Rukasha Maths

GCSE maths Beta

Averages, frequency tables and range · Foundation notes

Foundation · Statistics · AQA references S2, S4, S5

Averages summarise a data set, but different averages answer different questions. The mean shares the total equally, the median is the middle value when the data are ordered, and the mode is the most frequent value. The range describes spread rather than a typical value. An extreme value can change the mean and range a lot, while the median often changes less. A frequency table records repeated values efficiently: multiply each value by its frequency before finding the mean, and use the total frequency as the number of data values.

Learning objectives

  • Calculate mean, median, mode and range.
  • Find a mean from a frequency table.

Key ideas

The mean uses every value, including repetitions.

mean=sum of valuesnumber of values\text{mean}=\frac{\text{sum of values}}{\text{number of values}}

Order values for the median. For an even count, average the two middle values.

The mode is the most frequent value. There may be several modes or no unique mode.

Range measures the spread between the extremes.

range=maximum−minimum\text{range}=\text{maximum}-\text{minimum}

For a frequency table, multiply each value by its frequency before adding.

mean=∑xf∑f\text{mean}=\frac{\sum xf}{\sum f}

Worked examples

Example 1

Books read are zero, one and two, with respective frequencies two, three and five. Find the mean.

∑xf=0(2)+1(3)+2(5)=13\sum xf=0(2)+1(3)+2(5)=13
∑f=2+3+5=10\sum f=2+3+5=10
mean=1310=1.3 books\text{mean}=\frac{13}{10}=1.3\text{ books}

Answer

1.3 books1.3\text{ books}

Example 2

Find the median of the six given values.

Order the values.

9,2,6,3,8,4⟶2,3,4,6,8,99,2,6,3,8,4\quad\longrightarrow\quad2,3,4,6,8,9
median=4+62=5\text{median}=\frac{4+6}{2}=5

Answer

55

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Calculate a mean, median, mode or range from data or a frequency table.

AO2 · Reasoning

Compare data using an appropriate average and a measure of spread.

AO3 · Problem solving

Recover a missing value from a mean or interpret an average in context.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

A frequency counts repetitions. Do not divide a frequency-table total by the number of distinct values.

Try it yourself

Attempt each question before opening its answer.

Check 1: Find the median of 9, 2, 6, 3, 8, 4.

Answer: 5

Order the values: 2, 3, 4, 6, 8, 9. The middle values are 4 and 6, so the median is (4 + 6)/2 = 5.

Check 2: Find the range of -4, 1, 7, -2.

Answer: 11

The largest value is 7 and the smallest is -4. Range = 7 - (-4) = 11.

Explain it in your own words

  • Write one data set where the mean, median and mode are different.
  • Explain why the range is a measure of spread rather than an average.
  • Show how one very large outlier changes a mean more than a median.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All foundation notes

Specification source

AQA Statistics specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.