# How to calculate standard error of the mean of a set of correlation coefficients

I'm trying to figure out how to calculate the standard error of a mean correlation coefficient.

I have 6 bilateral correlation coefficients for 4 countries. I have transformed them using the Fisher z transformation in order to calculate their mean correlation coefficient. I'm trying to figure out what the standard error of this statistic is in order to create a confidence interval to test for significance against other mean correlation coefficients.

I think I'll need to use the Fisher std error formula 1/SQRT(N-3) but I'm not sure what the N here should be. The sample size, or the number of bilateral pairs used in the mean correlation coefficient statistic?

When I use the N=number of bilateral pairs, so that the standard error is the cross-sectional dispersion, my 95% confidence interval doesn't make sense. e.g. r = 0.7040 (-0.2510,0.9645). Using the n=number of observations would mean that the standard error is constant across all mean correlation coefficients. Would it make sense to bootstrap the errors instead?

It would be great if I could create a confidence interval for each of these mean statistics and report it in one table so that the reader is able to compare across these regional correlations as they like.

• what is a 'mean correlation coefficient'? May 24, 2011 at 4:22

Just a few thoughts:

• n is the sample size for each bivariate correlation, i.e. $n \neq 6$.

• I am not sure if this makes sense but you could run a small meta-analysis (based on the Fisher's transformed correlations). This would give you a pooled standard error (see page 4).

• Whatever you do, your effect sizes (correlations) are not independent because each country is part of three bilateral correlations. Using dependent effect sizes will probably lead to a biased pooled standard error (which means that running a simple meta-analysis isn't such a good idea but you could do a robust variance estimation).

• I do not understand the last paragraph of your post/question: "It would be great if I could create a confidence interval for each of these mean statistics and report it in one table so that the reader is able to compare across these regional correlations as they like." Can you give us an example?

• Thanks, I've decided to opt for bootstrapping the confidence intervals instead.
– SLT
May 31, 2011 at 4:14