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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