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S2.4StatisticsCore

Variance and Standard Deviation

Variance and standard deviation measure how far, on average, data lies from the mean. They use every data value, making them the most informative measures of spread — and the formula has a handy shortcut.

30 min Measures of Location and Spread
Edexcel AS Level Maths: 2.4 Variance and Standard DeviationCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Calculate variance using the formula
  • Calculate standard deviation
  • Use the Σx² shortcut formula
  • Work with frequency tables
1

The formulas

Variance is the mean of the squared distances from the mean. The exam shortcut avoids computing each deviation: subtract the mean squared from the mean of the squares. Standard deviation is the square root of the variance.

σ2=∑x2n−xˉ2,σ=σ2\sigma^{2} = \dfrac{\sum x^{2}}{n} - \bar{x}^{2}, \qquad \sigma = \sqrt{\sigma^{2}}
“Mean of the squares minus the square of the mean.”
Example 1Data: 2, 4, 6. Find the variance.
1Mean xˉ=4\bar{x} = 4.
2∑x2=4+16+36=56\sum x^2 = 4 + 16 + 36 = 56, so 563≈18.67\tfrac{56}{3} \approx 18.67.
3Variance =18.67−16= 18.67 - 16.
Answerσ2≈2.67\sigma^{2} \approx 2.67

Tip — Remember it as “mean of the squares − square of the mean”. Mixing the order gives a negative (impossible) variance.

2

With frequencies

For a frequency table, weight by the frequencies: use ∑fx2\sum fx^2 and ∑f\sum f in place of ∑x2\sum x^2 and nn.

σ2=∑fx2∑f−xˉ2\sigma^{2} = \dfrac{\sum fx^{2}}{\sum f} - \bar{x}^{2}
Frequency version of the variance formula.
3

Standard deviation

The standard deviation σ\sigma has the same units as the data (unlike variance, which is squared units), so it is the more interpretable measure of spread. A larger σ\sigma means more spread-out data.

Formula recap

σ2=∑x2n−xˉ2\sigma^{2} = \dfrac{\sum x^{2}}{n} - \bar{x}^{2}
Variance (shortcut).
σ=σ2\sigma = \sqrt{\sigma^{2}}
Standard deviation.
σ2=∑fx2∑f−xˉ2\sigma^{2} = \dfrac{\sum fx^{2}}{\sum f} - \bar{x}^{2}
With frequencies.

Common mistakes to avoid

Computing (square of the mean) − (mean of squares).
It is mean of squares − square of the mean; the other way gives a negative.
Reporting variance as the final “spread” in data units.
Variance is in squared units; take the square root for standard deviation.

Key takeaways

  • Variance σ² = Σx²/n − x̄² (mean of squares − square of mean).
  • Standard deviation σ = √variance, in the same units as the data.
  • For frequency tables use Σfx²/Σf.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Variance and Standard Deviation. Use the feedback to decide what needs another look.

5 past-paper questions · 10 marks

Practise this topic

Related lessons

All 5 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.