Chi-Square Calculator
Calculate chi-square statistics, p-values, and test for goodness of fit or independence between categorical variables
χ² = Σ[(O - E)² / E]
Chi-Square Formula
Actual frequencies observed in your data
Expected frequencies based on hypothesis (or single value for equal distribution)
Enter observed frequencies for each cell in the contingency table
Quick Examples
Chi-Square Statistic (χ²)
P-Value
Degrees of Freedom
Critical Value
Decision
Interpretation
Contribution Breakdown
| Category | Observed | Expected | Contribution |
|---|---|---|---|
| Total χ² | |||
Expected Frequencies
Critical Values Reference
| df | α=0.10 | α=0.05 | α=0.01 |
|---|---|---|---|
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About Chi-Square Calculator
What is a Chi-Square Test?
Chi-square (χ²) tests are statistical hypothesis tests used to determine if there is a significant difference between observed and expected frequencies in one or more categories. They are essential tools for analyzing categorical data.
Types of Chi-Square Tests
1. Goodness of Fit Test
Tests whether an observed frequency distribution differs from a theoretical distribution.
Formula: χ² = Σ[(Oᵢ - Eᵢ)² / Eᵢ]
Degrees of Freedom: df = k - 1 (where k is the number of categories)
Use when: Testing if your sample data follows an expected distribution (e.g., uniform, normal, or any specified proportions).
2. Test of Independence
Tests whether two categorical variables are related or independent.
Formula: χ² = Σ[(Oᵢⱼ - Eᵢⱼ)² / Eᵢⱼ]
Expected Frequency: Eᵢⱼ = (Row Total × Column Total) / Grand Total
Degrees of Freedom: df = (r - 1) × (c - 1) (where r = rows, c = columns)
Use when: Analyzing contingency tables to determine if there's an association between two categorical variables.
Key Concepts
| Term | Description |
|---|---|
| Observed (O) | Actual frequencies from your data |
| Expected (E) | Frequencies expected under the null hypothesis |
| Chi-square (χ²) | Sum of squared differences between observed and expected, divided by expected |
| Degrees of Freedom (df) | Number of values free to vary |
| p-value | Probability of getting results at least as extreme if null hypothesis is true |
Interpretation Guidelines
- p < 0.01: Highly significant - very strong evidence against null hypothesis
- p < 0.05: Significant - sufficient evidence to reject null hypothesis
- p < 0.10: Marginally significant - weak evidence
- p ≥ 0.10: Not significant - insufficient evidence to reject null hypothesis
Assumptions
- Random Sampling: Data should come from a random sample
- Independence: Observations should be independent
- Expected Frequencies: All expected frequencies should be ≥ 1, and at least 80% should be ≥ 5
- Categorical Data: Variables must be categorical (nominal or ordinal)
Note: If expected frequencies are too small, consider combining categories or using Fisher's exact test.