Transcription of Standard Errors of Mean, Variance, and Standard Deviation ...
1 1. Standard Errors of Mean, Variance, and Standard Deviation Estimators Sangtae Ahn and Jeffrey A. Fessler EECS Department The University of Michigan July 24, 2003. I. I NTRODUCTION Hence S8 E G J B ' . It is useful to note We often estimate the mean, variance, or Standard Deviation SEQG ' F8 EHG3' P B . from a sample of elements and present the estimates with . Standard Errors or error bars (in plots) as well. A Standard & and have the square of the units of , often it error of a statistic (or estimator ) is the (estimated) Standard Since . is preferable to report estimates of , as described next. Deviation of the statistic. An error bar is, in a plot, a line which is centered at the estimate with length that is double the Standard error.
2 Standard Errors mean the statistical fluctuation is T [1, p. 92] where C. Standard Deviation estimator of estimators, and they are important particularly when one compares two estimates (for example, whether one quantity The UMVU estimator of =< . is higher than the other in a statistically meaningful way).. T U P B V P B W3X YFZC[ \ G`. _ < G1_. In this note we review the Standard Errors of frequently used ]H^ X YSZa[ \. estimators of the mean, variance, and Standard Deviation . where the second form V is more numerically stable for large T b , is a common choice in practice but it is slightly values of when using the ln gamma function. By setting II. N ORMAL O NE S AMPLE P ROBLEM.]
3 Be a random sample from where Let are unknown parameters. Define, for conve- biased. Since 7 "O: =< . both and .. nience, two statistics (sample mean and sample variance): and ! [see (1)] and the chi distribution with c degrees of freedom :d=<& has variance [2, p. 49: typo corrected].. eF BUf R =< Ng . A. Mean estimator B V . the Standard error of T is V. " $ % & ' . #. The uniformly minimum variance unbiased (UMVU) es- timator of is . [1, p. 92]. Since , the Fh E J -/.10AL T M ; T P ei .. )* ( ,+ /- .1032 5 46 7 . Standard error of is \. To investigate the asymptotic behavior of h E , we need the 6( 98 ' 7 . For 8 , see Subsection II-C. Hence 8 *. following approximation [3, P.)]
4 602]: \. =< P j f k Ilnm f gog (2). V B j . V (2), it can be shown that is UMVU for & and that B. Variance estimator Using Note from [1, p. 92] that ;. " : =<& (1). T l5m f g Since the chi-squared distribution with > degrees of . and :9 =< A@ has a variance of BC D [1, p. 31], the ih E . freedom ? Standard error of is J BC j f l5m f Ig g \ . FEHGI KJ -/.10AL NM O QP B . R J CB j lnm f 7 g . 2. T. T7 e 7 e 1. replacements 0 20 40 60 80 100. n Fig. 1. This plot shows that and approach and , respectively, as increases. 8 h E ' T / T 7 7 e To summarize, h E ' .. \ \ . J BC j for large (3). Figure 1 shows a plot of T , 7 e , and T 7 e versus . For , it seems reasonable to use T and the approximation (3) for the Standard error.
5 R EFERENCES. [1] E. L. Lehmann and G. Casella, Theory of point estimation, Springer- Verlag, New York, 1998. [2] M. Evans, N. Hastings, and B. Peacock, Statistical distributions, Wiley, New York, 1993. [3] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete mathematics: a foundation for computer science, Addison-Wesley, Reading, 1994.
