1. What is a χ² Goodness of Fit Test?

Definition: This statistical test determines whether observed frequencies differ significantly from expected frequencies.

Purpose: Used to test hypotheses about distributions of categorical data and check how well theoretical distributions fit observed data.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( \chi^2 \) — Chi-square test statistic (dimensionless)
• \( O_i \) — Observed frequencies (dimensionless)
• \( E_i \) — Expected frequencies (dimensionless)

Explanation: For each category, the squared difference between observed and expected counts is divided by the expected count, then summed across all categories.

3. Importance of χ² Goodness of Fit

Details: This test helps determine whether sample data matches a population distribution, validates assumptions in statistical models, and tests theoretical predictions.

4. Using the Calculator

Tips: Enter comma-separated observed and expected values. Both lists must be same length and expected values must be > 0.

5. Frequently Asked Questions (FAQ)

Q1: What does the χ² value tell me?
A: Higher values indicate greater discrepancy between observed and expected distributions. Compare to critical values from χ² tables.

Q2: What are typical applications?
A: Testing genetic ratios, survey response distributions, quality control checks, and any categorical data analysis.

Q3: What are the assumptions?
A: Independent observations, adequate sample size (all expected counts ≥ 5), and categorical data.

Q4: How to interpret results?
A: Compare calculated χ² to critical value at your desired significance level (e.g., 0.05) with appropriate degrees of freedom.

Q5: What if expected counts are small?
A: Consider Fisher's exact test or combine categories if some expected counts are < 5.