All subjects · Mathematics · WST01 (S1)

Variance and standard deviation

The standard deviation measures how spread out data are about the mean. In S1 you calculate it from raw data, summary statistics and frequency tables.

Formulae

Variance σ2 = Σ(x - x̄)2 / n = Σx2 / n - x̄2 (the 'mean of the squares minus the square of the mean').

Standard deviation σ = sqrt(variance), in the same units as the data.

The sum Sxx = Σ(x - x̄)2 = Σx2 - (Σx)2 / n is used a lot, so σ2 = Sxx / n. S1 uses divisor n. Recent mark schemes also accept the sample value with divisor n - 1 (often written s), but the n version is the expected one, and it is the one used on this site.

From summary statistics

n = 10, Σx = 142, Σx2 = 2130: x̄ = 14.2, σ2 = 213 - 14.22 = 11.36, σ = 3.37.

Always use the exact mean in the formula, not a rounded one: examiners take marks off for answers spoilt by early rounding.

From frequency tables

Use σ2 = Σfx2 / Σf - x̄2. For grouped data, x is the class midpoint and the answer is an estimate. Most calculators do this in statistics mode, but show the substitution in the exam.

Goals, x01234
Frequency, f58632

Interpreting

A larger standard deviation means the values are more spread out from the mean. For many data sets most values lie within 2 standard deviations of the mean. Compare spreads in context: 'Group A's scores were more consistent (σ = 3.1 against 5.4).'

Worked example

n = 10, Σx = 142, Σx2 = 2130. Find the standard deviation.

  1. x̄ = 142/10 = 14.2
  2. σ2 = 2130/10 - 14.22 = 213 - 201.64 = 11.36
  3. σ = sqrt(11.36)

Answer: σ = 3.37 (3 s.f.)

Key idea: σ2 = Σx2/n - x̄2 = Sxx/n, with frequencies σ2 = Σfx2/Σf - x̄2. Keep the exact mean.

Check yourself

  1. n = 10, Σx = 142, Σx2 = 2130. Find the variance.
    Show answer

    11.4. 213 - 14.22 = 11.36.

  2. n = 8, Σx = 96, Σx2 = 1250. Find the standard deviation. Give your answer to 3 significant figures.
    Show answer

    3.50. x̄ = 12; σ2 = 156.25 - 144 = 12.25; σ = 3.5.

  3. n = 10, Σx = 142, Σx2 = 2130. Find Sxx.
    Show answer

    113.6. Sxx = 2130 - 1422/10 = 2130 - 2016.4 = 113.6.

  4. Find the standard deviation of the number of goals in the table. Give your answer to 3 significant figures.
    Show answer

    1.19. Σfx2 = 91, so σ2 = 91/24 - (37/24)2 = 1.4149; σ = 1.19.

  5. Estimate the standard deviation of the 85 times, using class midpoints. Give your answer to 3 significant figures.
    Show answer

    9.88 minutes. Σfx2 = 39268.8 and x̄ = 19.0882, so σ2 = 39268.8/85 - x̄2 = 97.62; σ ≈ 9.88.

  6. 15 values have mean 20 and standard deviation 4. Find Σx2.
    Show answer

    6240. σ2 + x̄2 = Σx2/n, so Σx2 = 15 × (16 + 400) = 6240.

  7. Group A has standard deviation 3.1 and group B has 5.4 (same units, similar means). What does this show?
    • Group A's values are larger
    • Group B has a higher median
    • Group B has fewer values
    • Group A's values are more consistent
    Show answer

    Group A's values are more consistent. A smaller standard deviation means less spread, so more consistent values.

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Last updated 7 November 2026. Spotted a mistake? Tell us.