Chi-Square Test Calculator

📚 How It Works

What is the Chi-Square Test?

The chi-square (χ²) test is a statistical hypothesis test used to determine if there's a significant difference between observed and expected frequencies, or if two categorical variables are independent.

Chi-Square Formula

χ² = Σ [(O - E)² / E]

Where:

• O = Observed frequency
• E = Expected frequency
• Σ = Sum over all categories

Goodness of Fit Test

Tests if observed data fits an expected distribution:

• Null Hypothesis (H₀): Observed data fits expected distribution
• Alternative (H₁): Observed data doesn't fit expected distribution
• Degrees of Freedom: df = k - 1 (where k = number of categories)

Example: Is a die fair? Roll 60 times, expect 10 per face.

Test of Independence

Tests if two categorical variables are independent:

• Null Hypothesis (H₀): Variables are independent
• Alternative (H₁): Variables are dependent (associated)
• Degrees of Freedom: df = (rows - 1) × (columns - 1)

Example: Is gender associated with product preference?

Expected Frequencies (Independence)

For a contingency table:

E = (Row Total × Column Total) / Grand Total

Interpreting Results

1. Calculate χ²: Sum of [(O-E)²/E] for all cells
2. Find df: Degrees of freedom
3. Compare: χ² statistic vs. critical value
4. Decision:
   • If χ² > critical value: Reject H₀ (significant difference)
   • If χ² ≤ critical value: Fail to reject H₀ (no significant difference)

Significance Levels

• α = 0.05 (5%): Standard in most fields
• α = 0.01 (1%): More stringent (reduce Type I error)
• α = 0.10 (10%): More lenient (exploratory research)

P-Value Interpretation

• p < 0.001: Very strong evidence against H₀
• p < 0.01: Strong evidence against H₀
• p < 0.05: Moderate evidence against H₀
• p > 0.05: Weak evidence, fail to reject H₀

Assumptions

• Data are frequencies (counts), not percentages
• All observations are independent
• Expected frequency in each cell should be ≥ 5
• Sample is random

Real-World Applications

• Genetics: Test Mendelian inheritance ratios (3:1)
• Medicine: Treatment effectiveness vs. outcomes
• Marketing: Product preference by demographic
• Quality Control: Defect rates across shifts
• Social Science: Survey response patterns
• Gaming: Testing if dice/cards are fair

Example Calculation

Question: Is a die fair? Rolled 60 times:

Face	Observed	Expected	(O-E)²/E
1	8	10	0.4
2	12	10	0.4
3	11	10	0.1
4	9	10	0.1
5	10	10	0.0
6	10	10	0.0
χ² =			1.0

df = 6 - 1 = 5, critical value (α=0.05) = 11.07

Conclusion: χ² (1.0) < 11.07, fail to reject H₀. Die appears fair.