Why Use the Smaller Tail?

Why define the two-sided p-value as 2 × min(p_left, p_right) instead of using max?

First, don’t worry about this for exams or daily use. Usually, p_left and p_right are fairly similar, say 0.009 and 0.012. So min and max often lead to the same conclusion.

But let’s dig a little deeper. What if p_left = 0.001 and p_right = 0.026? Then:

• 2 × min = 0.002 → reject the assumption.
• 2 × max = 0.052 → just fail to reject.

Why do we use the min? The symbol ⇔ means “is equivalent to.”

• 2 × min < 0.05 ⇔ p_left < 0.025 OR p_right < 0.025 ⇔ at least one direction is sufficiently extreme.
• 2 × max < 0.05 ⇔ p_left < 0.025 AND p_right < 0.025 ⇔ both directions must be sufficiently extreme.

The second rule asks for more than we want: one sufficiently extreme direction is already enough evidence against our assumption. A silly analogy:

Assumption: this person is innocent.

Evidence 1: He is holding a knife covered with the victim’s blood.

Evidence 2: He owns an old rusty gun that stopped working years ago.

Even if Evidence 2 is only borderline suspicious, Evidence 1 alone may already give us a strong reason to question the assumption. We don’t need both pieces of evidence to be equally suspicious!

This analogy is just about OR versus AND. In our symmetric permutation distribution, the two tails are equal in theory; the unequal values above are only a thought experiment to make the logic easier to see.