x2 test calculator

What is an X² (Chi-Square) test?

The X² test, usually written as χ² (chi-square), is a statistical test that compares what you observed in your data to what you would expect under a specific hypothesis. It helps answer questions like: “Do these counts look random, or is there a meaningful pattern?”

This page focuses on a goodness-of-fit version of the chi-square test, where you compare one set of observed category counts against expected category counts.

How this calculator works

The calculator uses the standard formula:

X² = Σ ((Observed - Expected)² / Expected)

It then computes:

• Degrees of freedom (df) = number of categories - 1
• p-value from the chi-square distribution
• Decision at your chosen significance level α

When to use this test

Use it when:

• Your data are counts/frequencies (not means or percentages by themselves).
• Each observation belongs to exactly one category.
• You want to test if observed category distribution matches an expected one.

Do not use it when:

• Your data are continuous measurements (e.g., height, weight, time).
• Expected counts are too small in many categories (common rule: expected should usually be at least 5).
• Observations are not independent.

Step-by-step example

Suppose you flip a coin 100 times and observe 58 heads and 42 tails. Under a fair-coin hypothesis, expected counts are 50 and 50.

• Observed: 58, 42
• Expected: 50, 50
• X² = (58-50)²/50 + (42-50)²/50 = 2.56
• df = 2 - 1 = 1
• p-value is about 0.11

Since p > 0.05, you would usually fail to reject the fair-coin assumption.

Interpreting your result

A small p-value means your observed counts are unlikely under the expected pattern. In that case, you reject the null hypothesis. A larger p-value means your data are reasonably consistent with the expected distribution.

Remember: “not significant” does not prove the null hypothesis true. It only means the data did not provide strong enough evidence against it.

Practical tips for better analysis

• Combine sparse categories if expected counts are very low.
• Use a clear hypothesis before running the test.
• Report X², df, p-value, and sample size together.
• Context matters: statistical significance is not the same as practical importance.

Frequently asked questions

Why does the calculator rescale expected counts?

For a goodness-of-fit test, expected and observed totals should match. If they do not, this calculator rescales expected values proportionally to the observed total so the test remains valid.

What if I need a chi-square test of independence?

That test uses a contingency table (rows × columns) and computes expected values from row/column totals. This page is specifically for one-sample goodness-of-fit.

Can I use percentages?

Yes, but convert percentages into expected counts using your total sample size, or enter a proportional set that can be scaled to the observed total.