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Unbiased estimates of the population mean and variance

Edexcel IAL Mathematics · WST03 Statistics 3

The sample mean X̄ is an unbiased estimator of μ, and the sample variance S2 with divisor n - 1 is an unbiased estimator of σ2. These estimates feed every confidence interval and large-sample test in S3.

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The formulae

x̄ = Σx/n (unbiased for μ).

s2 = Σ(x - x̄)2/(n - 1) = (Σx2 - nx̄2)/(n - 1) = Sxx/(n - 1) (unbiased for σ2).

These are in the formulae booklet. The mark scheme uses exactly this form, for example s2 = (Σx2 - nx̄2)/(n - 1).

Why n - 1?

Deviations are measured from x̄, which is itself fitted to the data, so Σ(x - x̄)2 tends to be smaller than Σ(x - μ)2. Dividing by n would underestimate σ2 on average (E(Σ(X - X̄)2/n) = (n - 1)σ2/n). Dividing by n - 1 removes the bias.

If a variance with divisor n has already been found, multiply by n/(n - 1).

From summary statistics

n = 20, Σx = 364, Σx2 = 6870: x̄ = 18.2, s2 = (6870 - 3642/20)/19 = 245.2/19 = 12.9053.

Keep full calculator accuracy in the working, and give final answers to 3 s.f. (awrt).

Estimated standard error

When σ is unknown, the standard error of the mean is estimated by s/sqrt(n). For the summary above: sqrt(12.9053/20) = 0.8033.

Worked example

Find unbiased estimates of the mean and variance from the sample 12.4, 15.1, 13.8, 14.6, 11.9, 16.2.

  1. Σx = 84, n = 6, so x̄ = 14.0000.
  2. Σx2 = 1189.42
  3. s2 = (Σx2 - (Σx)2/6)/5 = 2.6840

Answer: x̄ = 14.0, s2 = 2.68

Key idea: x̄ = Σx/n and s2 = (Σx2 - nx̄2)/(n - 1) are unbiased for μ and σ2. Divisor n gives a biased variance.

Check yourself

  1. Find an unbiased estimate of the population mean from the sample 12.4, 15.1, 13.8, 14.6, 11.9, 16.2. Give your answer to 3 significant figures.
    Show answer

    14.0. 84/6 = 14.0000.

  2. Find an unbiased estimate of the population variance from the sample 12.4, 15.1, 13.8, 14.6, 11.9, 16.2. Give your answer to 3 significant figures.
    Show answer

    2.68. s2 = Sxx/5 = 2.6840.

  3. A sample has n = 20, Σx = 364 and Σx2 = 6870. Find an unbiased estimate of the population variance. Give your answer to 3 significant figures.
    Show answer

    12.9. (6870 - 3642/20)/19 = 12.9053.

  4. For a sample of 10, the variance calculated with divisor n is 5.6. Find the unbiased estimate of the population variance. Give your answer to 3 significant figures.
    Show answer

    6.22. 5.6 × 10/9 = 6.2222.

  5. A sample has n = 20, Σx = 364 and Σx2 = 6870. Estimate the standard error of the sample mean. Give your answer to 3 significant figures.
    Show answer

    0.803. s/sqrt(n) = sqrt(12.9053/20) = 0.8033.

  6. A sample of 15 has mean 8 and unbiased variance estimate 2.5. Find Σx2.
    Show answer

    995. Σx2 = (n - 1)s2 + nx̄2 = 35 + 960 = 995.

  7. Why is the divisor n - 1 used in s2?
    • Dividing by n would underestimate σ2 on average
    • It makes the arithmetic easier
    • It is only used for small samples to make s2 smaller
    • It gives the variance of the sample mean
    Show answer

    Dividing by n would underestimate σ2 on average. Deviations from x̄ are on average smaller than deviations from μ, so divisor n gives a biased (too small) estimate.

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Last updated 8 November 2026. Spotted a mistake? Tell us.