Chi-square p Value Calculator

P-Value Calculator for Chi-Square Test

Use this calculator to find the p-value from a chi-square test statistic given degrees of freedom. This is useful after you’ve calculated your χ² statistic and need to determine statistical significance.

When to Use

After calculating chi-square test statistics (goodness of fit, independence, variance tests)
Testing associations in categorical data or claims about variance
Determining statistical significance by comparing p-value to α
Examples: Testing if observed frequencies fit expected distribution, testing independence of variables, testing variance claims

How to Use

Step 1: Enter your calculated chi-square test statistic (always non-negative)

Step 2: Enter degrees of freedom (depends on test type):

Goodness of fit: df = k - 1 (k = number of categories)

Independence test: df = (rows - 1) × (columns - 1)

Variance test: df = n - 1

Step 3: Select tail type:

Left-tailed: Testing if variance is less than claimed

Right-tailed: Testing if variance is greater than claimed (most common)

Two-tailed: Testing if distribution differs from expected

Step 4: Click “Calculate”

Step 5: Interpret the result:

If p-value < 0.05: Reject H₀, results are statistically significant

If p-value ≥ 0.05: Fail to reject H₀, insufficient evidence

Chi-square p Value Calculator
Chi-square Value : ($\chi^2$)
Degrees of Freedom : (n)
Tail :	Left tailedRight tailedTwo tailed

Results
p-value:

Understanding Chi-Square P-Values

The p-value for a chi-square test represents the probability of observing a test statistic as extreme as (or more extreme than) yours, assuming the null hypothesis is true. It uses the chi-square distribution.

Key Formula

p-value calculation using chi-square distribution:

Left-tailed: p-value = P(χ² ≤ χ²_obs)
Right-tailed: p-value = P(χ² ≥ χ²_obs) (most common)
Two-tailed: p-value = P(χ² ≤ χ²_lower) + P(χ² ≥ χ²_upper)

Where the distribution depends on degrees of freedom.

P-Value Interpretation

p-value Range	Decision	Meaning
p < 0.001	Reject H₀	Extremely strong evidence against H₀
0.001 ≤ p < 0.01	Reject H₀	Very strong evidence
0.01 ≤ p < 0.05	Reject H₀	Strong evidence
0.05 ≤ p < 0.10	Borderline	Weak evidence
p ≥ 0.10	Fail to reject H₀	No significant evidence

Chi-Square Distribution Properties

The chi-square distribution used here:

Always non-negative (χ² ≥ 0)
Right-skewed with positive values only
Shape depends on df: Higher df → more symmetric
Used for: Variance tests, goodness of fit, categorical association tests

Common Chi-Square P-Value Misconceptions

❌ WRONG: “p-value = 0.02 proves the categories are associated” ✓ RIGHT: “p-value = 0.02 means if variables were independent, this result occurs 2% of the time”

❌ WRONG: “Small p-value means large effect size” ✓ RIGHT: “Small p-value means results unlikely under H₀; effect size is measured separately”

❌ WRONG: “Chi-square tests work with normally distributed data” ✓ RIGHT: “Chi-square tests work with categorical/count data, not continuous”

Worked Examples

Example 1: Goodness of Fit Test

Scenario: Testing if die is fair (equal probability for each face)

Given:

Observed χ² = 8.4
Degrees of freedom = 5 (6 categories - 1)
H₀: Observed frequencies match expected (fair die)

Interpretation:

For χ² = 8.4, df = 5: p-value ≈ 0.136
Since p-value (0.136) > 0.05, fail to reject H₀
No evidence die is biased

Example 2: Test of Independence

Scenario: Testing if gender is associated with product preference

Given:

Observed χ² = 12.7
Degrees of freedom = 2 (2 rows - 1) × (2 columns - 1)
H₀: Gender and preference are independent

Interpretation:

For χ² = 12.7, df = 2: p-value ≈ 0.002
Since p-value (0.002) < 0.05, reject H₀
Strong evidence gender and preference are associated

Types of Chi-Square Tests

Test Type	Use	H₀	Alternative
Goodness of fit	Compare to expected distribution	Observed matches expected	Observed differs
Independence	Test association in 2-way table	Variables independent	Variables associated
Homogeneity	Compare distributions across groups	Distributions equal	Distributions differ
Variance test	Test population variance claim	σ² = σ²₀	σ² ≠ σ²₀

Tips for Using This Calculator

Calculate χ² statistic first:
- Goodness of fit: χ² = Σ(O - E)² / E
- Independence: χ² = Σ(O - E)² / E (same formula)
- Variance: χ² = (n-1)s² / σ²₀
Count degrees of freedom carefully:
- Watch for each test’s df formula
- Common error: forgetting to subtract 1
Expected cell counts: For categorical tests, all expected counts should be ≥ 5
Choose appropriate tail type:
- Most chi-square tests are right-tailed
- Variance tests can be left, right, or two-tailed
Remember: Chi-square only measures association, not causation