I have two variables and 1000 cases. How can I statistically find representative cases from total of 1000, based on statistical properties of both variables and correlation between them. Perhaps something based on T-test and 95% (or 99%) interval but for both variables? I would like to know which statistical method can find cases that have both values (simultaneously) statistically the most significant. I know that this deals with sample distribution and estimating the proportions.
I like multivariate Gaussian Mixture Models because, with just a few dimensions to the data, they can show something about the interaction between the values.
Here is an example of a Gaussian Mixture Model in plot. http://www.mathworks.com/help/stats/gaussian-mixture-models.html#bra9fvn
When you say "most representative" to me that says "expection" or "mean". The figure shown in step 3 of the preceding link gives you the two modes/peaks of "higher expectation". If you were looking for central tendency then you should pick there.
I think that you are looking for domain-spanning sub-sampling. It sounds like you want data, not only at the edges, but with some measure of "uniformity" across the potential values. This could be for some experiment or characterization. If you know something about the actual distribution by fitting or knowledge of the system that produced the data then you could use methods from "design of experiments" to determine which samples are most informative. A useful sampling strategy might be to find "D-optimal" sampling locations within the domain.