The coefficient of variation is a dimensionless number
that quantifies the degree of variability relative to the mean.
The population coefficient of variation is defined as
K
S
M
,
(1)
where À is the population standard deviation and µ is the pop-
ulation mean. The typical sample estimate of is given as
k
s
M
,
(2)
where s is the sample standard deviation, the square root
of the unbiased estimate of the variance, and M is the sam-
ple mean. Equations 1 and 2 are sometimes multiplied by
100 so that the ratio of the standard deviation to the mean
is expressed in terms of a percentage.
The coefficient of variation has long been a widely
used descriptive and inferential quantity in various areas
of the biological and medical sciences. Compared with
several other effect size measures, the coefficient of varia-
tion has not historically been widely used in behavioral,
educational, or social sciences. However, some research
questions within the behavioral, educational, and social
sciences lend themselves to being addressed with the co-
efficient of variation. At least in part due to the current
emphasis in many parts of the literature on the interaction
of biological and psychological systems as explanatory
factors of behavior (e.g., Frith & Frith, 2001; Kosslyn
et al., 2002; Salmon & Hall, 1997; etc.), use of the coef-
ficient of variation will likely continue to increase.
1
An example of where the coefficient of variation is
often used is for assays, which are procedures that mea-
sure certain designated properties of biological compo-
nents. The results of repeated trials of such assays are
many times reported in terms of the coefficient of varia-
tion, because the standard deviations of assays generally
increase (or decrease) proportional to the mean increase
(or decrease; Reed, Lynn, & Meade, 2002). Coefficients
of variation are generally based on a single measurement
from different individuals (i.e., an interindividual coeffi-
cient of variation). However, intraindividual coefficients
of variation, where repeated measures for the same indi-
vidual are obtained, are also possible. One place where an
intraindividual coefficient of variation is of interest is be-
fore and after a treatment to evaluate the effectiveness of
the treatment (Reed et al., 2002). In a psychiatry setting,
Volkow et al. (2002) used the coefficient of variation to
compare patterns of homogeneity/heterogeneity in brain
metabolism for Alzheimer’s disease patients with those in
a control group without Alzheimer’s. It was shown that the
coefficient of variation was larger across the entire cor-
tex, but that there were smaller coefficients of variation in
temporal and parietal cortices for Alzheimer’s disease pa-
tients. As more researchers move to a biology– psychology
model of behavior, assays will likely become more preva-
lent in the behavioral sciences.
Of course, the coefficient of variation need not be re-
stricted to biological systems. In a classic experimental
psychology setting, Babkoff, Kelly, and Naitoh (2001)
755 Copyright 2007 Psychonomic Society, Inc.
Sample size planning for the coefficient
of variation from the accuracy in
parameter estimation approach
KEN KELLEY
Indiana University, Bloomington, Indiana
The accuracy in parameter estimation approach to sample size planning is developed for the coefficient of
variation, where the goal of the method is to obtain an accurate parameter estimate by achieving a sufficiently
narrow confidence interval. The first method allows researchers to plan sample size so that the expected width of
the confidence interval for the population coefficient of variation is sufficiently narrow. A modification allows
a desired degree of assurance to be incorporated into the method, so that the obtained confidence interval will
be sufficiently narrow with some specified probability (e.g., 85% assurance that the 95% confidence interval
width will be no wider than units). Tables of necessary sample size are provided for a variety of scenarios that
may help researchers planning a study where the coefficient of variation is of interest plan an appropriate sample
size in order to have a sufficiently narrow confidence interval, optionally with some specified assurance of the
confidence interval being sufficiently narrow. Freely available computer routines have been developed that allow
researchers to easily implement all of the methods discussed in the article.
Behavior Research Methods
2007, 39 (4), 755-766
K. Kelley, [email protected]
756 KELLEY
related to the variance of the estimator and bias is the
systematic discrepancy of an estimator and the param-
eter it estimates (Rozeboom, 1966). As the confidence
interval width decreases, holding constant the confidence
interval coverage, the estimate is contained within a nar-
rower set of plausible parameter values and the expected
accuracy of the estimate improves (i.e., the root mean
square error is be reduced). Thus—provided the confi-
dence interval procedure is exact—when the width of the
(1 ()100% confidence interval decreases, the expected
accuracy of the estimate necessarily increases. The goal
of this approach is to plan the necessary sample size so
that the estimated coefficient of variation accurately re-
flects the corresponding population value by achieving a
sufficiently narrow confidence interval. This approach to
sample size estimation has been termed accuracy in pa-
rameter estimation (AIPE), because when the width of the
(1 ()100% confidence interval decreases, the expected
accuracy of the estimate increases (Kelley & Maxwell,
2003, in press; Kelley, Maxwell, & Rausch, 2003; Kelley
& Rausch, 2006).
Probabilistically, holding everything else constant, the
narrower a confidence interval, the higher the degree of
expected accuracy of the obtained parameter estimate.
The use of the term “accuracy” in this context follows
the usage of the term given by Neyman (1937) when he
discussed the “accuracy of estimates” in his seminal work
on the theory of confidence intervals. He stated that “the
accuracy of estimation corresponding to a fixed value of
1 ( may be measured by the length of the confidence in-
terval” (Neyman, 1937, p. 358; notation changed to reflect
current usage). The problem that the present article solves
is planning sample size so that the confidence interval for
the coefficient of variation is sufficiently narrow, and thus
that the obtained estimate has a sufficient degree of ex-
pected accuracy. The overarching goal of AIPE is thus to
avoid obtaining confidence intervals that are “embarrass-
ingly large” (Cohen, 1994, p. 1002) and of limited useful-
ness. Note that the general goal of AIPE has nothing to
do with rejecting a null hypothesis, which is the stated
goal of power analysis (e.g., Cohen, 1988; Kraemer &
Thiemann, 1987; Lipsey, 1990; Murphy & Myors, 1998).
The goal of planning sample size so that the confidence
interval is sufficiently narrow (i.e., the AIPE approach) is
a fundamentally different process from the power analytic
approach—that is, planning sample size so that the confi-
dence interval does not contain the null value.
