The question came from Detlef Axmann.

This note explains the formulae for the standard error and 95% confidence
interval for the estimated within subject standard deviation,
*s _{w}*. (This is described in a
Statistics Note in the

The within-subject standard deviation is found by dividing a sum of squares by its degrees of freedom, to get the estimate of variance. The square root of this is the estimate of the standard deviation.

Assume that the observations themselves follow a Normal distribution, and are
identically distributed about the subject mean, within-subject SD =
*sigma _{w}*. The within-subject variance is estimated by

The distribution followed by the sum of squares divided by
its degrees of freedom is that of a
Chi-squared random variable with *d* degrees of freedom, multiplied by
*sigma _{w}*

The square root of a Chi-squared variable has an approximately Normal
distribution, with mean approximately root(*n* - 1/2) and variance
approximately 1/2, provided *d* is reasonably large.

The distribution followed by the estimated within-subject standard deviation,
*s _{w}*, is the distribution followed by the square root of

If we have *n* subjects with *m* observations per subject, we have
*n*(*m*-1) degrees of freedom. Hence the standard error of
*s _{w}* will be estimated by

If the sample is large, the 95% confidence limits for *s _{w}* are
the observed value minus and plus 1.96 times this standard error. If the
sample is small, we should replace 1.96 by the corresponding value of the t
Distribution with

There is an alternative way to find a confidence interval which does not use a standard error.
For a Normal population, a sample variance follows a Chi-squared distribution
multiplied by the the population variance and divided by the degrees of freedom.
Find the 2.5% and 97.5% points of Chi-squared with the d.f. for the estimate.
Multiply these by the sample variance and divide by the d.f., *d*.
This gives the 95% CI for the variance.
Now take the square roots.
This gives the CI for the standard deviation.
This is another large-sample approximation.

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Last updated: 4 August, 2011