Chi-Square Test Calculator

Calculate chi-square statistic for goodness-of-fit and test of independence. Determine if observed frequencies significantly differ from expected, or if two categorical variables are associated.

📊 Statistics📐 χ² = Σ (O - E)² / E🔢 Math
Test type
Observed count 1
Expected count 1
Observed count 2
Expected count 2
Observed count 3 (optional)
Expected count 3 (optional)
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Chi-Square Test Calculator—χ² = Σ (O - E)² / Edimensionless

Step-by-Step Examples

Example 1
Die Fairness

Roll a die 60 times. Observed: 12,8,13,10,9,8. Expected: 10 each (fair die).

  • χ² = (12-10)²/10 + (8-10)²/10 + (13-10)²/10 + (10-10)²/10 + (9-10)²/10 + (8-10)²/10
  • = 0.4+0.4+0.9+0+0.1+0.4 = 2.2
  • df = 6-1 = 5, critical χ² = 11.07
  • 2.2 < 11.07: die appears fair
✓ χ² = 2.2, not significant — die is fair
Example 2
Marketing Campaign

Expected: 25 clicked from each of 3 groups (n=75 total). Observed: 30, 20, 10.

  • χ² = (30-25)²/25 + (20-25)²/25 + (10-25)²/25
  • = 1 + 1 + 9 = 11
  • df = 2, critical χ² = 5.991
  • 11 > 5.991: significant difference between groups
✓ χ² = 11, significant at α=0.05
Example 3
Goodness of Fit

Expected by genetics: 75 tall, 25 short (3:1 ratio). Observed: 70, 30.

  • χ² = (70-75)²/75 + (30-25)²/25
  • = 0.333 + 1.0 = 1.333
  • df = 1, critical = 3.841
  • 1.333 < 3.841: consistent with 3:1 ratio
✓ χ² = 1.33, consistent with Mendelian 3:1

Real-World Applications

🧬
Genetics
Chi-square tests if observed offspring ratios match expected Mendelian ratios.
💼
Marketing
Is click-through rate independent of user segment or day of week?
🗳️
Political Science
Are voting patterns independent of demographic characteristics?
💊
Medical Research
Is disease incidence independent of risk factor exposure (2×2 contingency table)?

Common Mistakes to Avoid

⚠️
Using chi-square when expected counts are too small

Each expected cell must be ≥5 for valid chi-square test. If not: combine categories, use Fisher's exact test (for 2×2 tables), or collect more data.

⚠️
Wrong degrees of freedom

GOF: df = categories - 1 (minus estimated parameters). Independence: df = (rows-1)(cols-1). Always check df before looking up critical value.

⚠️
Chi-square only tests association, not causation

Significant chi-square means the variables are related — it does not tell you why or in which direction the effect goes.

Frequently Asked Questions

What does the chi-square test do? ▾
Tests if categorical data fits an expected distribution (GOF) or if two categorical variables are independent. It compares OBSERVED frequencies to EXPECTED frequencies.
What is the test of independence? ▾
Tests if two categorical variables are associated. Example: are gender and product preference independent? Based on a contingency table of counts. df = (r-1)(c-1).
What is goodness of fit? ▾
Tests if a single variable's distribution matches a hypothesized distribution. Example: are die rolls uniform? Are genetics data consistent with Mendel's ratios?
What is Fisher's exact test? ▾
Alternative to chi-square for 2×2 tables with small expected counts. Computes exact p-value without approximation. Always valid; chi-square is an approximation that works for large samples.
What is Yates' continuity correction? ▾
A small-sample correction to chi-square for 2×2 tables: χ² = Σ(|O-E|-0.5)²/E. Reduces Type I error for small samples. Some statisticians always apply it for 2×2; others prefer Fisher's exact test.

Related Math Calculators

📊 P-Value📊 T-Test📊 Confidence Interval📊 Binomial Distribution📊 Expected Value

Formula Explorer connections

Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.

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