Chi-Square Goodness of Fit Calculator
Calculate Chi-Square Goodness-of-Fit statistic ($\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$), degrees of freedom ($df = k-1$), $p$-value, and Cramér's V.
Calculate Goodness-of-Fit Observed vs Expected
Enter observed counts (O) and expected counts (E) separated by commas.
Calculation Results
Calculated using Chi-Square Goodness-of-Fit equations: \chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}
Quick Summary
The Chi-Square Goodness of Fit Calculator evaluates $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$, degrees of freedom ($df = k-1$), $p$-value, and Cramér's V effect size.
Formula Explanation
\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}, \quad df = k - 1
\text{Cramér\'s V} = \sqrt{\frac{\chi^2}{N(k - 1)}}How It Works
Goodness-of-Fit compares observed category counts $O_i$ against theoretical expected counts $E_i$. The Chi-Square Goodness of Fit Calculator evaluates normalized squared residuals and checks if departures are statistically significant.
Step-by-Step Worked Example
Practical Problem: A die is rolled $N = 100$ times with observed counts: 25, 30, 20, 25 across $k = 4$ categories. Test if outcome distribution is uniform ($E_i = 25$).
- Step 1: Calculate $(O_i - E_i)^2 / E_i$ for category 1: $(25-25)^2 / 25 = 0 / 25 = \mathbf{0.00}$.
- Step 2: Calculate for category 2: $(30-25)^2 / 25 = 25 / 25 = \mathbf{1.00}$.
- Step 3: Calculate for category 3: $(20-25)^2 / 25 = 25 / 25 = \mathbf{1.00}$.
- Step 4: Calculate for category 4: $(25-25)^2 / 25 = 0 / 25 = \mathbf{0.00}$. Sum $\chi^2 = \mathbf{2.0000}$.
- Step 5: Determine df ($df = 3$) & p-value: $p$-value for $\chi^2 = 2.00, df = 3 \rightarrow \mathbf{0.5724\text{ (Fail to reject } H_0)}$.
Real-World Calculation Examples
Scenario 1: 4-Category Uniform Test (O = 25, 30, 20, 25)
Parameters: k = 4, E = 25 each
Result: χ² = 2.0000, p = 0.5724 (Distribution fits uniform model).
Scenario 2: Mendelian Genetics 9:3:3:1 Pea Inheritance
Parameters: k = 4 categories, expected ratio 9:3:3:1
Result: Goodness-of-fit test.
Scenario 3: Customer Preference Across 5 Products
Parameters: Equal 20% market share model
Result: Goodness-of-fit evaluation.
Scenario 4: Days of Week Hospital Admission Distribution
Parameters: Mon-Sun uniform admissions
Result: Goodness-of-fit evaluation.
Key Benefits of Using This Calculator
Custom Observed & Expected Inputs
Supports any custom array of observed $O_i$ and expected $E_i$ counts.
Cramér's V Effect Size
Calculates Cramér's V effect size to measure magnitude of divergence.
Automatic df Calculation
Automatically determines degrees of freedom $df = k - 1$.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a Chi-Square Goodness-of-Fit test?
The Chi-Square Goodness-of-Fit test checks if sample categorical frequencies fit a specified theoretical probability distribution.
What is null hypothesis ($H_0$) in Goodness-of-Fit test?
$H_0$: The observed frequency distribution matches the expected distribution.
How do I calculate expected frequencies if given probabilities?
Expected count $E_i = N \times p_i$, where $N$ is total sample size and $p_i$ is theoretical probability.
How do I calculate Goodness-of-Fit in Excel?
Use formula =CHISQ.TEST(observed_range, expected_range).
What are degrees of freedom for Goodness-of-Fit?
$df = k - 1 - m$, where $k$ is number of categories and $m$ is number of estimated distribution parameters.
What minimum expected count is required per category?
Each category should have expected count $E_i \ge 5$.
Can Goodness-of-Fit test continuous distributions (Normal/Poisson)?
Yes — by binning continuous data into discrete interval bins ($k$ bins).
What alternative test exists for continuous distributions?
The Kolmogorov-Smirnov (K-S) test and Anderson-Darling test do not require binning continuous data.
What is Cramér's V formula for Goodness-of-Fit?
$\text{Cramér's V} = \sqrt{\frac{\chi^2}{N(k - 1)}}$.
Why does $O_i - E_i$ get squared?
Squaring prevents positive and negative deviations from cancelling out and penalizes larger discrepancies.