Chi-Square Calculator

Chi-square formula

The chi-square statistic compares observed counts with the counts expected under the null hypothesis:

chi-square = sum((O - E)^2 / E)

For a goodness-of-fit test with k categories, the basic degrees of freedom are k - 1. If parameters were estimated from the same data to build the expected counts, subtract those estimated parameters as well. For a test of independence in an r x c table, the degrees of freedom are (r - 1)(c - 1).

Worked example

Suppose a five-category process should be evenly distributed and you observe:

The chi-square statistic is 1.3 with 4 degrees of freedom. The right-tail p-value is about 0.861, so this sample is not unusual enough to reject the equal-distribution null at the common 0.05 level.

Goodness of fit vs independence

Use goodness of fit when you have one categorical variable and a known or assumed distribution. Examples include testing whether dice rolls are fair, whether support tickets arrive equally across weekdays, or whether observed survey answers match a planned split.

Use the independence table when every count sits at the intersection of two categorical variables. Examples include device type by conversion outcome, department by response choice, or treatment group by side-effect category. The calculator computes each expected cell as:

expected = row total x column total / grand total

Assumption checks

Chi-square tests use an approximation that improves as expected counts get larger. A common classroom rule is to watch for expected counts below 5. If many cells are small, combine sensible categories, collect more data, or use an exact test where appropriate.