The Geometric Mean
Understand The Geometric Mean for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. The geometric mean multiplies n values and takes the nth root, and is used for rates of change and index numbers.
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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.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
The geometric mean is used for data measuring rates of change or ratios, where the ordinary mean would give a misleading answer. This is a Higher-only topic.
It is calculated by multiplying all \(n\) values together and taking the \(n\)th root: \(\sqrt[n]{x_1 \times x_2 \times \dots \times x_n}\).
It suits growth factors and index numbers because those combine by multiplication rather than addition. Averaging growth factors of 1.2 and 0.8 arithmetically gives 1.0, suggesting no overall change — but multiplying gives 0.96, so the true average factor is \(\sqrt{0.96} \approx 0.98\), correctly showing an overall decrease.
Revision notes
The formula
Geometric mean \(= \sqrt[n]{x_1 \times x_2 \times \dots \times x_n}\).
Multiply all \(n\) values together, then take the \(n\)th root. For two values this is the square root of their product; for three it is the cube root.
When to use it
Use it for rates of change, growth factors, ratios and index numbers.
These quantities combine by multiplication rather than addition, so the arithmetic mean gives a misleading result. The geometric mean respects how they actually combine.
Why the arithmetic mean fails
Growth factors of 1.2 and 0.8 have an arithmetic mean of 1.0, suggesting no overall change.
But applying both gives \(1.2 \times 0.8 = 0.96\), a decrease. The geometric mean \(\sqrt{0.96} \approx 0.98\) correctly reflects this.
Key points
- The geometric mean multiplies then takes a root.
- \(\sqrt[n]{x_1 \times \dots \times x_n}\).
- It suits rates of change and ratios.
- Such quantities combine by multiplication.
- The arithmetic mean misleads for growth factors.
- Use the nth root for n values.
Worked examples
Example 1
Work out the geometric mean of 4 and 9. [2 marks]
Working
Example 2
Work out the geometric mean of 2, 4 and 8. [2 marks]
Working
Example 3
Explain why the geometric mean is used for growth factors rather than the arithmetic mean. [2 marks]
Working
Common mistakes
Adding the values instead of multiplying.
The geometric mean multiplies, then takes a root.
Using the wrong root.
Take the nth root where n is the number of values.
Using it for ordinary measurements.
It is for rates, ratios and growth factors, not heights or masses.
Forgetting the root entirely.
The product alone is not the geometric mean.
Exam tips
- Multiply first, then take the nth root.
- Count the values to determine which root to take.
- Use it only for rates, ratios and growth factors.
- Remember this is a Higher-only topic.
Key terms
- Geometric mean
- The nth root of the product of n values.
- Growth factor
- A multiplier representing proportional change.
- Index number
- A value comparing a quantity to a base value.
- nth root
- The inverse of raising to the power n.
Related topics
Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.