The first sample size planning method developed deter-
mines the necessary sample size so that the expected width
of the confidence interval for the coefficient of variation
is sufficiently narrow. A modified sample size procedure
is then developed so that there is some specified level of
assurance (i.e., a probabilistic statement) that the obtained
confidence interval will be sufficiently narrow (e.g., 85%
assurance that the 95% confidence interval for will be
no wider than 0.10 units). General methods are developed
and algorithms given so that the necessary sample size
can be determined in any particular situation. A series of
tables for necessary sample size are provided for a variety
of scenarios that may help researchers planning studies in
reported the effects of sleep deprivation on a four-choice
reaction time (RT) experiment to assess performance
stability over time between a placebo and two treatment
groups that received stimulants. The coefficient of varia-
tion was used for a subset of the analyses because mean
performance level is postulated to be less affected by sleep
deprivation than is the trial-to-trial variance (Babkoff et al.,
2001; Dinges & Kribbs, 1991). Babkoff et al. went on to
model the coefficient of variation (the measure of perfor-
mance stability) over time as a function of group member-
ship. Similarly, Hayashi (2000) examined the coefficient
of variation for choice RT when participants were using
benzodiazepine in an effort to manipulate their cognitive
states. In an educational setting, Monchar (1981) used the
coefficient of variation to quantify educational inequality
in the context of political instability in 39 countries at vari-
ous times from the late 1950s until the early 1970s. The idea
was that countries with greater political instability would
have more educational inequities (larger coefficients of
variation), using the rate of enrollment for reporting re-
gions within the country, than would more-stable coun-
tries.
2
The coefficient of variation is also used for quan-
tifying risk sensitivity, which is especially helpful when
comparing diverse populations (Weber, Shafir, & Blais,
2004). With two experiments, Weber et al. (2004, p. 430;
see also Shafir, 2000) show that risk sensitivity of human
beings, as with other types of animals, becomes strongly
proportional to the coefficient of variation when they learn
about choice alternatives. Finally, a comprehensive review
of the demography and diversity literature shows that the
coefficient of variation is one of the most widely used ways
of assessing group-based demographic differences (Wil-
liams & O’Reilly, 1998). As can be seen, the coefficient of
variation has been used in a wide variety of contexts.
Even though the estimated coefficient of variation can
be a useful measure, perhaps the greatest use of it as a
point estimate (like most point estimates) is to construct
a confidence interval for the population quantity.
3
A con-
fidence interval provides much more information about
the population value of the quantity of interest than does
a point estimate (e.g., Cohen, 1994; Hunter & Schmidt,
2004; Kirk, 2001; Meehl, 1997; Schmidt, 1996; Smith-
son, 2001; Steiger, 2004; Task Force on Reporting of Re-
search Methods in AERA Publications, 2006; Thompson,
2002; Wilkinson & the American Psychological Associa-
tion Task Force on Statistical Inference, 1999). The more
accurate the estimated quantity, the more information is
known about the population quantity. Because the goal of
most research is to implicitly or explicitly learn as much
as possible about the parameter of interest, obtaining an
accurate parameter estimate should generally be of utmost
concern in applied research.
4
The purpose of the present article is to offer an ap-
proach to sample size planning for the coefficient of
variation, where the goal is to obtain a sufficiently nar-
row confidence interval that illustrates the expected ac-
curacy with which the parameter has been estimated. In
the context of parameter estimation, accuracy is defined
as the square root of the mean square error, which is a
function of both precision and bias. Precision is inversely
SAMPLE SIZE FOR THE COEFFICIENT OF VARIATION 757
where
ˆ
is the estimated noncentrality parameter. Given the
estimated noncentrality parameter, a (1 ()100% confi-
dence interval can be formed for the population noncen-
trality parameter by making use of the confidence interval
transformation principle and the inversion confidence in-
terval principle. These two principles are nicely delineated
in Steiger and Fouladi (1997) and in Steiger (2004).
6
To illustrate the inversion confidence interval princi-
ple, let 1 ( be the confidence interval coverage with
(
L
(
U
(, where (
L
is the proportion of times that
will be less than the lower confidence limit and (
U
the
proportion of times that will be greater than the upper
confidence limit in the confidence interval procedure. In
most cases (
L
(
U
(/2, but that need not be the case
(e.g., when (
L
0 and (
U
.05, or vice versa, for one-
sided confidence intervals). The confidence bounds for
are determined by finding the noncentrality parameter
whose 1 (
L
quantile is
ˆ
(for the lower bound of the
confidence interval) and by finding the noncentrality pa-
rameter whose (
U
quantile is
ˆ
(for the upper bound of
the confidence interval). Stated another way, the lower
confidence bound for is the noncentrality parameter that
leads to t
(1(
L
,,
L
)
ˆ
and the upper confidence bound for
is the noncentrality parameter that leads to t
((
U
,,
U
)
ˆ
,
where t
(1(
L
,,
L
)
is the value of the noncentral t distribu-
tion at the 1 (
L
quantile with noncentrality parameter
L
and t
((
U
,,
U
)
is the value of the noncentral t distribu-
tion at the (
U
quantile with noncentrality parameter
U
,
respectively.
The confidence interval transformation principle
becomes useful, because a confidence interval for the
population noncentrality parameter is not generally of in-
terest in and of itself. However, since there is a one-to-one
monotonic relation between and , a confidence interval
for can be transformed into a confidence interval for .
If one rearranges Equation 5, it can be seen that
K
N
L
.
(7)
Therefore, transforming the limits of the confidence inter-
val for , by dividing the limits by ·N and then taking the
inverse, confidence limits for can be obtained:
p
NN
L
L
A
U
L
¤
¦
¥
³
µ
´
aa
¤
¦
¥
³
µ
´
§
©
¨
¨
¶
¸
·
·
1
1
1K ,
(8)
where p represents probability. Notice that the lower con-
fidence limit for is obtained by dividing the upper confi-
dence limit for by ·N and that the upper confidence limit
for is obtained by dividing the lower confidence limit for
by ·N. The reversal of limits is a necessary part of the
transformation procedure in the present context.
