This might be a someone basic question, but it is one I still do not really grasp satisfacotrily.

So I have been told, and I do not disagree, that when the H0 is true the P values will be uniformly distributed, and that this can be used to diagnose potential violations of the assumptions in the test.

If we use the normal distribution as an example, I think that if you have a one sided test where you want to see how extreme your test statistic X is, then you can take 1 - Pr(X <= a), (a is the significance level) - which is the cumulative distribution function. And since this is a function that is monotonically increasing, it kind of maps values uniquely, meaning that for no 2 different values of X will you get the same function value. Is that so?

But then again will not most X values center around the mean, thereby getting P-values around 0.5?

And how would the distribution and the cumulative distribution function look when the P-values are not uniform? I cannot really imagne this.

I apologise if this post is bit rambling, I hope you can provide some clarity!