1. What is the Chi-Square Test?

The Chi-Square (χ²) test is a statistical method used to determine if there is a significant difference between the expected and observed frequencies in categorical data. It's commonly used for goodness-of-fit tests and tests of independence.

2. How Does the Calculator Work?

The calculator uses the Chi-Square formula:

Where:

• \( O_i \) — Observed frequency for category i
• \( E_i \) — Expected frequency for category i

Explanation: The test compares the observed frequencies with the expected frequencies under the null hypothesis. A large chi-square value indicates a significant difference.

3. When to Use Chi-Square Test

Details: Use the Chi-Square test when:

• Testing goodness of fit (how well observed data match expected distribution)
• Testing independence between two categorical variables
• Working with frequency counts in contingency tables

4. Using the Calculator

Tips:

• Enter observed and expected frequencies as comma-separated values
• Both lists must have the same number of values
• All expected frequencies should be ≥5 for reliable results

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between goodness-of-fit and test of independence?
A: Goodness-of-fit compares observed to theoretical distribution, while test of independence examines relationship between two categorical variables.

Q2: What are degrees of freedom in Chi-Square test?
A: For goodness-of-fit: df = (number of categories - 1). For test of independence: df = (rows - 1) × (columns - 1).

Q3: When is Chi-Square test not appropriate?
A: When expected frequencies are too small (<5), or when dealing with continuous data or paired samples.

Q4: How to interpret the Chi-Square value?
A: Compare your calculated χ² to critical values from Chi-Square distribution table based on your degrees of freedom and significance level.

Q5: What are alternatives when expected frequencies are too small?
A: Fisher's exact test is often used when sample sizes are small.