Unbiased Estimation of the Standard Deviation
for Non-Normal Populations
DAVID E. GILES
Department of Economics
University of Victoria, Victoria, B.C.
CANADA
Abstract: - The bias of the sample standard deviation as an estimator of the population standard deviation, for
a simple random sample of size N from a Normal population, is well documented. Exact and approximate bias
corrections appear in the literature for this case. However, there has been less discussion of the downward bias
of this estimator for non-Normal populations. The appropriate bias correction depends on the kurtosis of the
population distribution. We derive and illustrate an approximation for this bias, to , for several
common distributions.
Keywords: - Standard deviation, unbiased estimation, bias approximation
Received: May 26, 2021. Revised: January 28, 2022. Accepted: February 22, 2022. Published: March 23, 2022.
1 Introduction
Let X follow a distribution, F, with integer moments
that are finite, at least up to fourth order. Denote the
population central moments by
,
j = 1, 2, 3, …. ; where
and
, say; and the kurtosis coefficient is
.
Based on a simple random sample of size N, the
sample variance is
, where
. For any F with finite first and second
moments, and
. In the
special case where F is Normal, the sampling
distributions of both and s itself are well known.
For example, for the latter see Holtzman (1950). In
particular, the bias of s as an estimator of σ, and
various approximations to this bias, have been
examined in detail in the Normal case – e.g., see
Bolch [1], Brugger [2], Cureton [3], D’Agostino [4],
Gurland and Tripathi [5], Markowitz [6] and Stuart
[7],
However, if F is non-Normal, then although
is still an unbiased estimator of , s is a
downward-biased estimator of σ in finite samples,
by Jensen’s inequality. The magnitude of this bias is
not easily
determined, in general, and we explore this problem
here.
2 Main Result
Under standard regularity conditions, both
and are ; and note that we can
write . So, by the
generalized binomial theorem (or using the
Maclaurin expansion), we have:
.
(1)
Convergence of the infinite series in (1) requires
that , and this condition will be
satisfied for large N as is a consistent estimator
of . However, convergence is not required for the
approximation that follows.
Retaining terms in the expected value of (1) up
to , we have
. (2)
Now, , and from eq. (19) of
Angelova [8],
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DOI: 10.37394/23206.2022.21.18
David E. Giles
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. (3)
This yields the approximation,
, (4)
where
. (5)
So, our main result is that
is an unbiased
estimator of σ, to . For a Normal
population, the corresponding scale factor for to
be exactly unbiased for s is known to be
. (6)
Using (4), and the fact that , we also
see immediately that
and
, each to .
3 Discussion
Some early tabulations for by various authors
are discussed by Jarrett [9]. Also, see Holtzman
[10],
Bolch [1], and Gurland and Tripathi [5]. Table 1
compares the exact value of with two
approximations to suggested by Gurland and
Tripathi, for the Normal case. Values of
, for the
Normal and three other common population
distributions, and various values of N, appear in
Table 2. An extended table that provides values of
, for several other well-known distributions can
be downloaded as an Excel spreadsheet from
https://github.com/DaveGiles1949/My-Documents.
For a Normal population, the accuracy of
relative to the exact is apparent in Tables 1 and
2 – even for sample sizes as small as N = 15. This
lends credence to the accuracy of the
values for
the other distributions, which show that this bias
adjustment factor increases with the degree of
kurtosis, but decreases (to 1) rapidly as N increases.
In practice, the form of the population
distribution, and hence the value of κ, may be
unknown. In this case an estimate of κ – such as the
fourth standardized central sample moment, b2 – can
be used. Johnson and Lowe [11] show that ,
so the corresponding estimate of
satisfies
for . In particular,
, as
required, but the order of magnitude of our main
unbiasedness result is then only approximate.
Table 1. CN Values for Normal Population
N CN
Exact GT(5)(6) GT(7)
2 1.2533 1.2649 1.2500
3 1.1284 1.1314 1.1250
4 1.0854 1.0864 1.0833
5 1.0638 1.0643 1.0625
6 1.0509 1.0512 1.0500
7 1.0424 1.0425 1.0417
8 1.0362 1.0363 1.0357
9 1.0317 1.0317 1.0313
10 1.0281 1.0282 1.0278
11 1.0253 1.0253 1.0250
12 1.0230 1.0230 1.0227
13 1.0210 1.0210 1.0208
14 1.0194 1.0194 1.0192
15 1.0180 1.0180 1.0179
16 1.0168 1.0168 1.0167
17 1.0157 1.1057 1.0156
18 1.0148 1.0148 1.0147
19 1.0140 1.0140 1.0139
20 1.0132 1.0132 1.0132
21 1.0126 1.0126 1.0125
22 1.0120 1.0120 1.0119
23 1.0114 1.0114 1.0114
24 1.0109 1.0109 1.0109
25 1.0105 1.0105 1.0104
26 1.0100 1.0100 1.0100
27 1.0097 1.0097 1.0096
28 1.0093 1.0093 1.0093
29 1.0090 1.0090 1.0093
30 1.0087 1.0087 1.0086
Note: .
