P-Value Calculator

A p-value is the probability of the test statistic being at least as extreme as the one you observed, given that the null hypothesis is true. It is a single number that summarizes how surprising your result would be if the null hypothesis were true. ¹

A small p-value is a sign that the null hypothesis is false. That is the basic logic behind every test this page runs, whatever statistic you started from. ¹

You reject the null hypothesis when the test statistic passes a critical value. Or, in the same way, you reject it when the p-value is smaller than your chosen significance level. Both routes lead to the same conclusion. ¹

Choosing a significance level of 0.05 has a cost. The null hypothesis gets rejected 5% of the time, even when it is actually true. That is the price of picking that particular cutoff. ¹

The choice of significance level is somewhat arbitrary. In practice, 0.1, 0.05, and 0.01 are common choices. Because of that, this page also checks something else. It checks whether your conclusion would flip at one of those other common levels. It flags the result as borderline when it would. ¹

A Z score compares your statistic to the standard normal curve. Tables for that curve list the area under it. This area runs from zero up to a given value. ³

Common Z cut-offs include 1.960 for a two-tailed test and 1.645 for a one-tailed test. Both are tied to a significance level of 0.05. More extreme cut-offs, such as 2.576 and 3.090, go with smaller levels, like 0.01 and 0.001. ³

A T score comes from Student's t distribution. This distribution is symmetric. So its upper and lower cut-offs mirror each other. The most commonly used significance level is 0.05. For a two-sided t-test, you compare against the 0.975 point of the curve. This is because 1 minus 0.05 divided by 2 equals 0.975. ⁴

A chi-square statistic comes from squaring and summing independent variables. Each of these variables follows a standard normal distribution. It only takes values of zero or more. Its spread grows with its degrees of freedom. ⁷

The chi-square distribution is not symmetric. So separate tables exist for its upper and lower tails. That is why this page looks up different values for a lower-tail test, an upper-tail test, and a two-tailed test. It does this on the same chi-square statistic. ⁶

An F ratio is the ratio of two chi-square distributions. Each one is first divided by its own degrees of freedom, numerator and denominator. That is why this page asks for two separate degrees-of-freedom values for an F test. ⁸

F tables such as this one cover one-sided tests. They commonly use the 0.05, 0.10, and 0.01 levels. The F distribution is not symmetric. So a proper two-sided test needs its own set of tables. ⁵

This page produces the same figures as printed tables for the normal, t, chi-square, and F distributions. These figures match what general-purpose statistical software returns for these tests. ¹⁰

A two-tailed test guards against your result being either too small or too large. Pick it when either direction would matter to you. Pick a lower or upper tail only when you care about one direction. ²

It is good practice to decide in advance how small a p-value has to be. Only then will you reject the null hypothesis. Set your significance level before you look at the result, not after. ¹

Pearson's r measures the linear correlation between two sets of data. It always falls between -1 and 1. Unlike covariance, r has no units. ⁹

Take pairs drawn from an uncorrelated bivariate normal distribution. Here, r converts into a t-statistic. The degrees of freedom equal your sample size minus 2. The formula is t equal to r times the square root of that degrees of freedom over 1 minus r squared. Once you have that t-statistic, enter it above as a t score with the matching degrees of freedom. This gives you its p-value. ⁹

Lowering your significance level reduces the risk of a false rejection. But it raises a different risk, called beta. There is a trade-off between the two. So an especially small alpha is not automatically the safer choice. ²

Critical values depend on which test statistic you used and which significance level you picked. ¹

You find your p-value by locating where your statistic falls on the matching distribution's table and reading off the area beyond that point ¹. For a z score, the standard normal table gives the area between zero and your value, and you add 0.5 to cover the half of the curve below zero; a z of 1.53 reads as 0.43699 in the table, so the area works out to 0.93699 ³. Rejecting the null hypothesis whenever the p-value falls under your significance level gives the same call as checking whether the test statistic exceeds the matching critical value for that level ¹.

Some things a calculator page like this might cover are not addressed by the material behind this page. What the sources do say that touches on this: the risk of rejecting a true null hypothesis, alpha, is chosen in advance and is typically 0.10, 0.05, or 0.01, and a related risk, beta, rises as alpha falls ².