Chi Square Calculator

A chi-square goodness-of-fit test checks whether observed counts across a set of categories match what you’d expect under some hypothesis. This chi square calculator takes your observed and expected counts and returns the chi-square statistic, the degrees of freedom, and the upper-tail p-value.

How to use this calculator

1. Enter your observed counts as a comma- or space-separated list, one number per category (for example, 20, 30).
2. Enter the matching expected counts in the same order, one per category (for example, 25, 25).
3. Read the chi-square statistic, degrees of freedom, and p-value the calculator returns.

The two lists must have the same length — each position pairs an observed count with the expected count for that same category.

Worked example

Take observed counts 20, 30 against expected counts 25, 25 (the calculator’s defaults).

For each category, find the squared difference between observed and expected, divided by expected:

category 1: (20 − 25)² / 25 = 25 / 25 = 1
category 2: (30 − 25)² / 25 = 25 / 25 = 1

Sum the per-category contributions to get the chi-square statistic:

χ² = Σ (observed − expected)² / expected
χ² = 1 + 1 = 2.00

Degrees of freedom is the number of categories minus one:

df = 2 − 1 = 1

Looking up chi-square = 2.00 with df = 1 in the chi-square distribution gives an upper-tail p-value of about 0.157. At a typical 0.05 significance level, that p-value is too large to reject the hypothesis that the observed counts match the expected ones. For the full method and reference tables, see the NIST/SEMATECH e-Handbook section on hypothesis testing.

Frequently asked questions

What does the chi square statistic measure?

The chi square statistic sums, across all categories, the squared gap between observed and expected counts scaled by the expected count. A larger statistic means the observed data deviates more from what was expected.

When should I use a chi square test?

Use a chi square test when you have counts (not means) sorted into categories and want to check goodness of fit — whether the observed distribution matches a hypothesized or expected distribution.

How do I interpret the p-value?

A small p-value (typically below 0.05) means the observed counts are unlikely under the expected distribution, so you’d reject the hypothesis that they match. A larger p-value, like the ≈ 0.157 in the worked example, means the observed counts are reasonably consistent with what was expected.

Why is degrees of freedom categories minus one?

One degree of freedom is used up because the observed counts are constrained to sum to a fixed total, so only k − 1 of the k category counts are free to vary once that total is fixed.