Chi-Square Calculator
Chi-Square Test Results:
Interpretation:
Fail to reject null hypothesis (p ≥ 0.05). No significant difference.
Step-by-step Solution:
What is the Chi-Square Test?
The chi-square test is a statistical hypothesis test used to determine whether there is a significant association between categorical variables or whether observed data fits an expected distribution. It's one of the most widely used statistical tests in research and data analysis.
The test calculates a chi-square statistic (χ²) by comparing observed frequencies with expected frequencies, helping researchers make decisions about their hypotheses based on the probability of observing such differences by chance.
How It Works
Goodness of Fit Test:
Tests whether observed data follows an expected distribution:
Where O = observed, E = expected frequencies
Test of Independence:
Tests whether two categorical variables are independent:
Where Eᵢⱼ = (row total × column total) / grand total
Interpretation:
- • Calculate chi-square statistic and degrees of freedom
- • Determine p-value from chi-square distribution
- • Compare p-value with significance level (α)
- • If p < α: reject null hypothesis (significant result)
- • If p ≥ α: fail to reject null hypothesis (not significant)
Common Examples
Goodness of Fit Examples
Independence Test Examples
Calculation Reference Table
| Test Type | Data | χ² Statistic | df | P-value | Result (α=0.05) |
|---|---|---|---|---|---|
| Goodness of Fit | O:[10,15,20] E:[15,15,15] | 3.333 | 2 | 0.189 | Not significant |
| Independence | 2×2 table | 5.476 | 1 | 0.019 | Significant |
| Goodness of Fit | Dice test (6 categories) | 2.154 | 5 | 0.827 | Not significant |
| Independence | 3×3 table | 12.847 | 4 | 0.012 | Significant |
| Goodness of Fit | Normal distribution test | 8.921 | 7 | 0.259 | Not significant |
| Independence | Treatment effectiveness | 7.234 | 2 | 0.027 | Significant |
Frequently Asked Questions
What is the chi-square test used for?
The chi-square test is used to test relationships between categorical variables. The goodness of fit test determines if observed data matches an expected distribution, while the test of independence determines if two categorical variables are related or independent of each other.
What are the assumptions of the chi-square test?
Key assumptions include: (1) Data must be categorical, (2) Observations must be independent, (3) Expected frequencies should be at least 5 in each category/cell, (4) Sample size should be reasonably large. Violating these assumptions can lead to inaccurate results.
How do I interpret the p-value in chi-square tests?
The p-value represents the probability of observing the calculated chi-square statistic (or larger) if the null hypothesis is true. If p < α (typically 0.05), reject the null hypothesis, indicating a significant relationship or difference. If p ≥ α, fail to reject the null hypothesis.
What's the difference between goodness of fit and independence tests?
Goodness of fit tests compare observed frequencies to expected frequencies for one variable (e.g., testing if a die is fair). Independence tests examine the relationship between two categorical variables using contingency tables (e.g., testing if gender affects product preference).
How are degrees of freedom calculated?
For goodness of fit: df = (number of categories - 1). For independence tests: df = (number of rows - 1) × (number of columns - 1). Degrees of freedom are crucial for determining the critical value and p-value from the chi-square distribution.
How accurate is this chi-square calculator?
Our calculator uses precise mathematical algorithms including gamma function approximations for p-value calculations. Results are accurate to 6 decimal places for p-values and 4 decimal places for chi-square statistics, suitable for academic and professional statistical analysis.