Chi-Square To P-Value Calculator

What the chi-square to p-value calculator does

Once you have a chi-square statistic, the p-value tells you whether it is large enough to be meaningful. This calculator converts a chi-square value and its degrees of freedom into that p-value, using the chi-square distribution, so you can judge significance without reading it off a printed table.

It is the step that turns a chi-square total from a goodness-of-fit or independence test into a decision. Below is how it works and the details that make it correct.

How to use it

1. Enter the chi-square statistic from your test.
2. Enter the degrees of freedom.
3. Press Calculate for the p-value, or Reset to clear it.

How the p-value is worked out

The chi-square distribution describes how a chi-square statistic behaves when the observed counts differ from the expected ones by chance alone. The p-value is the area under that distribution beyond your statistic, out in the upper tail.

So the calculator places your chi-square value on the distribution for your degrees of freedom and measures how much area lies above it. A large chi-square sits far out in the tail with little area beyond it, giving a small p-value, while a small chi-square leaves a lot of area, and a large p-value.

Why the chi-square test is right-tailed

Unlike a z-test or a t-test, a chi-square test looks at only one tail, the upper one. The reason is in how the statistic is built. Every category adds a squared, and so positive, contribution, so the chi-square value only grows as observed and expected counts drift apart. It can never go negative.

That means a big chi-square is the only kind of surprising, since it signals a large departure from what was expected. A chi-square near zero means the counts matched expectations almost perfectly, which is not evidence against anything. So all the interesting probability sits in the upper tail, which is why this calculator reports the right-tailed p-value.

Getting the degrees of freedom right

The chi-square distribution changes shape with its degrees of freedom, so the same chi-square value gives different p-values depending on them. Getting them right is essential. For a goodness-of-fit test, the degrees of freedom are usually the number of categories minus one. For a test of independence on a table, they are the number of rows minus one times the number of columns minus one.

If you are unsure of the right figure for your test, our degrees of freedom calculator works it out for the common cases, so you can be confident the p-value here rests on the correct value.

A worked example

Suppose a test gives a chi-square value of 3.84 with 1 degree of freedom.

The area beyond 3.84 in the upper tail of the chi-square distribution with 1 degree of freedom is about 0.05. So this result sits right at the usual significance threshold, which is no coincidence, since 3.84 is the well-known chi-square cut-off for significance at the 0.05 level with one degree of freedom.

Entering your values

Enter the chi-square statistic, which cannot be negative, and the degrees of freedom, which must be at least one. The p-value is returned directly. A p-value below your chosen level, commonly 0.05, means the observed counts depart from the expected ones by more than chance readily explains.

Questions people ask

What does this calculator find?

The p-value for a chi-square statistic, given its degrees of freedom. It is the probability of a chi-square that large or larger if only chance were at work.

Why only the right tail?

Because the chi-square statistic is a sum of squared terms, it only grows as counts depart from expectations. A large value is the only surprising outcome, so the test uses the upper tail.

Why do the degrees of freedom matter?

The chi-square distribution changes shape with them, so the same statistic gives different p-values for different degrees of freedom. The right value is needed for the right answer.

What p-value is significant?

By common convention, below 0.05, though this is an agreed threshold rather than a hard rule. The right cut-off depends on the context.

References

A quick note on where the methods here come from. The chi-square distribution and its tail probabilities 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 the chi-square distribution.

1. NIST/SEMATECH e-Handbook of Statistical Methods (the chi-square distribution). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (the chi-square distribution and its tests). https://openstax.org/details/books/introductory-statistics-2e