Chi-Square (χ²) Distribution Table

Interactive Chi-Square Table Calculator

Chi-Square Distribution Table

The chi-square table provides critical values for the χ² distribution, used in hypothesis testing for categorical data and variance.

How to Use This Table

1. Determine degrees of freedom (df) based on your test type
2. Choose your significance level (α)
3. Find the critical value at the intersection

Critical Values (Upper Tail)

Values where P(χ² > critical value) = α

Common Chi-Square Tests

Goodness-of-Fit Test

df = k - 1 where k = number of categories

Test of Independence

df = (r - 1)(c - 1) where r = rows, c = columns

Quick Reference: Common Critical Values

Most Used Values (α = 0.05)

Chi-Square Test Formula

$\chi^2 = \sum \frac{(O - E)^2}{E}$

Where:

• O = Observed frequency
• E = Expected frequency

Decision Rule

• If χ² ≥ critical value → Reject H₀
• If χ² < critical value → Fail to reject H₀

Example: Test of Independence

Problem: Testing if gender and product preference are independent using a 2×3 table with α = 0.05.

Solution:

1. df = (2-1)(3-1) = 2
2. Look up df=2, α=0.05: Critical value = 5.991
3. If calculated χ² > 5.991, reject independence

Assumptions

For valid chi-square tests:

1. ✓ Data must be frequencies (counts)
2. ✓ Categories must be mutually exclusive
3. ✓ Expected frequency ≥ 5 in each cell (ideal)
4. ✓ Observations must be independent

Related Resources

• Chi-Square Lesson - Complete guide to chi-square tests
• t-Table - For comparing means
• F-Distribution Table - For ANOVA tests
• Goodness of Fit - Testing distributions