P-Value Calculator

What a p-value is, and what it is not

A p-value is the probability of seeing a result at least as extreme as yours if the null hypothesis were true. That conditional is the whole thing. It is not the probability that the null hypothesis is true, it is not the probability your result is a fluke, and it says nothing about how large or important the effect is.

A tiny p-value on a huge sample can accompany an effect too small to matter. A large p-value on a small sample often means you simply did not collect enough data to tell. Neither is settled by the p-value alone — report the effect size and the confidence interval next to it.

Choosing the tail before you see the data

Two-tailed asks whether there is a difference at all. One-tailed asks whether there is a difference in one specific direction, and in exchange gives you a p-value half the size. That discount is only legitimate if the direction was committed to in advance. Picking the tail after seeing which way the data went doubles your false-positive rate, and it is one of the most common forms of accidental p-hacking.

z, t or chi-square

Use z when the population standard deviation is known or the sample is large. Use t when you estimated the standard deviation from the sample itself — its heavier tails are the price of that uncertainty, which is why the same statistic gives a larger p-value under t. The two converge as degrees of freedom rise; past about 200 they are practically identical.

Use chi-square for goodness-of-fit and independence tests on counts. It is naturally one-tailed: only the right tail indicates a poor fit, because the statistic can never be negative.

Frequently Asked Questions