# Confidence intervals for aggregated statistics

Given min, mean, max and var of scores of several subjects

                 min mean max var
Subject 01 X_01:   2    3   5   1
Subject 02 X_02:   2    4   7   2
...
Subject 30 X_30:   1    3   5   1


where it is assumed that each $X_i$ is normally distributed, but the $X_i$ are not necessarily identically distributed.

How can 95% confidence intervals for the aggregated statistics be computed? For example, the minimum has mean $(2+2+..+1)/30$. But what is its confidence interval? Same questions for the mean of all runs, the maximum of all runs, and the variation of all runs.

The amount of measurements taken per subject is not constant, but varies between, say, 80 en 100.

Quite likely this is a standard question in statistics. Therefore a couple of key words and pointers to the literature would probably do.

• It seems multilevel model may bt the answer. Have you considered it? If not, why doesn't it suite?
– Tim
Mar 8, 2016 at 8:56
• Considered it, but MLM doesn't seem to address confidence intervals of extreme values and confidence intervals of variance. Mar 8, 2016 at 9:19
• If the individual realizations are not from an identical distribution, then does it even make sense taking a mean over the observed quantities? Mar 8, 2016 at 9:20
• Hierarchical Bayesian model can deal with that.
– Tim
Mar 8, 2016 at 9:21

I would try with inverse variance weighting. Otherwise you can use bootstrap, but it would help knowing also how many measurements were taken per subject.

• Amount of measurements per subject added to question. Mar 8, 2016 at 9:39