Chi-Square Table (χ² Distribution)

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

Step-by-Step: How to Use the Chi-Square Table

Step 1: Identify Your Test Type

• Goodness-of-fit test: Does your data fit an expected distribution?
• Test of independence: Are two categorical variables related?
• Test of homogeneity: Do groups have the same distribution?

Step 2: Calculate Degrees of Freedom

• Goodness-of-fit: df = k - 1 (k = number of categories)
• Independence/Homogeneity: df = (rows - 1) × (columns - 1)

Step 3: Compute the Chi-Square Statistic

Use the formula: χ² = Σ[(O - E)² / E]

Step 4: Find the Critical Value

Look up df and α in the table to find the critical value.

Step 5: Make Your Decision

If your calculated χ² ≥ critical value, reject the null hypothesis.

Frequently Asked Questions

What is a chi-square table?

A chi-square table (χ² table) shows the critical values of the chi-square distribution for different degrees of freedom and significance levels. It’s used to determine whether to reject the null hypothesis in chi-square tests.

When do I use the chi-square test?

Use the chi-square test when analyzing categorical data to: (1) Test if observed frequencies match expected frequencies (goodness-of-fit), (2) Test if two categorical variables are independent, or (3) Compare distributions across groups.

What is the chi-square critical value for df=1 at α=0.05?

For df=1 and α=0.05, the critical chi-square value is 3.841. This is one of the most commonly used critical values.

How do I calculate degrees of freedom for chi-square?

For goodness-of-fit: df = (number of categories) - 1. For tests of independence: df = (number of rows - 1) × (number of columns - 1). For a 2×2 table, df = 1.

What is the difference between chi-square and t-test?

The chi-square test is used for categorical data (frequencies/counts), while the t-test is used for continuous numerical data (means). Use chi-square for questions like “Is there an association between gender and voting preference?” and t-test for “Is there a difference in average test scores between groups?”

What does it mean if chi-square is significant?

If your calculated chi-square value exceeds the critical value from the table, the result is statistically significant. This means the observed frequencies differ significantly from expected frequencies, or the variables are not independent.

What if expected frequency is less than 5?

If more than 20% of expected frequencies are below 5, the chi-square approximation may be inaccurate. Consider combining categories, using Fisher’s exact test (for 2×2 tables), or collecting more data.

What is the chi-square critical value for α=0.01?

Common critical values at α=0.01: df=1 → 6.635, df=2 → 9.210, df=3 → 11.345, df=4 → 13.277, df=5 → 15.086.

Related Resources

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