Sanity check: how low can a p-value go? I am using a ranksum test to compare the median of two samples ($n=120000$) and have found that they are significantly different with: p = 1.12E-207. Should I be suspicious of such a small $p$-value or should I attribute it to the high statistical power associated with having a very large sample? Is there any such thing as a suspiciously low $p$-value?
 A: P-values on standard computers (using IEEE double precision floats) can get as low as approximately $10^{-303}$.  These can be legitimately correct calculations when effect sizes are large and/or standard errors are low.  Your value, if computed with a T or normal distribution, corresponds to an effect size of about 31 standard errors.  Remembering that standard errors usually scale with the reciprocal square root of $n$, that reflects a difference of less than 0.09 standard deviations (assuming all samples are independent).  In most applications, there would be nothing suspicious or unusual about such a difference.
Interpreting such p-values is another matter.  Viewing a number as small as $10^{-207}$ or even $10^{-10}$ as a probability is exceeding the bounds of reason, given all the ways in which reality is likely to deviate from the probability model that underpins this p-value calculation.  A good choice is to report the p-value as being less than the smallest threshold you feel the model can reasonably support: often between $0.01$ and $0.0001$. 
A: A p-value can achieve a value of 0.
Suppose I am testing the composite hypothesis about the value of a range of a uniform 0, $\theta$ random variable. If I set $\mathcal{H}_0: \theta = 1$ and sample a value of $X=1.1$, you see it's impossible to observe such a value or higher under the null hypothesis. The p-value is 0.
A: There is nothing suspicious -- extremely low p-values like yours are pretty common when sample sizes are large (as yours is for comparing medians). As whuber mentioned, normally such p-values are reported as being less than some threshold (e.g. <0.001).
One thing to be careful about is that p-values only tells you whether the difference in median is is statistically significant. Whether the difference is significant enough in magnitude is something you will have to decide: e.g. for large sample sets, extremely small differences in means/medians can be statistically significant, but it might not mean very much.
