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 Video by Zeeshan Zamurred Measures of Location and Spread

Edexcel AS Level Maths: 2.4 Variance and Standard DeviationWatch the full walkthrough 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

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.

σ^{2} = \frac{\sum x ^{2}}{n} - \overset{x}{ˉ}^{2}, σ = σ^{2}

“Mean of the squares minus the square of the mean.”

1Mean

\overset{x}{ˉ} = 4

\sum x^{2} = 4 + 16 + 36 = 56

, so

\frac{56}{3} \approx 18.67

3Variance

= 18.67 - 16

Answer

σ^{2} \approx 2.67

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

With frequencies

For a frequency table, weight by the frequencies: use $\sum f x^{2}$ and $\sum f$ in place of $\sum x^{2}$ and $n$ .

σ^{2} = \frac{\sum f x ^{2}}{\sum f} - \overset{x}{ˉ}^{2}

Frequency version of the variance formula.

Standard deviation

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

Formula recap

σ^{2} = \frac{\sum x ^{2}}{n} - \overset{x}{ˉ}^{2}

Variance (shortcut).

σ = σ^{2}

Standard deviation.

σ^{2} = \frac{\sum f x ^{2}}{\sum f} - \overset{x}{ˉ}^{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.

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