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FRM Part I · FRM Exam Part I · Sample Moments

Which statement about the sample variance estimator s² = Σ(Xi − X̄)²/(n−1) for i.i.d. observations is correct?

The estimator is unbiased because dividing by n-1 offsets the fact that the sample mean is fitted to the same data, making deviations around it smaller than deviations around the true mean. Dividing by n would understate the population variance on average.

  1. AIt is unbiased because the sample mean X̄ is subtracted, which removes all estimation error
  2. BIt is unbiased because dividing by n-1 compensates for the sample mean being fitted to the same data, which makes deviations from X̄ smaller than deviations from the true meanCorrect
  3. CIt is biased downward, and the bias is eliminated only when the population mean is known and n-1 is used
  4. DIt is biased upward, and dividing by n corrects the bias

Explanation

Deviations from the sample mean are on average smaller than deviations from the true mean, so dividing by n understates variance. Using n-1 corrects this exactly, making E[s²] = σ². The option about a known population mean is wrong since then n should be used.

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