Chi Square Test Calculator

Q: What is the chi-square test of independence?

The chi-square test of independence determines whether there is a statistically significant association between two categorical variables. It compares observed frequencies (your actual data) to expected frequencies (what you would expect if the variables were independent). A large chi-square statistic indicates that observed frequencies differ substantially from expected frequencies, suggesting the variables are associated.

Q: What are the assumptions of the chi-square test?

Key assumptions: (1) Data are frequencies/counts, not percentages or means. (2) Categories are mutually exclusive — each observation falls in exactly one cell. (3) Observations are independent. (4) Expected frequency in each cell should be at least 5 (Cochran's rule). If expected values are below 5, consider Fisher's exact test or combining categories.

Q: When should I use a t-test versus a z-test?

Use a z-test when the population standard deviation is known and the sample size is large (n > 30). Use a t-test when the population SD is unknown and you estimate it from the sample. For small samples (n < 30), the t-distribution accounts for the extra uncertainty in estimating SD.

Q: What is a chi-square test used for?

The chi-square test compares observed frequencies to expected frequencies in categorical data. A goodness-of-fit test checks if data follows an expected distribution. A test of independence checks if two categorical variables are related. The test statistic increases as observed and expected frequencies diverge.

Calculate chi square test instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.

Reviewed by Daniel Agrici, Founder & Lead Developer

Formula

Chi-Square = Sum[(Observed - Expected)^2 / Expected]

The chi-square statistic sums the squared differences between observed and expected frequencies, each divided by the expected frequency. Degrees of freedom = (rows - 1) x (columns - 1). Compare the statistic to the chi-square distribution to find the p-value.

Worked Examples

Example 1: Gender and Product Preference

Problem:Survey: 50 males prefer Product A, 30 prefer B. 20 females prefer A, 40 prefer B. Is there an association between gender and preference?

Solution:Observed: [[50,30],[20,40]], Grand total = 140\nExpected: [[35,35],[35,35]] (if independent, no difference)\nChi-square = (50-35)^2/35 + (30-35)^2/35 + (20-35)^2/35 + (40-35)^2/35 = 6.43 + 0.71 + 6.43 + 0.71 = 14.29\ndf = 1, p < 0.001

Result:Chi-square = 14.29, df = 1, p < 0.001 — Significant association

Frequently Asked Questions

What is the chi-square test of independence?

The chi-square test of independence determines whether there is a statistically significant association between two categorical variables. It compares observed frequencies (your actual data) to expected frequencies (what you would expect if the variables were independent). A large chi-square statistic indicates that observed frequencies differ substantially from expected frequencies, suggesting the variables are associated.

What are the assumptions of the chi-square test?

Key assumptions: (1) Data are frequencies/counts, not percentages or means. (2) Categories are mutually exclusive — each observation falls in exactly one cell. (3) Observations are independent. (4) Expected frequency in each cell should be at least 5 (Cochran's rule). If expected values are below 5, consider Fisher's exact test or combining categories.

When should I use a t-test versus a z-test?

Use a z-test when the population standard deviation is known and the sample size is large (n > 30). Use a t-test when the population SD is unknown and you estimate it from the sample. For small samples (n < 30), the t-distribution accounts for the extra uncertainty in estimating SD.

What is a chi-square test used for?

The chi-square test compares observed frequencies to expected frequencies in categorical data. A goodness-of-fit test checks if data follows an expected distribution. A test of independence checks if two categorical variables are related. The test statistic increases as observed and expected frequencies diverge.

References

Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy