# “Averaging” variances

I need to obtain some sort of "average" among a list of variances, but have trouble coming up with a reasonable solution. There is an interesting discussion about the differences among the three Pythagorean means (arithmetic, geometric, and harmonic) in this thread; however, I still don't feel any of them would be a good candidate. Any suggestions?

P.S. Some context - These variances are sample variances from $n$ subjects, each of whom went through the same experiment design with roughly the same sample size $k$. In other words, there are $n$ sampling variances $\sigma_1^2$, $\sigma_2^2$, ..., $\sigma_n^2$, corresponding to those $n$ subjects. A meta analysis has been already performed at the population level. The reason I need to obtain some kind of "average" or "summarized" sample variance is that I want to use it to calculate an index such as ICC after the meta analysis.

P.P.S. To keep the discussion more concrete, let me explain the issue with the following example in R:

library(metafor)
dat <- get(data(dat.konstantopoulos2011))
dat$district <- as.factor(dat$district)
dat$school <- as.factor(dat$school)


In the dataset there is a variance associated with each school's performance score:

str(dat)
Classes ‘escalc’ and 'data.frame':  56 obs. of  6 variables:
$district: Factor w/ 11 levels "11","12","18",..: 1 1 1 1 2 2 2 2 3 3 ...$ school  : Factor w/ 11 levels "1","2","3","4",..: 1 2 3 4 1 2 3 4 1 2 ...
$year : int 1976 1976 1976 1976 1989 1989 1989 1989 1994 1994 ...$ yi      : atomic  -0.18 -0.22 0.23 -0.3 0.13 -0.26 0.19 0.32 0.45 0.38 ...