This note describes how to choose the sample size to estimate the within
subject standard deviation, *s _{w}*. This is described in a
Statistics Note in the

The precision with which we can estimate *s _{w}* depends on both
the number of subjects,

on either side of the estimate.
(This is explained in another FAQ answer:
"What is the standard error of the within-subject standard deviation, *s _{w}*?".)
A convenient way to deal with the dependence of the standard error, and hence the
sample size, on the quantity we wish to estimate, is to think of estimating it
to within some fraction of the population value, such as 10%. To find the
sample size, we set

This is an equation with two unknown quantities, so there are many combinations
of *n* and * m * which will give the required precision.

One solution would be to take all the observations on a single subject, i.e. to
put * n* = 1. This is not an attractive idea, as some individuals may
have more variation than others, and the variation may not be the same
throughout the range of measurement. The standard error formula was derived
under the assumptions of a simplistic model: that the variation is the same for
everyone. This is quite adequate to enable us to estimate the mean variation,
but we must collect the data allowing for the possibility that the data to not
follow it exactly. We need several subjects with measurements covering the
whole range in which we are interested.

Returning to the example, suppose we know that we can make repeated
measurements on 20 subjects. How many measurements should we take on each?
We put *n* = 20 and solve for * m*, giving

and so

Hence * m* = 11, to the nearest integer, and we require 11 measurements on
each of 20 subjects.

Alternatively, we may decide that there is an upper limit to the number of
measurements per subject, perhaps because of the discomfort caused. If we can
take only two measurements per subject, for example, we have * m* = 2
giving us

hence

As so often happens in statistics, accuracy requires a large sample size.

We could set a less stringent target for accuracy. For example, we could set
the confidence interval as 20% either side of the estimate of
*s _{w}*. For

subjects.

I am very grateful to Garry Anderson for pointing out errors on this page.

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Last updated: 17 May, 2010.