Sigma2= Population variance = Sum (X - u)2

N

Desire to estimate this, using x instead of u

(sample mean instead of population mean)

Estimate (Sigma2) =

Est {Sum [(x - x) - (u - x)]2}

N

Estimate (Sigma2) =

Est {Sum [(x - x)2- 2(x - x)(u - x) + (u - x)]2}

N

N

N

continued

Since the sum of (x - x) = 0, the middle term is zero:

Estimate (Sigma2) = Est {Sum [(x - x)2

+ (u - x)]2}

N

N

The last term is the definition of standard error, or the variance of sample means

about the population mean:

Standard error = Sum [(u - x)2] = Sigma2

N

N

continued

(Fromthe central limit theorem,standard erroris equal to the population

variance / N)

Estimate(Sigma2) =

Est {Sum [(x - x)2+ Sigma2]}

N

N

Multiply both sidesby N, then combineSigma terms:

Est [(Sigma2)(N - 1)] = Est {Sum [(x - x)2]}

Divide both sidesby (N - 1) to obtain samplevariance