# p-value definition and inadequacy of tail events [duplicate]

Wikipedia defines p-values as

(The probability of obtaining a test statistic) equal to or more extreme than what was actually observed

For starters, usage of extreme there is subjective. To make matters more concrete,

$$Pr ( X ≥ x | H )$$ for right tail event
$$Pr ( X ≤ x | H )$$ for left tail event
$$2 min \{ Pr ( X ≤ x | H ) , Pr ( X ≥ x | H ) \}$$ for double tail event.

This diagram explains it for a right tailed event:

Now these definitions work well for normal distribution, t distribution, F distribution, and chi-square distributions.

But consider a bimodal distribution.

     ⏜       ⏜
/   \    /   \
/     \_/     |\
/              | \
x


For the observation x, won't we also have to consider the middle region as "extreme values" since their probability is low? So our extreme region will look something like the shaded region in:

     ⏜       ⏜
/   \    /   \
/|    |\_/|   |\
/.|    |...|   |.\
x


In this case we can't just consider tailed events. What exactly is the definition of "extreme values" then?

Any of these links can be of any help? Maybe you need to look for something other than p-value. Maybe even the fact that having bimodal distribution can be a sign of two overlapping distribution and maybe a t-test should be performed.

• @PeeyushKushwaha don't forget! you can accept the answer if it helped :) Commented Apr 21, 2019 at 14:24