P Value Calculator

What the p-value calculator does

A p-value is the probability of getting a test statistic at least as extreme as yours if nothing but chance were at work. This calculator turns a test statistic into that probability, and it handles the four you meet most often: a z-score, a t-score, a chi-square score, and an F-ratio.

It is the last step of most hypothesis tests, the point where a computed statistic becomes a decision about significance. Below is how it works across all four tests.

How to use it

1. Choose the statistic you have: z-score, t-score, chi-square, or F-ratio.
2. Enter the value, and the degrees of freedom the test needs. A t-score and a chi-square need one; an F-ratio needs two, a numerator and a denominator.
3. Press Calculate for the p-value, or Reset to clear it.

How the p-value is worked out

Each test statistic has a distribution that describes how it behaves when only chance is operating. The p-value is the area under that distribution out in the tail beyond your statistic, which is the probability of landing that far out or further by luck alone.

So the calculator places your statistic on the right curve, the normal curve for a z-score, the t-distribution for a t-score, and so on, and measures the tail area past it. A statistic deep in the tail leaves a small area, and so a small p-value, which is the signal that the result is hard to write off as chance.

The four test statistics it handles

A z-score uses the standard normal curve and needs no degrees of freedom. A t-score uses the t-distribution and needs its degrees of freedom, which shape how heavy the tails are. These two come from tests about means.

A chi-square score uses the chi-square distribution with its degrees of freedom, and turns up in tests of independence and goodness of fit. An F-ratio uses the F-distribution, which needs two separate degrees of freedom, a numerator and a denominator, and appears in analysis of variance and in comparing two variances. Feeding in the right degrees of freedom is what makes each p-value correct.

Left, right, and two-tailed

The calculator reports the tail areas so you can pick the one your test calls for. The left-tailed p-value is the area below your statistic, and the right-tailed is the area above it. For a z-score or a t-score it also gives the two-tailed value, which counts both tails, for a test that only asks whether there is a difference rather than in which direction.

Chi-square and F tests are usually read as right-tailed, since those statistics measure a size of departure that only gets larger, so a two-tailed value is not reported for them. As always, the tail you use should be decided by your hypothesis before you see the data, not chosen afterward to suit the result.

A worked example

Suppose a one-sample t-test gave a t-score of 2.5 with 24 degrees of freedom.

Choosing the t-score option and entering those, the area in one tail beyond 2.5 is about 0.01, so the two-tailed p-value is about 0.02. Being below the usual 0.05 threshold, this would be called statistically significant, meaning a difference this large would be unlikely if only chance were at work.

What the p-value does and does not say

A small p-value says your result would be surprising under pure chance, which is treated as evidence against the idea that nothing is going on. The usual line is 0.05, though that is a shared convention rather than a hard boundary.

What it does not say is just as important. The p-value is not the probability that your hypothesis is true, and it is not a measure of how big or important the effect is. A very small p-value can come from a tiny effect measured in a huge sample. To see how large the effect actually is, you need a separate measure like an effect size, which answers a different question than significance.

Questions people ask

What is a p-value?

The probability of a test statistic at least as extreme as yours if only chance were operating. A small p-value suggests the result is unlikely to be chance alone.

Why does it need degrees of freedom?

Because the t, chi-square, and F distributions change shape with their degrees of freedom. The right value is needed to measure the correct tail area. A z-score does not need any.

Which tail should I use?

Match it to your hypothesis. Use one tail for a directional question and two tails for a difference in either direction. Chi-square and F tests are normally right-tailed.

What p-value is significant?

By common convention, below 0.05, though this is an agreed threshold rather than a law. The right cut-off depends on the field and the stakes.

References

A quick note on where the methods here come from. The p-value and the distributions of the z, t, chi-square, and F statistics are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering hypothesis testing and p-values.

1. NIST/SEMATECH e-Handbook of Statistical Methods (test statistics and their distributions). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (hypothesis testing, the t, chi-square, and F distributions). https://openstax.org/details/books/introductory-statistics-2e