Chi-Square Test Calculator
Calculate chi-square statistics for goodness-of-fit and independence tests with observed vs expected frequencies
| Category | Observed Frequency | Expected Frequency | Action |
|---|---|---|---|
Calculation Results
Comprehensive Guide to Chi-Square Test Example Problems
The chi-square (χ²) test is a fundamental statistical method used to determine whether there is a significant association between categorical variables or whether observed frequencies differ from expected frequencies. This guide provides detailed example problems for both goodness-of-fit and independence tests, along with step-by-step solutions.
1. Understanding Chi-Square Tests
Chi-square tests are non-parametric tests that compare categorical data with expected distributions. There are two main types:
- Goodness-of-Fit Test: Determines if a sample matches a population with a specific distribution
- Test of Independence: Assesses whether two categorical variables are independent
2. Goodness-of-Fit Test Example Problem
A geneticist wants to test whether a new plant breed produces flowers in the ratio 9:3:3:1 (purple:red:white:yellow) as predicted by Mendelian genetics. She grows 200 plants and observes:
| Flower Color | Observed Count | Expected Ratio | Expected Count |
|---|---|---|---|
| Purple | 110 | 9 | 112.5 |
| Red | 35 | 3 | 37.5 |
| White | 30 | 3 | 37.5 |
| Yellow | 25 | 1 | 12.5 |
| Total | 200 | 16 | 200 |
Step-by-Step Solution:
- State the hypotheses:
- H₀: The flower colors follow a 9:3:3:1 ratio
- H₁: The flower colors do not follow a 9:3:3:1 ratio
- Calculate expected frequencies:
- Total observed = 200
- Purple expected = (9/16) × 200 = 112.5
- Red expected = (3/16) × 200 = 37.5
- White expected = (3/16) × 200 = 37.5
- Yellow expected = (1/16) × 200 = 12.5
- Compute chi-square statistic:
χ² = Σ[(O – E)²/E]
= (110-112.5)²/112.5 + (35-37.5)²/37.5 + (30-37.5)²/37.5 + (25-12.5)²/12.5
= 0.056 + 0.167 + 1.35 + 10.8
= 12.373
- Determine degrees of freedom:
df = number of categories – 1 = 4 – 1 = 3
- Find critical value:
At α = 0.05 with df = 3, critical value = 7.815
- Make decision:
Since 12.373 > 7.815, we reject the null hypothesis
3. Test of Independence Example Problem
A researcher wants to determine if there’s an association between gender and preference for three different smartphone brands. She surveys 300 people with these results:
| Brand Preference | Male | Female | Total |
|---|---|---|---|
| Brand A | 45 | 35 | 80 |
| Brand B | 50 | 60 | 110 |
| Brand C | 35 | 75 | 110 |
| Total | 130 | 170 | 300 |
Step-by-Step Solution:
- State the hypotheses:
- H₀: Brand preference is independent of gender
- H₁: Brand preference is not independent of gender
- Calculate expected frequencies:
Expected count = (row total × column total) / grand total
Brand Preference Male Female Brand A 34.7 45.3 Brand B 48.3 61.7 Brand C 47.0 63.0 - Compute chi-square statistic:
χ² = Σ[(O – E)²/E] for all cells
= (45-34.7)²/34.7 + (35-45.3)²/45.3 + … + (75-63.0)²/63.0
= 3.0 + 2.3 + 0.1 + 0.0 + 1.2 + 1.6
= 8.2
- Determine degrees of freedom:
df = (rows – 1) × (columns – 1) = (3-1) × (2-1) = 2
- Find critical value:
At α = 0.05 with df = 2, critical value = 5.991
- Make decision:
Since 8.2 > 5.991, we reject the null hypothesis
4. Common Applications of Chi-Square Tests
- Genetics: Testing Mendelian ratios in offspring
- Market Research: Analyzing consumer preferences across demographics
- Quality Control: Comparing defect rates across production lines
- Social Sciences: Examining relationships between categorical variables
- Medicine: Assessing treatment effectiveness across patient groups
5. Key Assumptions and Requirements
For valid chi-square test results, these conditions must be met:
- Categorical Data: Variables must be categorical (nominal or ordinal)
- Independent Observations: Each subject contributes to only one cell
- Expected Frequencies: No more than 20% of expected counts < 5, and none < 1
- If violated, combine categories or use Fisher’s exact test
- Sample Size: Generally requires at least 5 observations per cell
6. Comparing Chi-Square with Other Statistical Tests
| Test | Data Type | Purpose | When to Use Instead of Chi-Square |
|---|---|---|---|
| Chi-Square | Categorical | Compare observed vs expected frequencies | Baseline for categorical data |
| Fisher’s Exact | Categorical (2×2) | Test independence in small samples | Expected counts < 5 in 2×2 tables |
| McNemar’s | Paired categorical | Compare paired proportions | Before-after measurements on same subjects |
| Cochran’s Q | Categorical (3+ related samples) | Test for differences across multiple trials | Repeated measures with binary outcomes |
7. Practical Tips for Conducting Chi-Square Tests
- Data Collection:
- Ensure categories are mutually exclusive and exhaustive
- Use random sampling to maintain independence
- Table Construction:
- Include row and column totals (marginal distributions)
- Label categories clearly and consistently
- Calculation:
- Double-check expected frequency calculations
- Verify degrees of freedom formula for your test type
- Interpretation:
- Report chi-square statistic, df, and p-value
- Include effect size measures like Cramer’s V for independence tests
- Software Validation:
- Cross-validate manual calculations with statistical software
- Use our calculator above to verify your results
8. Advanced Considerations
For more complex analyses:
- Post-hoc Tests: After significant chi-square results, perform standardized residual analysis to identify which cells contribute most to the significance
- Effect Sizes: Calculate Cramer’s V (for tables larger than 2×2) or phi coefficient (for 2×2 tables) to quantify association strength
- Simpson’s Paradox: Be aware that associations can reverse when data is aggregated differently
- Power Analysis: Calculate required sample size before data collection to ensure adequate power (typically 0.8)