find. If $5$ numbers have mean $12$ , their total is $60$ .

To combine groups, combine totals, not means: a class of $20$ with mean $70$ and a class of $30$ with mean $80$ have combined mean $\frac{1400 + 2400}{50} = 76$ .

Measures of spread

The range is the difference between the largest and smallest values. Standard deviation describes how spread out values are around the mean: $10, 20, 30, 40, 50$ and $28, 29, 30, 31, 32$ both have mean $30$ , but the first set has a much larger standard deviation.

Worked examples

Example 1 · Combined mean

A class of 20 has a mean score of 70 and a class of 30 has a mean score of 80. What is the mean of all 50 students?

Step 1.Totals: $20 \times 70 = 1400$ and $30 \times 80 = 2400$ .
Step 2.Combined mean $= \frac{3800}{50} = 76$ .

Answer: 76

Example 2 · Effect of changing a value

For the data $3, 5, 6, 8, 10$ , what happens if $10$ is replaced by $15$ ?

Step 1.The total increases by 5, so the mean increases by 1.
Step 2.The middle value is still 6, so the median does not change.

Answer: Mean increases from 6.4 to 7.4; median stays 6

Common mistakes

Finding the median without first ordering the data.
Averaging two group means directly when the groups are different sizes.
Confusing range (a spread) with an average.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Averages and spreadEasier

What is the mean of $4, 7, 9, 12, 8$ ?

Question 2Averages and spreadEasier

What is the median of $3, 9, 4, 7, 5, 8$ ?