I recently started reading Gelman and Hill's, "Data Analysis Using Regression and Multilevel/Hierarchical Models" and the question is based on that:

The sample contains 6 observations on proportions: $p_{1}, p_{2}, \dots, p_{6}$

Each $p_{i}$ has mean $\pi_{i}$ and variance $\frac{\pi_{i}(1-\pi_{i})}{n_i}$, where $n_{i}$ is the number of observations used to compute proportion $p_{i}$.

The test statistic is $T_{i} = $ sample standard deviation of these proportions.

The book says that Expected value of the sample variance of the six proportions, $p_{1}, p_{2}, \dots, p_{6}$, is $(1/6)\sum_{i=1}^{6} \pi_{i}(1-\pi_{i})/n_{i}$. I understand all this.

What I want to know is the distribution of $T_{i}$ and its variance? Would appreciate if someone could let me know what it is, or guide me to a book or article that contains this information.

Thanks a ton.