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MathematicsGCSE Higher & A Level

What is standard deviation?

Standard deviation measures the average spread of data values about the mean — a small value means tightly clustered data.

Standard deviation explained

It is the square root of the variance, so it shares the units of the original data, which makes it easier to interpret than variance itself.
In comparisons, quote both the mean and the standard deviation: one describes the typical value and the other describes consistency.

Key formula

σ=∑(x−xˉ)2n\sigma = \sqrt{\dfrac{\sum (x - \bar{x})^{2}}{n}}

Worked example

For 2,4,4,4,5,5,7,92, 4, 4, 4, 5, 5, 7, 9 the mean is 55 and the standard deviation is 22.

Examiner tip

Comparison questions need a sentence in context: 'Team B has a smaller standard deviation, so their scores are more consistent.'

Standard deviation: common questions

What is the difference between variance and standard deviation?
Variance is the mean squared deviation; standard deviation is its square root and is measured in the same units as the data.
Can standard deviation be negative?
No. It is a square root of a sum of squares, so it is always zero or positive — zero only when every value is identical.

Related terms

Normal distribution

The normal distribution is a symmetric, bell-shaped continuous probability distribution defined by its mean and standard deviation.

Mean, median and mode

The mean, median and mode are three averages: the arithmetic average, the middle value in order, and the most frequent value.

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