If correlation doesn’t imply causation, then what does? (2012)

> Overall: Democrat (152/248, 61 percent), Republican (138/172, 80 percent)

This example is often cited for Simpson's Paradox, but because the South headcount for Republicans (10) and Democrats (94) had such a wide disparity and the Republican headcount was so low, it always seemed more like twisting the data to fit a political narrative than to make a statistical argument. Like other statistical methods, there should be a minimum population size for the paradox to be meaningful.

Has larger sample sizes, all would be valid for statistical analysis. Although this paradox isn't really dependent on a sample size, it is not so much a statistical test. It's a trend that can be observed when looking at sub groups compared to overall averages.

In the above example open surgery was better for both small and large kidney stones and is a prime example of Simpson's paradox. However, one should note that Simpson's paradox can also be subject to further confounding variables. Perhaps patients that were suitable for open surgery were in better medical condition and therefore the open surgery sample was non-random and caused the higher success rate.

The presence of Simpson's paradox, similar to correlation, also does not imply causation.

The third meaning is the one that "correlation does not imply causation" refers to. There is no other meaning for that word in that context.