As an example of forming a confidence interval for ,
an example is given for a situation that deals with Alz-
heimer’s disease patients. Volkow et al. (2002) studied the
reduction of glucose metabolism in the brains of Alzheim-
er’s disease patients (as part of a larger study). The coef-
ficient of variation across the entire cortex for the glucose
metabolism of segmented cortical regions in the brains of
which the coefficient of variation is of interest to achieve
a sufficiently narrow confidence interval.
5
Estimation and Confidence Interval
Formation for
Sokal and Braumann (1980, p. 51) derive the expected
value of k for normally distributed data given and N as
E[ |( ,)]
()
()
kN
NN
N
KK K
§
©
¨
¶
¸
·
1
1
41
11
21
2
2
kk
N(, )
,
K
(3)
where N is the sample size. Equation 3 will become useful
momentarily when developing the sample size planning
procedure. Although Equation 2 is the estimate typically
reported for the coefficient of variation, as can be seen
from Equation 3, k is a (negatively) biased quantity. A
“nearly unbiased estimate” (Haldane, 1955, p. 484; Sokal
& Braumann, 1980) of can be obtained with a simple
correction to k,
k
N
k
u
1
1
4
,
(4)
where k
u
is the nearly unbiased estimate of . When report-
ing point estimates of , k
u
is the recommended quantity.
However, as will be discussed, k is used in the formation
of confidence intervals for .
Although k
u
provides a nearly unbiased estimate of , a
point estimate in and of itself does not convey any infor-
mation regarding the uncertainty with which the param-
eter has been estimated. As previously discussed, the use
of confidence intervals is especially helpful for conveying
information about the uncertainty with which an estimator
estimates the population parameter of interest. Although
will almost never be known exactly, a range of plausible
values can be obtained by forming a confidence interval
for . Until recently, confidence intervals for were pro-
hibitively difficult to obtain. However, as a result of recent
advances in statistical software, such intervals can be eas-
ily found with several software titles that have noncentral
t distribution routines. The remainder of this section dis-
cusses how a confidence interval can be formed for the
coefficient of variation.
It can be shown that the coefficient of variation follows
a noncentral t distribution, when the parent population
of the scores is normally distributed, with noncentrality
parameter
L
M
S
N
N
K
(5)
with degrees of freedom, where N 1 (Johnson
& Welch, 1940; McKay, 1932). An estimate of the non-
centrality parameter can be obtained by substituting the
typical estimate of (i.e., k) for the population value in
Equation 5:
ˆ
,
L
N
k
(6)
758 KELLEY
sired degree of statistical power is chosen a priori when
one is planning necessary sample size in a power analytic
context (e.g., Cohen, 1988; Kraemer & Thiemann, 1987;
Lipsey, 1990; Murphy & Myors, 1998).
As is true with essentially all sample size planning pro-
cedures (e.g., power analysis), in order to plan sample size
under the goals of AIPE, must be known or its value
estimated. This complication is the same as almost all
sample size planning procedures (e.g., power analysis).
Conceptually, the way in which sample size is determined
so that the expected width of the confidence interval for
is sufficiently narrow is to substitute for k and system-
atically evaluate different sample sizes until the expected
width of the confidence interval is no wider than desired
(i.e., E[w] ). Actually, since k systematically under-
estimates (recall Equation 3), basing the sample size
planning procedure on instead of on E[k] would lead
to sample size estimates larger than necessary. For this
reason,
˜
k
(,N)
from Equation 3 is substituted for when de-
termining the necessary sample size so that the expected
width is no larger than . Note that since
˜
k
(,N)
is in part a
function of N,
˜
k
(,N)
will be updated for each iteration of
the sample size procedure, as each iteration is based on
a different sample size. This method of planning sample
size is consistent with other sample size planning meth-
ods so that the expected width is sufficiently narrow (e.g.,
Guenther, 1981; Hahn & Meeker, 1991; Kelley & Max-
well, 2003; Kelley et al., 2003; Kelley & Rausch, 2006;
Kupper & Hafner, 1989).
Although the overarching approach to sample size
planning from an AIPE approach is not new (e.g., Mace,
1964), it has not been discussed much in the behavioral,
educational, and social sciences. Furthermore, it has only
recently been discussed for effect sizes that follow non-
central distributions (e.g., Kelley, 2007b; Kelley & Max-
well, in press; Kelley & Rausch, 2006; see also Algina
& Olejnik, 2000, for a similar goal in a related context).
Given the importance placed on effect sizes and confi-
dence intervals in the literature, sample size planning
from an AIPE approach will almost certainly increase in
importance and frequency of usage.
Operationally, an algorithm that guarantees finding
the appropriate sample size is to start at some minimal
sample size, say N
0
, and determine E[k |(, N
0
)] so that
E[w |
˜
k
(,N
0
)
] can be determined. If the width is greater than
desired, the sample size should be increased by one, and
the confidence interval width determined again. This it-
erative process of increasing sample size and recalculating
the expected confidence interval width should continue
until E[w |
˜
k
(,N
i
)
] is equal to or less than , where i repre-
sents the particular iteration number.
One issue that might not be obvious is that the method
for planning sample size just discussed plans the neces-
sary sample size so that the expected width of the con-
fidence interval is sufficiently narrow; however, it does
not guarantee that for any particular confidence interval
the observed width will be sufficiently narrow. The con-
fidence interval width, w, is a random variable that will
fluctuate from sample to sample. The fact that sample size
is determined so that E[w] is no larger than implies that
the 35 Alzheimer’s patients was .271. Using Equation 6,
the estimated noncentrality parameter (
ˆ
) in this situation
is 21.831 (i.e., ·(35)/.271). A 95% confidence interval for
the noncentrality parameter is given as
CI
.95
[16.282 27.331],
where CI
.95
represents a 95% confidence interval. Con-
verting the confidence limits for to by way of Equa-
tion 8 leads to the following confidence interval for :
CI
.95
[.216 .363].
Note that the confidence interval is not symmetric about
k, where the lower confidence interval width is .055
(.271 .216) and the upper confidence interval width
is .092 (.363 .271). In general, noncentral distribu-
tions are not symmetric, and confidence intervals based
on noncentral distributions, even when (
L
(
U
, tend to
have different lower and upper confidence interval widths.
