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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