FRM Part I · FRM Exam Part I · Sample Moments
Which statement about the unbiased sample variance s^2 = [1/(n-1)] * sum(x_i - x̄)^2 is correct for i.i.d. observations?
The n-1 divisor corrects for estimating the mean from the same sample, making s squared an unbiased estimator of population variance for i.i.d. data with finite variance. The square root s remains slightly biased for the standard deviation, and normality is not required for unbiasedness.
- AIts square root, s, is an unbiased estimator of the population standard deviation.
- BDividing by n-1 corrects for the fact that the sample mean is estimated from the same data, making s^2 unbiased for the population variance.Correct
- CDividing by n-1 makes the estimator's variance equal to zero as n grows, whereas dividing by n does not.
- DIt is unbiased only if the underlying data are normally distributed.
Explanation
Because deviations are measured from the sample mean, which minimizes squared deviations, the n-divisor understates variance. Using n-1 removes this bias for any i.i.d. data with finite variance. The square root of an unbiased variance estimator is biased (Jensen's inequality), so the first option is wrong.
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