Thus, confidence intervals for , which are based on the
transformed confidence limits for the noncentral param-
eter, tend not to be symmetric about k.
AIPE for the Coefficient of Variation
When planning sample size, in order for the expected
width of the obtained confidence interval to be sufficiently
narrow for the population coefficient of variation, it is
necessary to use an iterative process.
7
Because the con-
fidence interval width for is not symmetric (due to the
nature of the noncentral t distribution), the desired width
can pertain to the full confidence interval width, the lower
width, or the upper width. Let
L
(X) be defined as the
random lower confidence limit for and
U
(X) be defined
as the random upper confidence limit for at the speci-
fied confidence level, where X represents the observed
data matrix on which the confidence interval is based. For
notational ease, the lower and upper random confidence
interval limits will be written as
L
and
U
, respectively,
with the understanding that they are random because they
are based on the observed data. The full width of the ob-
tained confidence interval is thus given as
w
U
L
, (9)
the lower width of the obtained confidence interval is
given as
w
L
k
L
, (10)
and the upper width of the obtained confidence interval
is given as
w
U
U
k. (11)
The goals of the research study will dictate for which con-
fidence interval width sample size should be determined.
In general, w will be the width of interest. Although the
methods discussed are directly applicable to determining
sample size for the lower or the upper confidence interval
width (i.e., w
L
or w
U
), the focus of the present work is
on the full confidence interval width (i.e., w). Let be
defined as the desired confidence interval width, which
is specified a priori by the researcher, much as the de-
SAMPLE SIZE FOR THE COEFFICIENT OF VARIATION 759
Meeker, 1991; Kelley & Maxwell, 2003; Kelley et al.,
2003; Kelley & Rausch, 2006; Kupper & Hafner, 1989).
Tables of Necessary Sample Size
Although the Appendix provides information on imple-
menting the methods proposed and discussed in order to
obtain sufficiently narrow confidence intervals for any
combination of , , (, and o using the Methods for the
Behavioral, Educational, and Social Sciences (MBESS;
Kelley, 2007b, 2007c) R package (R Development Core
Team, 2007), tables of selected conditions are provided.
The tables are not meant to include all potentially inter-
esting conditions; rather, they are intended to provide re-
searchers (1) a convenient way to plan sample size when
the situation of interest is approximately that included in
the table, and (2) a way to illustrate the relation between
, , o, (, and necessary sample size. The tabled values
are based on values thought to be useful for areas of the
behavioral, educational, and social sciences.
Sample size is tabled for values of 0.05 to 0.50 by
0.05, values of .01, .025 to .20 by .025, a subtable where
the expected value of w equals , along with subtables
with o values of .80 and .99, each for confidence interval
coverages of .90 (Table 1), .95 (Table 2), and .99 (Table 3).
Each of the conditions is crossed with all other conditions
in a factorial manner, and thus there are a total of 810 sit-
uations (10 9 3 3) for planning an appropriate
sample size.
10
Any time a necessarily positive quantity follows a normal
distribution, it is unlikely that k values larger than
1
⁄3 can be
obtained. Values of k larger than
1
⁄3 imply that the standard
deviation is more than three times larger than the mean,
which would further imply that the lower end of the distri-
bution would be expected to contain some proportion less
than zero. For example, if µ 30 and À 10 from a nor-
mal distribution, and thus
1
⁄3, .135% of the distribution
would fall below zero. This small proportion of expected
negative scores, when the normally distributed quantity is
necessarily positive, may be ignorable in this situation; but
as increases arbitrarily large, the proportion of expected
negative scores increases arbitrarily close to 50% of the dis-
tribution. The same discussion applies to quantities that are
necessarily negative and that follow a normal distribution.
Thus, for necessarily positive quantities k values tend to be
less than
1
⁄3. Values of k greater than
1
⁄3 tend to arise when the
distribution consists of both negative and positive values, or
when the distribution consists only of positive or negative
values but is not normally distributed.
Suppose a researcher is interested in estimating a coef-
ficient of variation that has a corresponding confidence
for that is sufficiently narrow. After a literature review of
studies that examined a similar phenomenon under similar
conditions, it was hypothesized that the population coef-
ficient of variation was .25. With the desire to obtain a
99% confidence interval that excludes 0.20 and 0.30, the
researcher sets to 0.10.
11
Application of the methods
leads to a necessary sample size of N 99. This sample
size is contained in the first subtable of Table 3, specifi-
cally (from the top left) five cells down (the 0.10 row)
and five cells over (the 0.25 column).
roughly 50% of the sampling distribution of ws will be less
than .
8
A modified sample size procedure, discussed in
the next section, can be implemented so that one can have
a desired degree of assurance that the obtained w will be
no larger than ; that is, a probabilistic statement can be
incorporated into the sample size planning procedure that
guarantees the obtainment of a sufficiently narrow confi-
dence interval with some desired degree of assurance.
Ensuring a Confidence Interval No Wider Than
Desired With a Specified Degree of Assurance
One property of the method for forming confidence in-
tervals using the noncentral t distribution for is that as k
increases, so does the width of the confidence interval for
. This implies that when k is larger than the on which
the standard sample size procedure is based, w will be
larger than . In order to avoid obtaining a k larger than
the value the sample size procedure is based on with some
specified degree of assurance, denoted o, and thus a w
wider than , a modified sample size procedure can be
used that substitutes for from the standard procedure
o
,
where k will not exceed
o
o100% of the time. That is,
o
is
the value at the oth quantile from the particular sampling
distribution of k. Thus, given the specified degree of as-
surance,
o
is the largest plausible value of k expected to
be obtained with probability o for a particular sample size
and value. The value 1 o thus represents the probabil-
ity that k will be greater than , which implies that w will
be greater than for the particular sample size.
When
o
is found based on the necessary sample size
from the standard procedure,
o
can be substituted for
in the standard procedure and sample size determined in
the same manner as before.
o
is obtained by transform-
ing the oth quantile from a noncentral t distribution with
noncentrality parameter and degrees of freedom. The
value
o
is the value of the particular noncentral t distribu-
tion that satisfies
ft dt
(;)
,
NL
L
G
G
c
¯
(12)
where f(t
(;)
) represents the noncentral t distribution prob-
ability function, and is based on the sample size from the
standard procedure.
