# Best way to report mean±SD of p-values (all values are positive and SD is larger than the mean)

I have 50 p-values. I want to show the mean of these 50 and the standard deviation:

0.06 ± 0.19


The standard deviation is quite large, because although almost all of the time, the p-value is close to 0, occasionally, there is a large value, close to 1.

0.06 ± 0.19


doesn't seem quite right though, because it seems to imply that the p-value could drop below zero. Is there a better way to state the mean and standard deviation in this situation?

Example of p-values:

[0.00001,0.03,0.0007,0.1,0.00005,0.78 ...]


The p-values come from testing a correlation between 2 variables in a simulation I have written. There are a few elements of randomness and so even if the variables are actually correlated, the results can sometimes show no correlation and hence I get a p-value close to 1.

Due to this random nature, I run the simulation 50 times and then I know I am getting a more reliable p-value. I then want to say something about the spread, which is where this question came from.

• Why are you trying to construct confidence intervals on $p$-values? $p$-values are known and don't have to be estimated so confidence intervals don't apply. Are you instead trying to estimate unknown probabilities? – dsaxton Mar 2 '16 at 15:17
• @dsaxton: Where does OP talk about confidence intervals? Perhaps they simply want to describe a bunch of p-values from various experiments; providing mean and SD could seem as a reasonable idea. – amoeba says Reinstate Monica Mar 2 '16 at 15:45
• Perhaps a mean and standard deviation of the logarithms of p-values? – amoeba says Reinstate Monica Mar 2 '16 at 16:04
• You have a bunch of numbers in [0, 1] and want to describe them. How about a plot of some sort, e.g. histogram or kernel density estimate? Or, if you prefer giving numbers, how about some quantiles? – Adrian Mar 2 '16 at 16:34
• It would help if you could provide a bit more information on where these "p-values" come from. If these are p-values of statistical significance tests, then averaging them doesn't serve much of a useful purpose that I can see off-hand. If they are instead ratios, say, of successes to total attempts at some task, then there may be better ways to summarize and display results. – EdM Mar 2 '16 at 16:54

If your 50 $p$-values are $p=(p_1,p_2,...,p_{50})$, is sd$(p)=0.19$? If so, this is incorrect. As you are reporting the mean of a sample of $p$-values you need to report the SD of the mean. Thus you should report $$0.06\pm \frac{0.19}{\sqrt{50}}=0.06\pm 0.027$$
• One cannot do mean $\pm$ SD. That is simply incorrect, since that supports that idea that there error in the estimate for the mean is SD, which it is not. – Greenparker Mar 2 '16 at 21:52