# How do I estimate the average size of objects

We have a large number of (physical) legal files, and want to get an estimate of the average number of pages per file. I do not know what distribution of pages to expect, the number of pages will range from 1 to a large finite number.

My thought is to take a number of samples, count the number of pages in those files, and then use the average from that, but how many samples should I take to get a mildly accurate answer (no more than 20% off)?

Does it matter which distribution I assume? I assume it does, but given that those files grow until they are viewed as done, at uneven rates (some letters contain more pages than others), what would most likely be the distribution to describe this?

• Don't forget to give some consideration of why you want to know the average. It's often not a very useful summary of a distribution. – D L Dahly Feb 18 '14 at 10:03

This is the kind of problem treated in finite population sampling theory, as presented in the book http://www.amazon.com/Finite-Population-Sampling-Inference-Prediction/dp/0471293415/ref=sr_1_1?s=books&ie=UTF8&qid=1401276486&sr=1-1&keywords=finite+population+sampling+theory (and many others).

First of all, you will want a practical way to do the sampling! So we will need to know how your physical files are stored! You should also think about if they are stored in completely random order, or in some other definite order (alphabetical? chronological? what???). A simple random sampling will often be impractical --- could be practical if you have some list of all the files in your collection, wheather that list is on a computer or on paper (and if it is practical to access the files out of order). If the answer to that question is NO, so simple random sampling is impractical, you can use some kind of cluster or stratified sampling. You can find explanations of this and related terms here: https://en.wikipedia.org/wiki/Statistical_sampling

Without knowing more about the practical situation, only some brief hints: sampling could be on the level of --- shelfes --- drawers or some other units in which the physical storage is organized. You should think about questions such as: If the storage is is say, chronological order, might there be some trend in document size with time? Or some cyclical variation, that is , in some parts of the year, documents of a specific type are produced, which typically differs in length?

We can help more if you tell us more about the situation!

(Then, depending on the sampling plan choosen, there will be some specific formulas/methods to use to construct estimates and confidence intervals)

An interval for the mean will

(i) be probabilistic - that is, with random sampling, you could compute a mean that's no more than 20% off 99% of the time (or some other percentage) -- but not 100% unless your sample is your population.

(ii) depend on the standard deviation of the number of pages; since this will be unknown, you'll need some information about that (perhaps an upper bound, perhaps an estimate from a pilot sample) before you can calculate an estimate of the required sample size.

I would explore graphically the population by construncting some random samples of documents from your repository, then plot histograms of the number of pages for each document in the sample.

If the distributions do not resembles like a normal distribution, the mean alone is not so informative (as said in comments) and maybe you should need to estimate higher order moments like variance, kurtosis and skewness.

Another good tool to get a feel of the distributions are boxplots.

If data does not fit any theoretical distribution, you can still estimate these moments at some confidence interval by bootstrap.

• I disagree that mean alone is non-informative for non-normal distributions. Consider Poisson. – Aksakal May 28 '14 at 13:00