📈 Chi-Square Goodness-of-Fit Test
Test whether your observed categorical counts match an expected distribution. Enter observed frequencies, choose how to define expected values, and receive the χ² statistic, degrees of freedom, p-value, critical region, Cramér’s V, and an interpretation with step-by-step guidance.
1. Provide Observed Counts
Provide one non-negative integer per category. At least two categories required.
Expected Distribution
Probabilities may be decimals or fractions; they will be scaled by the total sample size.
Significance Level (α)
Custom α must fall between 0.001 and 0.25.
Parameter Adjustments
Chi-Square Output
Results readyKey Values
- χ² statistic
- Degrees of freedom
- p-value
Decision
Critical & Effect
- Critical value
- Cramér’s V
- Effect interpretation
Step-by-Step Workflow
Observed vs Expected
Formula Reference
The chi-square goodness-of-fit statistic is
\\[ \chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}, \qquad \text{df} = k - 1 - m \\]
where \\(O_i\\) and \\(E_i\\) denote observed and expected counts for category \\(i\\), \\(k\\) is the number of categories, and \\(m\\) is the number of parameters estimated from data to compute \\(E_i\\). Effect size can be summarised by \\(V = \sqrt{\chi^2 / (n(k-1))}\\).
How to Use This Calculator
- Enter observed counts for each category (all non-negative, at least two categories, total > 0).
- Select how to derive expected counts: uniform distribution, custom counts, or custom probabilities.
- Specify the significance level \\(\alpha\\) and any parameters estimated from data (e.g., when estimating mean from the sample).
- Click “Run χ² Test” to obtain the statistic, degrees of freedom, p-value, critical value, and Cramér’s V.
- Review the step-by-step notes, residuals, and references to interpret and report the result.
References
- McHugh, M. L. (2013). “The chi-square test of independence.” Biochemia Medica, 23(2), 143–149.
- Agresti, A. (2013). Categorical Data Analysis (3rd ed.). Wiley.
- Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
- Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers (6th ed.). Wiley.
Disclaimer
Ensure expected counts are sufficiently large (rule of thumb ≥ 5) and observations are independent. When expected counts are small, consider exact or simulation-based methods.