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The figure below shows $2000$ points in $(x,y)$ coordinates that are supposed to be high quality pseudorandom numbers.

2000 "pseudorandom" points in 2-D.

However, when I zoom in on any area lots of points are lined up along line segments, and there are relatively large areas with no points. This is illustrated in the next figure where I drew red lines to show where some of them are arranged in a short line. I also drew ellipses where there are larger gaps.

Zoomed in on the above figure with non-random features indicated

Does this look like high quality pseudorandom numbers? What quatitative tests can be used to verify or disprove my hypothesis?

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Random does not mean "uniform"; nor does it mean "does not contain any small runs" etc. In fact, in a truly random sample, you can expect a certain amount of "clumping" and a certain amount of "valleys". A truly uniform distribution is not random. – Arturo Magidin Jun 30 '12 at 19:48
I would be more concerned about your generator if the distribution of points did not show such patterns... – D. Thomine Jun 30 '12 at 19:52
3 has a bit of statistical analysis on their true random number generator, and includes a comparison of their random bitmap generator and one which used a bad pseudorandom number generator. Your data doesn't appear to be anything near as bad as this. Bad pseudorandom number generators will often give global structure; local structure will appear even in truly random situations. – arjafi Jun 30 '12 at 19:58
possible duplicate of How to compare randomness of two sets of data? – Listing Jun 30 '12 at 20:08
those points arent on a line – TROLLKILLER Jun 30 '12 at 20:11

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