Averaging binomial confidence intervals I have a general statistical question. Three schools A, B, and C have 0.23, 0.56, and 0.8 fraction of female teachers. The upper and lower confidence intervals (CI) for the three schools can be tabulated as shown below:
school  female_instructors  upper_CI  lower_CI
A       0.33                0.53      0.13
B       0.55                0.65      0.45
C       0.80                0.90      0.70

If I were asked to report the average fraction of female instructors in the three schools combined I would have informed: (0.33+0.55+0.8)/3 or 0.56. How would I define the CI for the average fraction?? Average given CIs as well i.e. upper: 0.69 and lower:0.43?? Is this is the best way to do such calculations?
 A: Two points
First point: You could also perform a weighted average of the proportion of female instructors at the three schools (e.g. if A has ten times as many total instructors as B and C combined, one might argue that it should count more heavily). For example, $(n_{A}\times 0.23 + n_{B}\times 0.56 + n_{C} \times 0.8)/(n_{A} + n_{B} + n_{C})$. If the total number of instructors at each institution is the same size, then the weighted average reduces to the simple arithmetic mean as in your example.
Second point: If you recall that for Bernoulli distribution, the variance is determined by $p$ (the proportion), as in $\sigma^{2}_{p}=p(1-p)$, so that $\sigma_{p}=\sqrt{p(1-p)}$, and $\sigma_{\hat{p}}=\sqrt{p(1-p)/n}$ (where $n = n_{A} + n_{B} + n_{C}$), one could readily generate a CI (say $\hat{p} \pm z_{\alpha/2}\sigma_{\hat{p}}$) under the assumption that A, B, and C are all drawn from the same population.
Bonus point: Agresti and Coull have shown that the nominal coverage of the CI I just indicated performs suboptimally for small $n$. They provide an alternative which gives better nominal CI coverage for $\hat{p}$ thus:
$$\tilde{p} \pm z_{\alpha/2}\sqrt{\tilde{p}(1-\tilde{p})\tilde{n}}$$
where $\tilde{n} = n+2z_{\alpha/2}$, and $\tilde{p} = \frac{\left(\Sigma x\right) +z_{\alpha/2}}{\tilde{n}}$. Understand that the use of $\tilde{p}$ is purely instrumental, and the Agresti-Coull confidence interval is for $\hat{p}$. As sample size gets big, the nominal coverage of the standard Wald-type CI and that of the Agresti-Coull CI converge. More details about these and other binomial proportion confidence intervals on Wikipedia.

**References**
Agresti, A. and Coull, B. A. (1998). Approximate is better than “exact” for interval estimation of binomial proportions. The American Statistician, 52(2):119–126.
