I have several sets of data, unfortunately the data comes to me in a "summary" form. My job is to consolidate the several data sources into one general summary. I'm currently using the median to summarise the data, but I don't know if this is statistically sound. Here's a description of my problem:
There are $N_P$ samples, each with varying sample sizes, but all from a single population. Neither the sample size or the standard variation are known. Each sample can be divided into $N_Q$ disjoint groups (or qualities). From each sample, the only data that is known is what percent of the sample falls within a group (or category). For example, population $A$ contains, $x\%$ of $a$, $y\%$ of $b$ and $z\%$ of $c$.
The different samples are not disjoint, so a single item might be in several of the samples; but I don't know how much overlapping there is. There are 5-8 different samples with 5-7 categories. An example (smaller) table is the following.
cat. a cat. b cat. c sample A 47.34% 30.05% 11.92% sample B 41.60% 29.90% 11.90% sample c 47.74% 29.67% 12.69% -------- ------ ------ ------ median 47.34% 29.90% 11.92%
Now is it statistically sound to create this "median" summary, which takes each group from the different samples and finds the median? Maybe I should be using the mean? The problem I'm seeing is the "median sample" usually sums to less than 100%, even though the percentages from each sample sum to 100%. Should this matter?
Sample sizes: 100k - 100m Population size: ~1 billion