Chi-Square Test Calculator

What is the chi-square test used for?

A goodness-of-fit test compares observed category counts with the counts a hypothesis expects. A test of independence asks whether two categorical variables are related. This tool computes the statistic for a three-category goodness-of-fit comparison only.

How are the degrees of freedom set?

The tool always uses three categories, so df is fixed at 2 (categories minus 1) and the Degrees of freedom row always shows 2. There is no way to change it, so a table with another shape needs a different tool.

How is the p-value calculated, and is it exact?

For 2 degrees of freedom the upper-tail probability has a closed form, p = e^(−χ²/2), and the tool uses it, so the value is exact apart from display rounding to four decimals. For χ² = 5.00 it returns 0.0821, and below 0.0001 it prints “< 0.0001”. An earlier normal approximation returned 0.0800 for that case and 0.9962 instead of 1.0000 for a perfect match.

How is the decision line chosen?

It compares the p-value with the significance level in the last box. If p is below α the line reads “Reject H₀ at α=…”, otherwise “Fail to reject H₀ at α=…”. An α of 0, an emptied box, or any value of 1 or more returns “Significance level must be between 0 and 1”.

What inputs are rejected?

Any expected count at 0 or below returns “All expected frequencies must be positive”, because it would divide by zero, and an emptied expected box counts as 0. A negative observed count returns “Observed frequencies cannot be negative”. An observed count of 0 is allowed and contributes its full expected value to χ².

Should the observed and expected totals match?

They should, but the tool does not check. Entering 70, 20, 10 against expected 40, 40, 20 works because both total 100. Mismatched totals still produce a number, but it no longer describes a proper goodness-of-fit test. Enter counts, not percentages, since percentages change the statistic.

Can I use it for a contingency table?

Only to get the statistic. The tool does not build expected counts from row and column totals, and its p-value assumes 2 degrees of freedom, whereas a 2×2 table has 1 and a 3×3 table has 4. Compare your own total with the critical values on this page for the correct df.

What sample size does the test need?

The usual guideline is an expected count of at least 5 in each category. The tool does not enforce it, so a result with small expected counts still prints. A significant result also shows that a difference exists, not how large it is.