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FRM Part I · FRM Exam Part I · Sample Moments

Which statement best describes why the sample variance uses a divisor of n-1 rather than n when the population mean is unknown?

The n-1 divisor corrects bias. Because deviations are measured from the sample mean, which is fitted to the data, squared deviations are too small on average. Dividing by n-1 makes the sample variance unbiased, though the sample standard deviation remains slightly biased.

  1. ADeviations are taken from the sample mean, which makes the sum of squared deviations too small on average, so dividing by n would be biased downwardCorrect
  2. BIt guarantees the sample standard deviation is an unbiased estimator of the population standard deviation
  3. CIt corrects for fat tails in the return distribution
  4. DIt ensures the estimator is consistent, which dividing by n would not be

Explanation

The sample mean minimizes squared deviations, so deviations about it understate those about the true mean. Using n-1 removes this bias in the variance. Option B is false because the square root of an unbiased variance is still biased (Jensen's inequality). Both divisors give consistent estimators.

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