Interested in meta-analysis I found the lecture notes http://www.uazuay.edu.ec/cibse/lecture_notes.pdf
I understand that it makes sense in a meta-analysis to assign different weights to different studies. Personally I find it natural to weigh by the sample sizes of the particular studies. However, in the fixed effect model the inverse of the variances is used. As an explanation on page 17 I read the sentence "The inverse variance is roughly proportional to sample size, but has finer distinctions,( and serves to minimize the variance of the combined effect.)"
I do not understand that and have problems with the first claim.
Suppose $X_1,..,X_n$ are random variables iid. Then $\bar{X}=\frac{1}{n} \sum_n X_i$ has variance $\frac{Var(X)}{n}$. This is clear. So one could say that on the level of random variables the inverse variance of the arithmetic mean is proportional to the sample size.
However, on the level of samples, we always compute the sample means and the sample variances, and the sample variance is an estimator for the variance of an individual measurement and not the variance of the (sample) arithmetic mean.
Can somebody clear up my confusion