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Combined Mean

FoundationHigherAQAEdexcelOCR

Combined Mean is a key statistics topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on calculating a combined mean, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Multiply each mean by its count, add, then divide by the total count.

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

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A combined mean is found when two groups are merged. You cannot simply average the two means, because the groups usually contain different numbers of values.

The correct method reconstructs the totals. Multiply each mean by its group size to recover each group's total, add those totals, then divide by the combined number of values.

Averaging the means directly gives the right answer only when the groups happen to be the same size. In every other case it is wrong, and it is one of the most reliably tested misconceptions in the topic.

Revision notes

Recovering the totals

Multiply each mean by the number of values in that group.

If \(12\) students average \(60\), their total is \(720\). If \(8\) students average \(70\), their total is \(560\).

Combining

Add the totals and divide by the combined count.

Here \(720 + 560 = 1280\), and \(12 + 8 = 20\), so the combined mean is \(64\).

Why averaging the means fails

The simple average of \(60\) and \(70\) is \(65\), which is wrong.

The larger group pulls the combined mean towards its own value, so the correct answer of \(64\) sits closer to \(60\).

Key points

  • Multiply each mean by its group size to get totals.
  • Add the totals together.
  • Divide by the combined number of values.
  • Do not average the two means directly.
  • The larger group pulls the mean towards it.
  • Averaging means only works for equal group sizes.

Worked examples

Example 1

\(10\) values average \(8\) and \(15\) values average \(12\). Find the combined mean.

Working

\[10 \times 8 = 80 \text{ and } 15 \times 12 = 180\]recover each group's total
\[80 + 180 = 260 \text{, } 10 + 15 = 25\]combine the totals and counts
\[260 \div 25 = 10.4\]divide to find the combined mean

Example 2

\(5\) values average \(20\) and \(5\) values average \(30\). Find the combined mean.

Working

\[5(20) + 5(30) = 250\]recover and add the totals
\[250 \div 10 = 25\]the groups are equal, so this matches the simple average

Example 3

Explain why averaging two means is usually wrong.

Working

\[\text{Groups are usually different sizes}\]identify the issue
\[\text{The larger group has more influence}\]explain the effect

Common mistakes

  • Averaging the two means directly.

    This is only correct when the groups are the same size.

  • Forgetting to multiply by the group size.

    The totals must be recovered before combining.

  • Dividing by 2 instead of the combined count.

    Divide by the total number of values across both groups.

  • Mixing up which mean belongs to which group.

    Label the groups clearly before calculating.

Exam tips

  • Recover both totals before combining anything.
  • Divide by the combined count, never by 2.
  • Sense-check that the answer lies between the two means.
  • Expect the answer to sit nearer the larger group's mean.

Key terms

Combined mean
The mean of two groups merged together.
Total
The sum of all values in a group.
Group size
How many values a group contains.
Weighted
Influenced more by the larger group.

Related topics

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