The rationale for replacing from the standard sample
size procedure with
o
is so that w will exceed no more
than (1 o)100% of the time. Since
p(k
o
) o, (13)
no more than (1 o)100% percent of the time will w be
greater than :
p(w ) 1 o. (14)
Thus, the modified sample size procedure ensures that
the observed confidence interval will not be wider than
with probability no less than o.
9
The method of determin-
ing the modified sample size is consistent in theory with
other methods of sample size planning in order to attach
a probabilistic statement to the confidence interval width
being sufficiently narrow (e.g., Guenther, 1981; Hahn &
760 KELLEY
val will be sufficiently narrow (e.g., 99% certain compared
with 80% certain) also requires a larger sample size, because
increasing the probabilistic component implies that it will be
more difficult to achieve the goal satisfactorily. Similarly, as
the confidence interval coverage increases (i.e., a decrease
in (), for a desired confidence interval width the sample size
also increases. The reverse is also true: (1) a decrease in
(e.g., by reducing variability in the sample), (2) an increase
in , (3) smaller values of o, and (4) smaller confidence
interval coverages all lead to smaller sample sizes.
Sensitivity Analyses
In almost all situations, will be unknown; yet must
be specified in order to plan the appropriate sample size.
This is the conundrum for most sample size planning pro-
cedures. One question that arises is “What are the effects
on the width of the confidence interval if the value of
specified is not equal to the population value?” This is an
analogous problem when planning sample size in a power
analytic context, where sensitivity analyses are often sug-
gested. Such sensitivity analyses are also recommended in
the context of AIPE.
The idea of a sensitivity analysis in the present context
is to assess the effects of misspecifying the population
Realizing that an N of 99 will lead a sufficiently nar-
row confidence interval only about half of the time, the
researcher incorporates an assurance (i.e., o) of .99. A o of
.99 implies that the width of the 99% confidence interval
will be greater than desired (i.e., 0.10) no more than 1%
of the time. From the third subtable of Table 3 (again five
cells down and five cells over from the top left), it can be
seen that the modified sample size procedure yields a nec-
essary sample size of 141. Using a sample size of 141 will
thus provide 99% assurance that the obtained confidence
interval for will be no wider than 0.10 units.
A summary of the results contained in the table is pro-
vided. As can be seen, holding all other factors constant,
(1) larger values of , (2) smaller values of , (3) larger val-
ues of o, and (4) larger values of the confidence interval cov-
erage (i.e., 1 () all lead to larger necessary sample sizes.
These findings are intuitively reasonable, in that a quantity
in a population with more variability is estimated as in-
creases, implying larger standard deviations relative to the
mean (or smaller means relative to the standard deviation).
Smaller values of imply that a larger sample size is neces-
sary in order to reduce the standard error and increase the
degrees of freedom, so that the confidence interval becomes
narrower. Having more assurance that the confidence inter-
Table 1
Necessary Sample Size for 90% Confidence Intervals
for the Coefficient of Variation in Selected Situations
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
E[w]
.010 141 557 1,277 2,342 3,810 5,752 8,258 11,434 15,401 20,298
.025 26 93 208 379 614 925 1,326 1,834 2,469 3,253
.050 9 27 56 98 157 235 335 463 622 818
.075 6 14 26 45 71 106 151 207 278 365
.100 5 9 17 27 42 62 87 119 159 209
.125 – 7 12 19 29 41 57 78 104 136
.150 – 6 10 14 21 30 41 56 74 96
.175 – 5 8 12 17 23 32 42 56 72
.200 – 5 7 10 14 19 25 33 44 56
o .80
.010 154 585 1,322 2,406 3,896 5,865 8,402 11,616 15,627 20,576
.025 31 104 226 404 648 970 1,384 1,907 2,560 3,365
.050 11 32 64 111 174 257 364 499 668 874
.075 7 16 31 53 82 121 170 232 309 404
.100 6 11 20 33 50 73 102 138 182 237
.125 – 9 15 23 35 50 69 127 122 159
.150 – 7 12 18 26 37 51 68 89 115
.175 – 6 10 14 21 29 40 53 69 89
.200 – 6 8 12 17 24 32 43 55 71
o .99
.010 180 638 1,405 2,523 4,053 6,070 8,664 11,945 16,036 21,077
.025 42 126 261 453 714 1,055 1,493 2,044 2,730 3,573
.050 16 43 82 137 209 303 423 573 758 985
.075 10 25 45 72 108 154 212 285 374 483
.100 8 18 31 49 71 100 136 180 235 301
.125 – 14 23 36 52 73 98 129 167 213
.150 – 11 19 29 41 57 77 100 129 164
.175 – 10 16 24 34 47 63 82 105 133
.200 – 9 14 21 29 40 54 70 89 113
Note— is the population coefficient of variation, o is the desired degree of assurance of achieving
a confidence interval for no wider than desired, is the desired full confidence interval width, and
E[w] is the expected confidence interval width (i.e., when o is not specified).
SAMPLE SIZE FOR THE COEFFICIENT OF VARIATION 761
to be known before a statement about the effects of the
misspecification of can be known. In general, properties
of misspecification are not known analytically; hence the
need for a Monte Carlo simulation study. The Appendix
provides information on how a sensitivity analysis can be
implemented in the AIPE context for the coefficient of
variation using MBESS.
Discussion
If a point estimate is of interest, the confidence limits
that bracket the population quantity also should be. Re-
gardless of the value of a point estimate, the population
quantity will almost certainly differ. Since it is the popu-
lation value, not a sample estimate, that is ultimately of
interest, the limits of the confidence interval are arguably
more important than is the point estimate itself. The limits
of the confidence interval bracket what can be considered
plausible values of the population parameter with some
specified level of confidence. As the width of the interval
narrows, holding constant the level of confidence, more
and more values are excluded and thus are no longer con-
sidered plausible. The width of the confidence interval
is therefore a way through which the accuracy of the pa-
rameter estimate can be operationalized (e.g., Neyman,
value of directly (or indirectly by misspecifying µ and/
or À) on the typical confidence interval width and the pro-
portion of sufficiently narrow confidence intervals. For
example, suppose is specified to be 0.20 for purposes
of the sample size planning procedure, yet in actuality the
value of is 0.30. The question that a sensitivity analysis
addresses in this situation concerns the properties of the
confidence interval width for k when using the sample
size necessary when is based on 0.20, when in fact the
true value of is 0.30.