GT(5)(6) and GT(7) refer to values imputed from
equations (5) and (6), and equation (7), respectively in
Gurland and Tripathi [5].
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Table 2.
Values for Various Populations.
N
Normal Logistic Uniform Expon.
(κ = 3.0) (κ = 4.2) (κ = 1.8) (κ = 9.0)
2 1.3333 1.4815 1.2121 2.6667
3 1.1429 1.2121 1.0811 1.6000
4 1.0909 1.1474 1.0480 1.3714
5 1.0667 1.1019 1.0336 1.2698
6 1.0526 1.0811 1.0256 1.2121
7 1.0435 1.0673 1.0207 1.1748
8 1.0370 1.0576 1.0173 1.1487
9 1.0323 1.0503 1.0148 1.1129
10 1.0286 1.0447 1.0129 1.1146
11 1.0256 1.0402 1.0115 1.1028
12 1.0233 1.0365 1.0103 1.0932
13 1.0213 1.0335 1.0094 1.0842
14 1.0196 1.0309 1.0086 1.0785
15 1.0182 1.0287 1.0079 1.0728
16 1.0169 1.0267 1.0073 1.0679
17 1.0159 1.0251 1.0068 1.0635
18 1.0149 1.0236 1.0064 1.0597
19 1.0141 1.0223 1.0060 1.0564
20 1.0133 1.0211 1.0057 1.0534
21 1.0127 1.0200 1.0054 1.0507
22 1.0120 1.0191 1.0051 1.0482
23 1.0115 1.0182 1.0049 1.0460
24 1.0110 1.0174 1.0046 1.0440
25 1.0105 1.0167 1.0044 1.0421
26 1.0101 1.0160 1.0042 1.0404
27 1.0097 1.0154 1.0041 1.0388
28 1.0093 1.0148 1.0039 1.0374
29 1.0090 1.0143 1.0038 1.0360
30 1.0087 1.0138 1.0036 1.0348
Note:
.
“Expon.” denotes “Exponential”. GT(5)(6) and GT(7)
refer to values imputed from equations (5) and (6), and
equation (7), respectively in Gurland and Tripathi [5].
References:
[1] Bolch, B.W., More on Unbiased Estimation
of the Standard Deviation, The American
Statistician, 22 (3), 1968, 27.
[2] Brugger, R.M., A Note on Unbiased
Estimation of the Standard Deviation,
The American Statistician, 23 (4), 1969, 32.
[3] Cureton, E.E., Unbiased Estimation of the
Standard Deviation, The American
Statistician, 22 (1), 1968, 22.
[4] D’Agostino, R.B., Linear Estimation of the
Normal Distribution Standard Deviation,
The American Statistician, 24 (3), 1970,
14–15.
[5] Gurland, J. and Tripathi, R.C., A Simple
Approximation for Unbiased Estimation of
the Standard Deviation, The American
Statistician, 25 (4), 1971, 30-32.
[6] Markowitz, E., Minimum Mean-Square-
Error Estimation of the Standard Deviation
of the Normal Distribution, The American
Statistician, 22 (3), 1968, 26.
[7] Stuart, A., Reduced Mean-Square-Error
Estimation of in Normal Samples, The
American Statistician, 23 (4), 1969, 27.
[8] Angelova, J.A., On Moments of Sample
Mean and Variance, International
Journal of Pure and Applied Mathematics,
79 (1), 2012, 67-85.
[9] Jarrett, R.F., A Minor Exercise in History,”
The American Statistician, 22 (3), 1968,
25.
[10] Holtzman, W.H., The Unbiased Estimate of
the Population Variance and Standard
Deviation,” American Journal of
Psychology, 63 (4), 1950, 615-617.
[11] Johnson, M.E. and Lowe Jr., V.W.,
“Bounds on the Sample Skewness and
Kurtosis,” Technometrics, 21 (3), 1979, 377-
378.
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