# How to infer the distribution of a statistic (Bayesian inference?)

I have a list of approximately 30,000 venues in a major US city. These venues hold all kinds of events, sports, conferences, concerts etc. I want to know the distribution of the 'capacity' of these venues, that is, the maximum number of people that can attend an event held at the venue.

I have to do this manually, so I can't go through all 30,000 venues. I want to devise a process by which I can improve the estimate I have of the distribution of this capacity statistic with every iteration.

The naive approach I can think of would be to repeatedly sample (i.e. manually find the capacity for) a few venues at a time. For each sample, look at its distribution (called the 'sampling' distribution?), record the mean and variance. Then, look at the distribution of those means and variances and somehow use them to make an inference about the original distribution, say by just taking the mean of each, or perhaps something more subtle than that.

1) Is this process the correct approach?

2) How can I know the optimal size of the samples I'll be taking to converge to the distribution as quickly as possible without taking too large a sample that takes a long time and effort?

3) How can I get a sense of how good my estimate of the distribution is and how much it improves with each new sample I calculate?

I can imagine there is a huge amount of literature written about this, I am open to links referring me to sources where I can learn more, but I am also looking for a basic rule of thumb here to go about this in a way that wouldn't make Bayes turn in his grave.