Chi-Square Calculator
Compute the chi-square (χ²) statistic from observed and expected frequencies across two or more categories. Evaluate goodness-of-fit and calculate degrees of freedom.
About this Calculator
Compute the chi-square (χ²) statistic from observed and expected frequencies across two or more categories. Evaluate goodness-of-fit and calculate degrees of freedom.
Formula & Calculations
Formula
χ² = Σ((O − E)² / E); df = k − 1 (where k = number of categories)Where:
- O=Observed frequency for each category
- E=Expected frequency for each category
- χ²=Chi-square test statistic
- df=Degrees of freedom (number of categories minus 1)
Assumptions
- Each category's expected frequency should be at least 5 for the chi-square approximation to be reliable.
- Observations are independent of each other.
- The data represents count/frequency data, not percentages or proportions.
- Categories are mutually exclusive and collectively exhaustive.
Calculation Examples
Example 1
χ² = (50−40)²/40 + (30−40)²/40 + (20−20)²/20 = 2.5 + 2.5 + 0 = 5.0... wait, 100/40=2.5, 100/40=2.5, 0/20=0, sum = 5.
Example 2
χ² = (60−50)²/50 + (40−50)²/50 = 100/50 + 100/50 = 2 + 2 = 4.
Frequently Asked Questions
What is the chi-square test used for?
The chi-square test is used to determine if there is a significant difference between observed and expected frequencies. Common applications include goodness-of-fit tests (comparing data to a theoretical distribution) and tests of independence (checking if two categorical variables are related).
How do I interpret the chi-square result?
A larger chi-square value indicates a greater difference between observed and expected values. To determine statistical significance, compare your chi-square value against a critical value from a chi-square distribution table using your degrees of freedom and desired significance level (e.g., α = 0.05).