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I'm looking at correlation for a large number of vectors, and many (about 3000) of these pairwise comparisons appear to have a significant correlation even after Bonferroni correction. Plotting these vectors tells a different story, though. A few look like this:

but the vast majority are clearly spurious correlations arising from a few outliers, with plots that look like this:

I want to filter down to just the vectors which seem genuinely associated, but it's not realistic to go through 3000 plots. Are there statistics I can calculate for each pair of vectors that will help me distinguish between the first and second cases above?

I'm looking at correlation for a large number of vectors, and many (about 3000) of these pairwise comparisons appear to have a significant correlation even after Bonferroni correction. Plotting these vectors tells a different story, though. A few look like this:

but the vast majority are clearly spurious correlations arising from a few outliers, with plots that look like this:

I want to filter down to just the vectors which seem genuinely associated, but it's not realistic to go through 3000 plots. Are there statistics I can calculate for each pair of vectors that will help me distinguish between the first and second cases above?

I thought about doing something with (co)kurtosis, and have been playing around with that a bit, but haven't been able to figure out what to do with it exactly.