Determine X² Test Statistic Calculator

Calculator Form

Test type

Alpha level

Estimated parameters

Degrees of freedom override

Options

Scale expected list to observed total

Use Yates correction for 2 by 2 tables

Ratios or probabilities

Observed counts

Use a list for goodness of fit. Use rows for table tests.

Expected counts

Leave blank to build expected counts from ratios or a table.

Input examples

Goodness list: 30, 25, 45
Expected list: 25, 25, 50
Table input:
20, 30, 25
15, 35, 40

Example Data Table

Category	Observed	Expected ratio	Expected count	X² contribution
A	30	1	25	1.000
B	25	1	25	0.000
C	45	2	50	0.500
Total	100	4	100	1.500

Formula Used

The main formula is X² = Σ((O - E)² / E). O is the observed count. E is the expected count.

For an independence or homogeneity table, E = row total × column total / grand total.

For goodness of fit, df = categories - 1 - estimated parameters. For a table, df = (rows - 1)(columns - 1).

The p value is the right tail probability from the chi square distribution. The calculator also returns a critical value at the selected alpha.

How to Use This Calculator

Select the test type. Use goodness of fit for one categorical variable.
Enter observed counts. Separate list values with commas, spaces, or new lines.
For table tests, enter each row on a separate line.
Add expected counts, or leave them blank when the calculator can create them.
Set alpha, estimated parameters, or a degrees of freedom override when needed.
Press Submit. The result appears above the form and below the header.
Use the CSV or PDF button to save the result.

Understanding the X² Test Statistic

The X² test statistic measures how far observed counts move from expected counts. It is used with count data, not averages. A small value suggests the counts match the model closely. A large value suggests the gap is too large to ignore. This calculator supports common reporting needs. It can compare observed and expected lists. It can also build expected counts from a contingency table.

Why This Calculator Helps

Manual work is simple for one category. It becomes slower when many cells are used. Each cell needs a contribution. Each contribution must be added. A table test also needs row totals, column totals, and degrees of freedom. This page keeps those steps together. It also displays residuals. Residuals show which cells create the biggest difference. That helps explain the result, not just report it.

Goodness Of Fit Use

A goodness of fit test checks one categorical variable. You enter observed counts for each group. Expected counts may come from a fixed theory, a market share, a genetic ratio, or equal distribution. If expected counts are not known, enter weights or probabilities. The calculator scales them to the observed total. You may also subtract estimated parameters from the degrees of freedom.

Independence And Homogeneity Use

A table test checks two categorical dimensions. Independence asks whether two variables are associated. Homogeneity asks whether several groups share the same distribution. The arithmetic is the same. Expected counts are found from row totals and column totals. Each expected count equals row total times column total, divided by the grand total.

Reading The Result

The statistic is compared with a chi square distribution. The p value shows the chance of seeing a statistic this large, assuming the null model is true. When the p value is below alpha, the result is statistically significant. You should still check sample design, expected counts, and practical size. Cramer's V or Cohen's w gives extra context. A clear report should include the test type, statistic, degrees of freedom, p value, alpha, decision, and any important cell residuals. Always avoid using percentages alone. The test needs raw counts. For small samples, combine sparse categories when possible. Use exact methods when assumptions fail badly in practice.

FAQs

What does the X² statistic show?

It shows the total gap between observed and expected counts. Larger values mean the observed pattern is farther from the expected model.

Can I use percentages as observed values?

No. Use raw counts. Percentages hide sample size, and the chi square test needs frequency counts for correct inference.

When should I use goodness of fit?

Use it when one categorical variable is compared with a known or assumed distribution. Examples include ratios, market shares, and equal category expectations.

When should I use independence?

Use independence when two categorical variables come from one sample. The test checks whether the variables appear related.

What is a standardized residual?

It is the cell difference divided by the square root of expected count. Large residuals show cells that influence the result most.

What does a small p value mean?

A small p value means the observed pattern would be unusual under the null model. It often supports rejecting the null hypothesis.

Why are expected counts important?

Expected counts define the comparison model. Very small expected counts can make the approximation weak, so results need caution.

What does Yates correction do?

Yates correction adjusts 2 by 2 table contributions. It can reduce the statistic when sample sizes are modest.