# Two-tailed hypothesis test; Why do we multiply p-value by two?

I understand that in a two-tailed hypothesis test, we must multiply the p-value by two.

i.e. if z=1.95 and it's a one-tailed hypothesis test, our p-value is 0.0256. But, if it's a two-tailed hypothesis test and z=1.95, we must multiply the p-value of 0.0256 by two. Hence, the correct p-value is 0.0512 for the two-tailed hypothesis test.

I can draw it out on the standard normal curve and I understand that we must multiply the p-value by two. But, my question is why we have to multiply by two. What is the conceptual idea behind it?

• Because you want both sides. Think about doing a $<$ and $>$ in one. with $p$ for each Oct 23, 2015 at 13:28
• Consider what the p value corresponds to: areas under a PDF. When it's 2-tailed, you are taking both the area user the curve to the right of a certain point, and the same area on the other side of the PDF. By symmetry these are the same so you just times 2, for twice the area. May 22, 2016 at 17:36

$2$ because it is two-tailed.
The test is the probability of seeing that value or something more extreme if the null hypothesis is true. $2$ is more extreme than $1.95$; $-3$ is also more extreme.
So you want $\Pr(Z \ge 1.95)+\Pr(Z \le -1.95)$ which, by the symmetry of the normal distribution is equal to $2\times \Pr(Z \ge 1.95)$