I have several hundred measurements. Now, I am considering utilizing some kind of software to correlate every measure with every measure. This means that there are thousands of correlations. Among these there should (statistically) be a high correlation, even if the data is completely random (each measure has only about 100 datapoints).

When I find a correlation, how do I include the information about how hard I looked for a correlation, into it? 

I am not at a high level in statistics, so please bear with me.

This is an excellent question, worthy of someone who is a clear statistical thinker, because it recognizes a subtle but important aspect of multiple testing.

There are standard methods to adjust the p-values of multiple correlation coefficients (or, equivalently, to broaden their confidence intervals), such as the Bonferroni and Sidak methods (q.v.).  However, these are far too conservative with large correlation matrices due to the inherent mathematical relationships that must hold among correlation coefficients in general.  (For some examples of such relationships see the recent question and the ensuing thread.)  One of the best approaches for dealing with this situation is to conduct a permutation (or resampling) test.  It's easy to do this with correlations: in each iteration of the test, just randomly scramble the order of values of each of the fields (thereby destroying any inherent correlation) and recompute the full correlation matrix.  Do this for several thousand iterations (or more), then summarize the distributions of the entries of the correlation matrix by, for instance, giving their 97.5 and 2.5 percentiles: these would serve as mutual symmetric two-sided 95% confidence intervals under the null hypothesis of no correlation.  (The first time you do this with a large number of variables you will be astonished at how high some of the correlation coefficients can be even when there is no inherent correlation.)

When reporting the results, no matter what computations you do, you should include the following:


The size of the correlation matrix (i.e., how many variables you have looked at).
How you determined the p-values or "significance" of any of the correlation coefficients (e.g., left them as-is, applied a Bonferroni correction, did a permutation test, or whatever).
Whether you looked at alternative measures of correlation, such as Spearman rank correlation.  If you did, also indicate why you chose the method you are actually reporting on and using.

