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Mean, Median and Mode

FoundationHigherAQAEdexcel

Revise Mean, Median and Mode for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. The mean, median and mode are three measures of average, each summarising a data set with a single value.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

There are three averages, each summarising a data set in a different way, and each with situations where it is the most appropriate choice.

The mean is the total of all values divided by how many there are. The median is the middle value when the data is in order. The mode is the most frequently occurring value.

Each responds differently to extreme values. The mean uses every value, so a single outlier pulls it substantially. The median depends only on position, so it is unaffected by extremes. The mode may not exist at all, or there may be more than one, and it is the only average available for qualitative data.

Revision notes

Calculating each average

Mean \(= \frac{\text{sum of values}}{\text{number of values}}\).

Median: put the data in order and find the middle value. With an even number of values, take the mean of the middle two. Mode: the most frequently occurring value.

Effect of extreme values

The mean uses every value, so one outlier can pull it substantially.

The median depends only on position in the ordered list, so it is unaffected by extremes. This is why the median is preferred for skewed data such as house prices or salaries.

When each is available

The mean and median require quantitative data.

The mode is the only average available for qualitative data. There may be no mode if every value appears once, or several modes if values tie.

Key points

  • The mean is the total divided by the count.
  • The median is the middle value in order.
  • The mode is the most common value.
  • The mean is affected by extreme values.
  • The median is unaffected by extremes.
  • The mode is the only average for qualitative data.

Worked examples

Example 1

Work out the mean of 4, 7, 9, 12 and 8. [2 marks]

Working

\(4 + 7 + 9 + 12 + 8 = 40\)add all the values
\(\frac{40}{5} = 8\)divide by the number of values

Example 2

Work out the median of 3, 8, 5, 12 and 7. [2 marks]

Working

In order: 3, 5, 7, 8, 12put the values in ascending order
The middle value is 7identify the middle value

Example 3

Explain why the median is often preferred to the mean for house prices. [2 marks]

Working

House prices include a few very expensive properties that act as outliersidentify the feature of the data
The mean would be pulled upwards by these, whereas the median is unaffected and better represents a typical priceexplain why the median is better

Common mistakes

  • Forgetting to order the data before finding the median.

    The median is the middle value of the ORDERED list.

  • Dividing by the wrong number for the mean.

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

  • Saying there is always a mode.

    There may be none, or more than one.

  • Using the mean for heavily skewed data.

    The median is usually more representative when outliers are present.

Exam tips

  • Order the data first when finding the median.
  • Show the total and the division separately for the mean.
  • Check whether the data has outliers before choosing an average.
  • Remember the mode works for qualitative data.

Key terms

Mean
The total of values divided by the number of values.
Median
The middle value of an ordered data set.
Mode
The most frequently occurring value.
Outlier
An extreme value that can distort the mean.

Related topics

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