STATISTICS
where in the last line we have used again the fact that, since the population mean is zero,
μr=νr. However, result (31.47) holds even when the population mean is not zero.
From (31.43), we see thats^2 is abiasedestimator ofσ^2 , although the biasbecomes negligible for largeN. However, it immediately follows that an unbiased
estimator ofσ^2 is given simply by
σ̂^2 = N
N− 1s^2 , (31.48)where the multiplicative factorN/(N−1) is often calledBessel’s correction. Thus
in terms of the sample valuesxi,i=1, 2 ,...,N, an unbiased estimator of the
population varianceσ^2 is given by
σ̂^2 =^1
N− 1∑Ni=1(xi− ̄x)^2. (31.49)Using (31.47), we find that the variance of the estimatorσ̂^2 is
V[σ̂^2 ]=(
N
N− 1) 2
V[s^2 ]=1
N(
ν 4 −N− 3
N− 1ν 22)
,whereνr is therth central moment of the parent population. We note that,
sinceE[σ̂^2 ]=σ^2 andV[σ̂^2 ]→0asN→∞, the statisticσ̂^2 is also a consistent
estimator of the population variance.
31.4.3 Population standard deviationσThe standard deviationσof a population is defined as the positive square root of
the population varianceσ^2 (as, indeed, our notation suggests). Thus, it is common
practice to take the positive square root of the variance estimator as our estimator
forσ. Thus, we take
σˆ=(
σ̂^2) 1 / 2
, (31.50)whereσ̂^2 is given by either (31.41) or (31.48), depending on whether the population
meanμis known or unknown. Because of the square root in the definition of
σˆ, it is not possible in either case to obtain an exact expression forE[σˆ]and
V[σˆ]. Indeed, although in each case the estimator is the positive square root of
an unbiased estimator ofσ^2 ,itisnotitself an unbiased estimator ofσ. However,
the bias does becomes negligible for largeN.
Obtain approximate expressions forE[σˆ]andV[σˆ]for a sample of sizeNin the case
where the population meanμis unknown.As the population mean is unknown, we use (31.50) and (31.48) to write our estimator in