Many thanks to all who replied to my query (repeated below) in this
issue. I am sorry it has taken me so long to compile this summary - I
know at least two of you have been keen to see it!

For those of you who may have forgotten, my initial query was:

> The formula for the standard error of a sample mean is obviously
> well known, but can anyone point me towards a reference giving a
> formula for the standard error of a sample variance/coefficient of
> variation?

Several people recommended Kendall's Advanced Theory of
Statistics - Stuart, A., & Ord, J.K. (1997), Vol. 1. (I should have tried
this myself!) - in the 5th edition it's pp 327-328.

Other references given were:

Dudewicz & Mishra (1998). Modern Mathematical Statistics (Wiley)

Keeping, E.S. "Introduction to Statistical Inference" (Dover - ISBN 0-
486-68502-0)

Bland, M. (1995) An Introduction to Medical Statistics (Oxford
University Press), p.130.

The Encyclopedia of Statistics

Freund, J.E. & Walpole, R.E. (1980). Mathematical Statistics, p.380


A couple of people also suggested I tried bootstrapping. Mike
Weale and Peter Lane both gave (with working - others gave it
without working) the result that the standard error of the sample
variance s^2 as s^2*sqrt(2/(n-1)), and the variance as 2*sigma^4/
(n-1).

Two longer responses are as follows. Dick Brown wrote:

"An invaluable reference for somewhat out of the way topics like
this is
"Kendall's Advanced Theory of Statistics" M G Kendall, A Stuart
and
J K Ord.
In Volume 1, 5th Edition p338 there is a table of commonly required
standard errors including the sample varaince and the coefficient of
variation.

However the warnings on p327-328 shold be noted.
I quote:
"The standard error, strictly speaking, has been justified only in the
case where the statistic tends to normality..."
"... some statistics tend to normality more rapidly than others and
a given n may be large for some purposes but not for others ..."
"...the sampling variance of a moment depends on a population
moment
of twice the order, i.e. becomes very large for higher moments, even
when n is large. This is the reason why such moments have very
limited practical application."

At the risk of being accused of teaching grandmothers to suck
eggs:

One should also bear in mind that when sampling from a normal
population the sampling distribution is related to the chi-squared
distribution which is of course very asymmetrical for small and
moderate
sample sizes so one would not use the standard error to fudge up a
symmetrical "confidence interval" for the sample variance
along the lines used for the sample mean,

( sample variance +/- some number * standard error).

Instead one should exploit the above relationship to produce an
asymmetrical interval.
The coefficient of variation being the ratio of two statistics will
give rise to similar problems.

Another problem is that the sampling distribution of the sample
variance
is somewhat less robust to departures from normality than the
sampling
distribution for the sample mean. "

------------------------------------

and Robert Newcombe gave:


"Generally there are two reasons to want a standard error: to
construct a CI, or to do a hypothesis test. To construct a CI for a
variance estimated from a sample, use the alpha/2 and 1-alpha/2
quantiles of the chi-square distribution with the relevant number of
df, here n-1. To compare two variance estimates, refer their ratio
(higher/lower) to the F-distribution on n1-1 and n2-1 df, being
careful to be really two-tailed. It wouldn't make sense to do either of
these via a SE for the variance, as the sampling distribution isn't
close to Gaussian except for large sample sizes, whereas the
above
purpose-built methods are available, which assume the *original
variable* is Gaussian. Any formula for the SE of the variance would
also depend heavily on the distribution being Gaussian. Indeed,
testing equality of two variances using F is so Gaussian-dependent
that it practically also acts as a test for Gaussian distributional form -
 
homogeneity of variance is only a realistic possibility for Gaussian
data.

You mention the CV also. If you mean, sample SD / sample mean,
(often expressed in percentage terms), this can blow up when the
sample mean is zero, and potentially there are the same difficulties
as when a difference between two proportions is inverted to give a
number needed to treat - when the CI for the difference includes
zero (i.e. H0 is not rejected), the CI for the NNT is doubly infinite,
something that in my opinion only mathematicians can grasp. I
wouldn't recommend trying to put a CI on the CV as defined in this
way, that encompasses the uncertainty of both the mean and the
SD
- it's an artificial sort of thing, better to express the uncertainties of
the sample mean and SD separately (fortunately they're
independent!) Often, when such a CV is relevant, what is really
implied is that the data warrants log transformation, which will
improve both the fit of a Gaussian model and make SDs
independent of means instead of proportional to them. The SD on
a
natural log scale is practically the same as the CV, and is a much
more natural parameter to do things with - for example, you can
form a CI as described above.

Sometimes, people use CV to describe something else: the SD
for
reproducibility (whether intra- or inter-observer) divided by the
mean. Here, there should be no question of a zero or near -zero
mean. The dominant source of uncertainty is likely to be on the SD,
though it's important to check this. If so, then get a CI for the
reproducibility SD (using the above method, with the appropriate
no.
of df), and simply divide by the mean, and say that this interval
relates to the imprecision of the SD relative to an evidence-based
but assumed imprecision-free value for the mean. Alternatively, the
log transformation approach can again be used."


Once again, many thanks.


Jeremy
------------------------------------
Jeremy Dawson
Statistician
Organisation Studies Group
Aston Business School
(0121) 359 3611 ext. 4596
[log in to unmask]
------------------------------------


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