The way in which a sensitivity analysis can be imple-
mented is by performing a Monte Carlo simulation study
in a population where the value of is set to be the true
value but the sample size used is based on the misspeci-
fied value. Data is generated when all assumptions hold
(i.e., normality and independence of observations) and
a confidence interval is calculated for each of the large
number of generated data sets (e.g., 10,000). As the simu-
lation study progresses, information of interest is recorded
for subsequent analysis (e.g., mean confidence interval
width, proportion of confidence intervals less than the
desired width, etc.). In some cases, the effect of misspeci-
fying a minimal amount is trivial, whereas other times
it can be quite large. The combination of all factors needs
Table 2
Necessary Sample Size for 95% Confidence Intervals
for the Coefficient of Variation in Selected Situations
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
E[w]
.010 199 790 1,812 3,325 5,408 8,166 11,724 16,233 21,866 28,819
.025 37 131 295 537 871 1,312 1,882 2,603 3,505 4,618
.050 13 37 78 139 222 333 475 656 882 1,160
.075 8 18 36 63 100 149 213 296 396 520
.100 6 12 22 38 59 87 123 168 225 294
.125 5 9 16 26 40 58 81 110 146 191
.150 5 8 13 20 29 42 58 78 104 135
.175 – 7 10 16 23 32 44 59 78 101
.200 – 6 9 13 18 26 35 47 61 79
o .80
.010 215 823 1,866 3,401 5,511 8,300 11,896 16,450 22,136 29,150
.025 42 144 316 567 911 1,366 1,950 2,690 3,613 4,751
.050 16 43 88 153 242 360 510 700 936 1,227
.075 9 22 43 73 113 167 238 325 433 565
.100 7 14 27 45 69 100 140 190 253 328
.125 6 11 19 31 47 68 94 127 169 219
.150 5 9 15 24 35 50 69 93 122 158
.175 – 8 12 19 28 39 54 72 94 121
.200 – 7 11 16 23 32 43 57 75 97
o .99
.010 246 886 1,964 3,540 5,698 8,544 12,207 16,841 22,621 29,745
.025 55 170 357 625 989 1,466 2,079 2,852 3,814 4,996
.050 22 57 110 184 283 413 578 785 1,042 1,357
.075 13 31 58 94 142 205 286 385 507 656
.100 9 22 39 62 92 131 179 239 311 401
.125 8 17 30 46 67 93 127 168 219 281
.150 7 14 23 36 52 72 98 129 167 212
.175 – 12 20 30 43 59 79 104 134 170
.200 – 10 17 26 37 50 67 87 112 142
Note— is the population coefficient of variation, o is the desired degree of assurance of achieving a
confidence interval for no wider than desired, is the desired full confidence interval width, and E[w]
is the expected confidence interval width (i.e., when o is not specified).
762 KELLEY
analytic and the AIPE approach to sample size planning
have fundamentally different goals. Necessary sample
size to achieve a statistically significant estimate (i.e.,
power analysis) may be much different than necessary
sample size to achieve a narrow confidence interval (i.e.,
AIPE). Depending on the particular situation, the power
analytic approach or the AIPE approach could require a
larger sample size (see Kelley & Maxwell, 2003; Kelley
et al., 2003; Kelley & Rausch, 2006, for comparisons of
necessary sample sizes for the power analytic and AIPE
approach in difference situations).
From the outset, this article has made the assumption
that the coefficient of variation can be an important and
helpful quantity when trying to understand relative vari-
ability. If the coefficient of variation is indeed of interest,
the confidence interval for the population coefficient of
variation also should be. Since from a probabilistic per-
spective a wide confidence interval illustrates an estimate
with a low degree of expected accuracy—an obviously un-
desirable situation—an effort should be made to obtain an
estimate with a high degree of expected accuracy when-
ever possible. When planning studies and selecting an ap-
propriate sample size in situations where the coefficient
of variation is of interest, it is hoped that researchers take
1937). Since the value of interest is the population value,
designing a study to obtain an accurate estimate should be
a top concern for researchers.
At present, it is not clear to what extent violations of the
assumption of normality and independent observations
will have on the confidence interval coverage and/or the
necessary sample size from the procedure developed. Be-
cause the sample size planning procedure is based on the
parametric confidence interval procedure, in situations in
which the confidence interval procedure fails to provide
appropriate coverage (i.e., when the assumptions of the
procedure are violated), the sample size obtained from
the procedure will probably not be optimal. Thus, in order
for the sample size from the procedure to be appropri-
ate, the assumptions need to be satisfied. At present, the
robustness of the confidence interval procedure and the
sample size planning method are unknown. Issues of ro-
bustness, and appropriateness of sample size in violations
of assumptions, are certainly areas that could benefit from
additional research.
