Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICSwhere in the last line we have used again the fact that, since the population mean is zero,µ r = ν r . However, result (31.47) holds even when the population mean is not zero. ◭From (31.43), we see that s 2 is a biased estimator of σ 2 , although the biasbecomes negligible for large N. However, it immediately follows that an unbiasedestimator of σ 2 is given simply bŷσ 2 =NN − 1 s2 , (31.48)where the multiplicative factor N/(N − 1) is often called Bessel’s correction. Thusin terms of the sample values x i , i =1, 2,...,N, an unbiased estimator of thepopulation variance σ 2 is given bŷσ 2 = 1N − 1N∑(x i − ¯x) 2 . (31.49)Using (31.47), we find that the variance of the estimator ̂σ 2 is( 2V [ ̂σ N2 ]=V [sN − 1) 2 ]= 1 (ν 4 − N − 3 )N N − 1 ν2 2 ,where ν r is the rth central moment of the parent population. We note that,since E[ ̂σ 2 ]=σ 2 and V [ ̂σ 2 ] → 0asN →∞, the statistic ̂σ 2 is also a consistentestimator of the population variance.i=131.4.3 Population standard deviation σThe standard deviation σ of a population is defined as the positive square root ofthe population variance σ 2 (as, indeed, our notation suggests). Thus, it is commonpractice to take the positive square root of the variance estimator as our estimatorfor σ. Thus, we takeˆσ =( ̂σ2) 1/2, (31.50)where ̂σ 2 is given by either (31.41) or (31.48), depending on whether the populationmean µ is known or unknown. Because of the square root in the definition ofˆσ, it is not possible in either case to obtain an exact expression for E[ ˆσ] andV [ ˆσ]. Indeed, although in each case the estimator is the positive square root ofan unbiased estimator of σ 2 ,itisnot itself an unbiased estimator of σ. However,the bias does becomes negligible for large N.◮Obtain approximate expressions for E[ ˆσ] and V [ ˆσ] for a sample of size N in the casewhere the population mean µ is unknown.As the population mean is unknown, we use (31.50) and (31.48) to write our estimator in1248