# What is the pooled standard deviation of paired samples?

I am trying to do a priori sample size calculation based on published results. However, I am unable to obtain a reasonable estimate of the published effect size (which is not reported) as I am unable to obtain an estimate of the pooled standard deviation or the standard deviation of the difference.

The problem resides in the fact that it is a factorial experiment with two within-subjects factors ($2 \times 3$ levels). I only have the cell means and standard deviations (i.e., for the $2 \times 3$ table) but not the marginal means for the first factor (with 2 levels) which I need.

I know that the formula for the pooled standard deviation for independent samples is the square root of (taken from wikipedia) .
But what is the formula for pooled standard deviation for dependent samples?

Means:

     1A   1B
2a 3.24 3.01
2b 2.91 2.56
2c 3.01 3.05


Standard deviations:

     1A   1B
2a 0.65 0.70
2b 0.68 0.60
2c 0.46 0.53


I want to obtain the effect size between 1A and 1B (so pool over levels of factor 2). Sample size is 27.

If this is a completely balanced within-subject design with $27\times 6=162$ observations, then you can actually calculate the marginal means: simply average over the levels of the second factor. Of course, you have to be sure that averaging over different conditions is meaningful for your planned experiment - do you expect each of those conditions to be present with about 1/3 probability?
The real difficulty is with the variance of the difference. It is well known that $$Var(X-Y) = Var(X) + Var(Y) - 2 SD(X)SD(Y)Corr(X,Y)$$ The problem is that you don't know the within-subject correlation.
• Thanks a lot. I actually have some estimate of the correlation (we replicated the original study and used our correlation). From this I ended up using the mean of the sd (i.e., totally ignoring the $n$ in the formula) as it was indeed completely balanced. With means, sds and correlation you can make a power analysis using the fabulous G*Power 3: psycho.uni-duesseldorf.de/abteilungen/aap/gpower3 – Henrik Apr 16 '12 at 14:58
• $Cov(X,Y) = SD(X) SD(Y) Corr(X,Y)$, so you can write the formula with either covariance or correlation. – Aniko Jan 23 '17 at 15:22