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Averages and Range

FoundationChallengeAQAEdexcelOCR

Get to grips with averages and range using these Foundation GCSE Maths practice questions. The worksheet focuses on mean, median, mode and range, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. Mean, median and mode are averages; the range measures spread.

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Topic overview

Averages summarise a set of data with a single representative value. There are three: the mean, the median and the mode. The range is not an average — it measures spread instead.

The mean is the total divided by how many values there are. The median is the middle value once the data is in order. The mode is the value appearing most often.

Each has strengths. The mean uses every value but is distorted by extremes. The median ignores extremes, which makes it better for skewed data. The mode is the only average that works for non-numerical data such as favourite colours.

Revision notes

Finding each average

Mean: add all the values and divide by how many there are. Median: put the values in order and take the middle one. Mode: find the most frequent value.

For \(3, 7, 7, 9, 14\): the mean is \(8\), the median is \(7\) and the mode is \(7\).

The median with an even number of values

With an even count there are two middle values, so take their mean.

For \(2, 5, 8, 11\), the middle two are \(5\) and \(8\), so the median is \(6.5\). Ordering the data first is essential and is where most errors occur.

The range

The range is the largest value minus the smallest, and it measures spread rather than average.

For \(3, 7, 7, 9, 14\) the range is \(14 - 3 = 11\). A smaller range means the data is more consistent.

Key points

  • The mean is the total divided by the number of values.
  • The median is the middle value when ordered.
  • The mode is the most common value.
  • The range measures spread, not average.
  • With an even count, average the two middle values.
  • Always order the data before finding the median.

Worked examples

Example 1

Find the mean of \(4, 8, 9, 11, 13\).

Working

\[4+8+9+11+13 = 45\]add all the values
\[45 \div 5\]divide by how many there are
\[= 9\]state the mean

Example 2

Find the median of \(12, 5, 9, 3, 7\).

Working

\[3, 5, 7, 9, 12\]put the values in order first
\[= 7\]the middle value of five

Example 3

Find the range of \(6, 14, 9, 21, 8\).

Working

\[21 - 6\]largest minus smallest
\[= 15\]state the range

Common mistakes

  • Finding the median without ordering the data.

    The values must be in order first, or the middle one is meaningless.

  • Calling the range an average.

    It measures spread, not a typical value.

  • Taking only one middle value when the count is even.

    With an even number, average the two middle values.

  • Dividing by the wrong number for the mean.

    Divide by how many values there are, not by the largest one.

Exam tips

  • Order the data before finding the median.
  • Count the values carefully before dividing for the mean.
  • Remember the range is spread, not an average.
  • State which average you have found.

Key terms

Mean
The total divided by the number of values.
Median
The middle value when data is ordered.
Mode
The most frequently occurring value.
Range
The difference between the largest and smallest values.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.