Standard Errors of Mean, Variance, and Standard Deviation ...

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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. INTRODUCTION

We often estimate the mean, variance, or standard deviation from a sample of elements and present the estimates with standard errors or error bars (in plots) as well. A standard error of a statistic (or estimator) is the (estimated) standard 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. Standard errors mean the statistical fluctuation of estimators, and they are important particularly when one compares two estimates (for example, whether one quantity is higher than the other in a statistically meaningful way). In this note we review the standard errors of frequently used estimators of the mean, variance, and standard deviation.

II. NORMAL ONE SAMPLE PROBLEM

??????????????? ??! Let

be a random sample from

where

both and are unknown parameters. Define, for conve-

nience, two

?$" #&%

sta'( )t is? t?ics(

(sample

and 0

mean and

# 21% %

'(s)am? p? le(

variance):

1 ?3" ?

A. Mean Estimator

The uniformly

? "? timator of is

minimum variance

" [1, p. 92]. Since

?5u" nb4 iased6

(U?78M!V9 U) ,

esthe

standard error of is

AB @ #DC

EGFIHQP ?S" RT#

U

?

Hence TV B @ # WV 9 U V . For , see Subsection II-C.

B. Variance Estimator

0 Note from [1, p. 92] that 21 % 0

8

is UMVU for

4YX a`8? ?

and that (1)

b1 % Since the chi-squared distribution with

c XWa`?ed fg h1 % freedom

has a variance of

0 standard error of is

degrees of [1, p. 31], the

pirqs#ut

EGFIHev 0 xw #y

1f

? %

Hence V i q # 0 t f 9 21 % . It is useful to note

8

Since

and

0

iq!9?

have

# pV irqQ9 0

the square

# of

21f

? %

the units

of

?

(

, often it

is preferable to report estimates of , as described next.

C.

TSh teanUd# MarVd UD2ee1fvsitai% mti oantoa Er` so? tfim # atios r

21 f

0

[1, p. 92] where

% q ` qI Q pg??r

where the second form is more numerically stable for large

# % 0 values of when using the "ln gamma function." By setting , is a common choice in practice but it is slightly

biased. Since

U 2 1 % 0 4yX a`?

[t hsXdeeaes`8(t??1an)]dhaaanrspddhvetahrrireiepoac#rnhco#ietfd[fi2sEGt,frFIip0Heb.vu1fi4tsi9o%:n0 t1 yww pi#Ytoh coa 1r`re%? cx dtgehe2dg]?eiri1 e e%s

of

?

freedom

To investigate the asymptotic behavior of

, we need the

following a ` ?

approximation

# j1 % f

[3, P. 602]:

f % 1 k j%1

%

slnm

f

% gog

?

(2)

Using (2), it can be shown that

# % lSm f % g

and

ih i # #

t t

fg j 1 fg j1

% %

f % lSm lnm f

f % gsg U% g ?

2

1.2

U e U e

1

0.8

replacements

0.6

0

20

40

60

80

100

n

?? ? ??? ? ??? ? ? Fig. 1. This plot shows that

and

respectively, as increases.

approach and

,

To summarize,

h i 9?

#

V h

i 9 0 G#

U

U 21

e

FFoigrure 1

%shows

, it

a plot seems

t

of

fg

j% 1 U

,

e

%

reasonable

, to

for large

and use

U

#e

%?

(3)

% versus . and the

approximation (3) for the standard error.

REFERENCES

[1] E. L. Lehmann and G. Casella, Theory of point estimation, SpringerVerlag, 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.

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