Standard Deviation
Understand Standard Deviation for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. Standard deviation measures the average distance of the values from the mean.
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This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
Standard deviation measures how far the values typically lie from the mean. It uses every value, so it is the most complete measure of spread. This is a Higher-only topic.
The formula is \(\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}\), where \(\bar{x}\) is the mean and \(n\) the number of values. An equivalent form is \(\sigma = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}\), which is usually quicker to compute.
A larger standard deviation means the data is more spread out; a smaller one means the values cluster closely around the mean. Because every value contributes, standard deviation is affected by outliers, which is its main limitation compared with the interquartile range.
Revision notes
The formulae
\(\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}\), where \(\bar{x}\) is the mean.
Equivalently \(\sigma = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}\). The second form avoids calculating each deviation separately and is usually quicker.
Using the second formula
Square every value and add them to get \(\sum x^2\), then divide by \(n\).
Subtract the square of the mean, then take the square root. Forgetting the final square root is the most frequent error, and it gives the variance instead.
Interpreting the result
A larger standard deviation means the values are more spread out from the mean.
A smaller one means they cluster closely around it. Because every value contributes, standard deviation is affected by outliers — its main limitation compared with the interquartile range.
Key points
- Standard deviation measures spread about the mean.
- \(\sigma = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}\).
- It uses every value in the data set.
- A larger value means more spread out.
- The final square root is essential.
- It is affected by outliers.
Worked examples
Example 1
For a data set, \(\sum x^2 = 500\), \(n = 10\) and \(\bar{x} = 6\). Work out the standard deviation. [3 marks]
Working
Example 2
Two classes have the same mean but different standard deviations. Explain what this tells you. [2 marks]
Working
Example 3
Explain one disadvantage of using standard deviation as a measure of spread. [2 marks]
Working
Common mistakes
Forgetting the final square root.
Without it you have calculated the variance, not the standard deviation.
Subtracting the mean instead of its square.
The second formula subtracts \(\bar{x}^2\), not \(\bar{x}\).
Confusing \(\sum x^2\) with \((\sum x)^2\).
Square each value first, then add — not the other way round.
Saying standard deviation is unaffected by outliers.
It uses every value, so outliers do affect it.
Exam tips
- Write the formula before substituting.
- Take the square root as the final step, every time.
- Square each value before summing, not after.
- Remember this is a Higher-only topic.
Key terms
- Standard deviation
- A measure of spread about the mean, written \(\sigma\).
- Variance
- The square of the standard deviation.
- \(\bar{x}\)
- The symbol for the mean.
- \(\sum x^2\)
- The total of the squared values.
Related topics
Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.