Generally, when the term “sample size planning” is
used, it is taken to mean power analysis, where the goal
of the procedure is to be able to reject the null hypoth-
esis with some specified probability. Note that the power
Table 3
Necessary Sample Size for 99% Confidence Intervals
for the Coefficient of Variation in Selected Situations
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
E[w]
.010 342 1,362 3,129 5,742 9,340 14,102 20,248 28,037 37,766 49,774
.025 62 225 508 926 1,502 2,265 3,248 4,495 6,052 7,974
.050 21 63 134 238 383 573 820 1,132 1,522 2,003
.075 11 30 64 111 175 260 370 509 683 897
.100 9 19 37 63 99 147 209 287 389 510
.125 7 14 26 43 66 97 136 187 250 327
.150 6 12 20 32 49 70 98 134 177 232
.175 6 10 16 25 38 54 74 100 133 173
.200 5 9 14 21 30 43 59 79 104 135
o .80
.010 363 1,407 3,199 5,842 9,475 14,279 28,321 50,209 38,120 82,817
.025 69 242 536 966 1,556 2,335 3,339 4,609 6,194 8,149
.050 25 71 147 258 409 609 865 1,189 1,593 2,090
.075 13 35 73 123 193 283 400 547 730 956
.100 10 23 43 72 112 164 231 320 425 554
.125 8 17 30 50 76 110 154 210 279 363
.150 7 14 23 38 57 82 113 153 202 262
.175 6 11 19 30 44 63 87 117 154 199
.200 6 10 16 25 36 51 70 93 122 157
o .99
.010 404 1,489 3,327 6,023 9,719 14,597 20,881 28,832 38,753 50,986
.025 86 276 589 1,041 1,656 2,466 3,506 4,819 6,454 8,467
.050 32 88 174 297 461 677 952 1,298 1,728 2,256
.075 19 49 91 150 229 331 460 623 824 1,071
.100 13 31 58 93 141 201 279 379 498 644
.125 11 24 43 68 100 142 195 260 341 439
.150 9 19 34 53 77 109 148 196 255 327
.175 8 17 28 44 63 87 118 156 202 257
.200 8 15 24 37 53 73 98 128 166 211
Note— is the population coefficient of variation, o is the desired degree of assurance of achieving a
confidence interval for no wider than desired, is the desired full confidence interval width, and E[w]
is the expected confidence interval width (i.e., when o is not specified).
SAMPLE SIZE FOR THE COEFFICIENT OF VARIATION 763
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AUTHOR NOTE
This work was sponsored in part by a Proffitt Fellowship for Educational
Research. Correspondence concerning this article should be addressed to
K. Kelley, Inquiry Methodology Program, Indiana University, 201 North
Rose Avenue, Bloomington, IN 47405 (e-mail: kkiii@indiana.edu).
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ment Core Team, 2007) that implements all of the methods discussed in the
article, and many more. Both R and MBESS are Open Source and freely
available. R and MBESS can be obtained from the Internet at the Compre-
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page on the Comprehensive R Archival Network is cran.r-project.org/src/
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the documentation can be found (note that this Internet address is case
sensitive). The Appendix provides details on MBESS and example code
illustrating how each of the methods discussed can be easily implemented.
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cussed in the article can thus readily do so using MBESS.
6. It should be pointed out that the exact analytic confidence interval
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legitimate confidence interval for . One possibility is to make use of
the bootstrap technique, a nonparametric method that does not make the
assumption of normality (e.g., Efron & Tibshirani, 1993). Although the
bootstrap is a perfectly legitimate alternative to the analytic approach,
the sample size planning methods have been developed based on the
analytic approach. Of course, one could plan sample size based on the
analytic approach and compute confidence intervals with the bootstrap
approach. However, if normality held in the population, bootstrap con-
fidence intervals would tend to be wider than the normal theory analytic
approach confidence intervals, on which the sample size planning pro-
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developed here for sample size planning based on the population being
normal, then using the bootstrap approach to confidence interval forma-
tion in situations where normality is likely violated, will probably not
yield optimal sample size planning values.
7. Purely analytic solutions do not generally exist for finding quantiles
from the noncentral t distribution, and thus noncentral routines that deal
with noncentral t distributions are essentially always iterative in nature.
Given modern software, this poses no problem. However, the lack of
computing power once motivated a literature of approximate ways to
obtain necessary values from a noncentral t distribution. The difficulty in
finding the necessary confidence limits led to many tri-entry (probabil-
ity, degrees of freedom, and noncentrality parameter) tables (see a review
of such tables in Johnson, Kotz, & Balakrishnan, 1995, chap. 31) that
will almost certainly yield approximate results in any applied situation.
8. Typically just more than 50% of the sampling distribution of w will
be less than , because E[w |
˜
k
(,N)
] will typically be less than . Although
ideally E[w |
˜
k
(,N)
] , because sample size changes in whole numbers,
in order for E[w |
˜
k
(,N)
] to be no larger than , E[w |
˜
k
(,N)
] will typically
be just less than . Coupling this fact with the positively skewed nature
of the noncentral t distribution (when the noncentral parameter is posi-
tive) typically leads to just more than 50% of the sampling distribution
of w being less than .
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of only whole numbers; thus, the change in N is not a continuous and
smooth function. In order to ensure that the confidence interval is no
wider than , p(w ) will almost always be less than o. Although
Equations 13 and 14 theoretically should be equalities, maintaining
sample size at only whole numbers implies a need for inequalities.
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skewed nature of the noncentral t distribution (when 0), an expected
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NOTES
1. In fact, “many parts of the literature” can be identified with specific
journals, where the overarching goal of the journals combines various
aspects of biological systems and processes with psychological systems
and processes (e.g., Behavioral Ecology; Behavioural Brain Research;
Behavioural Pharmacology; Biological Psychology; Brain Research;
Developmental Psychobiology; European Neuropsychopharmacology;
Genes, Brain & Behavior; Hormones & Behavior; International Jour-
nal of Psychophysiology; Journal of Behavioral Medicine; Journal of
Clinical & Experimental Neuropsychology; Neurobiology of Learning
& Memory; Physiology & Behavior; Pharmacology Biochemistry & Be-
havior; and Psychoneuroendocrinology, to list a nonexhaustive set of
peer reviewed journals).
2. Using the work of Monchar (1981) as partial motivation, and im-
proving the quantitative measure of inequities as another, Sheret (1984)
discussed using a weighted coefficient of variation when the number of
observations within groups differed across a set of groups for which the
coefficient of variation was to be calculated.
3. It should be noted that there is some debate as to whether or not the
coefficient of variation should be computed only for nonnegative ratio
scaled data (e.g., Bedeian & Mossholder, 2000). Although there is a con-
siderable literature on the subject of what should and should not be done
to a set of numbers (i.e., the measurement scale debate; e.g., Velleman &
Wilkinson, 1993, for a review), the coefficient of variation can certainly
be computed whenever the mean and standard deviation are available.
Of course, whether or not the coefficient of variation is a meaningful
quantity depends on the particular situation and the question of interest.
Regardless of which side of the measurement scales debate one comes
down on, the methods discussed in the article can be used whenever there
is an interest in the coefficient of variation.
4. Assuming that the assumptions of the model are met, the correct
model is fit, and observations are randomly sampled, (1 () is the
probability that any given confidence interval from a collection of confi-
dence intervals calculated under the same circumstances will contain the
population parameter of interest. However, it is not true that a specific
confidence interval is correct with (1 () probability, since a computed
confidence interval either does or does not contain the value of the pa-
rameter. The confidence interval procedure refers to the infinite number
of confidence intervals that could theoretically be constructed, and the
(1 ()100% of those confidence intervals that correctly bracket the
population parameter of interest (see Hahn & Meeker, 1991, for a techni-
cal review of confidence interval formation and interpretation). Although
the meaning of confidence intervals has been given from a frequentist
perspective, the methods discussed in the article are equally applicable
under the Bayesian perspective of confidence interval interpretation.
5. Methods for the Behavioral, Educational, and Social Sciences
(MBESS; Kelley, 2007a; Kelley, in press) is an R package (R Develop-
SAMPLE SIZE FOR THE COEFFICIENT OF VARIATION 765
APPENDIX
Using MBESS to Implement the Methods
All of the methods and procedures discussed and the algorithms presented can easily be implemented in
the Methods for the Behavioral, Educational, and Social Sciences (Kelley, 2007a, 2007b, 2007c) R package
(R Development Core Team, 2007). This appendix provides a brief overview of the way in which the necessary
MBESS functions can be used when the coefficient of variation is of interest. Those not familiar with R will
see that R is a command driven language. The commands (which are case sensitive) are input directly into the
R console. Unlike some programming languages, R executes code line by line sequentially. R code that is directly
executable in this appendix is proceeded with R, where R is used to illustrate the R command prompt where
commands are input and then executed (by entering the command). Both R and MBESS are Open Source and thus
freely available. R and the MBESS package are available at the following Internet address for all commonly used
operating systems: cran.r-project.org. The specific Internet address for MBESS is cran.r-project.org/src/contrib/
Descriptions/MBESS.html. Because MBESS is an optional package, it must be loaded with each new R session
where its routines will be used. Packages in R are loaded with the library() command, which is illustrated
with MBESS as follows:
R> library(MBESS).
Confidence Intervals for a Noncentral t Parameter
For constructing confidence intervals for the noncentrality parameter from a noncentral t distribution, the
conf.limits.nct() function can be used. The lower and upper critical values from the noncentral t distri-
bution are returned by specifying the following arguments in the conf.limits.nct() function:
R> conf.limits.nct(ncp =
ˆ
, df=, conf.level= 1 (),
where ncp is the estimated noncentrality parameter, df is the degrees of freedom for the particular situation,
and conf.level is the desired level of confidence (i.e., 1 (). For the Volkow et al. (2002) example dis-
cussed previously, the function would be implemented as
R> conf.limits.nct(ncp=21.831, df=34, conf.level=.95).
Confidence Intervals for the Coefficient of Variation
Given the one-to-one relation between and , and the confidence interval transformation principle previ-
ously discussed, the confidence limits for can be found by transforming the confidence limits for given the
relation specified in Equation 7 and replacing what was the upper limit with what was the lower limit (and vice
versa). Alternatively and simpler, the ci.cv() function can be used directly to determine the confidence limits
for . The lower and upper critical value from Volkow et al. (2002) discussed previously are returned using the
following specifications
R> ci.cv(cv=.271, n=35, conf.level=.95),
where cv is the observed coefficient of variation (i.e., k), n is the sample size, and conf.level is the desired
level of confidence (i.e., 1 ().
Planning Sample Size for the Coefficient of Variation
Using the example provided in the Tables of Necessary Sample Size section, where was set to .25 and
set to .10 for a 99% confidence interval, the ss.aipe.cv() function can be used. The way in which the
ss.aipe.cv() can be used so that the expected width is sufficiently narrow is given as
R> ss.aipe.cv(C.of.V=.25, width=.10, conf.level=.99),
where C.of.V is the population coefficient of variation (i.e., ), width is the desired confidence interval
width, and conf.level is the confidence level (i.e., 1 (). Implementation of this function yields a neces-
sary sample size of 99 (as reported in the text and in Table 3).
The way in which the desired degree of assurance can be used in the function is by specification of assurance,
which is an optional argument in the ss.aipe.cv() function. The function would thus be specified as
R> ss.aipe.cv(C.of.V=.25, width=.10, conf.level=.99, assurance=.99),
which returns necessary sample size of 141 (as reported in the text and in Table 3).
Sensitivity Analysis for the Coefficient of Variation Given the Goals of AIPE
Sensitivity analysis to assess the effect of misspecifying on the width of the confidence interval can be per-
formed with the ss.aipe.cv.sensitivity() function. The function ss.aipe.cv.sensitivity()
allows one to specify the true population and an estimated but incorrect value, so that the effect of misspeci-
fying on the width of the obtained confidence intervals can be empirically determined. The function performs
a simulation where the empirical findings regarding the width of the confidence interval can be determined.
766 KELLEY
APPENDIX (Continued)
The results of the simulation within ss.aipe.cv.sensitivity() can be very helpful for determining
how discrepant an incorrectly specified value of can be from itself in order to still have an acceptably narrow
confidence interval for . The ss.aipe.cv.sensitivity() function can be specified as
R> ss.aipe.cv.sensitivity(True.C.of.V, Estimated.C.of.V, width,
conf.level, G),
where True.C.of.V and Estimated.C.of.V are the true and the estimated values, width is the de-
sired confidence interval width, assurance is the desired degree of assurance, conf.level is the desired
confidence level (i.e., 1 (), and G is the number of replications that take place within the simulation study
(e.g., G 10,000). Instead of specifying Estimated.C.of.V, a particular sample size can be specified using
Specified.N, so that the properties of the confidence interval can be readily determined for a particular
value at a specific sample size. A value of o can also be specified with the addition of the optional assurance
argument. The output of the function provides a thorough summary of the results, optionally with the full set
of results from the Monte Carlo simulation study for the other analyses not implemented in the function can be
performed on the results (e.g., visualization techniques).
(Manuscript received August 30, 2006;
revision accepted for publication February 1, 2007.)
