Chi-Square (χ²) Goodness-of-Fit Calculator
Enter the observed and expected frequencies for each category below. Click "Add Category" to include more data points. The calculator will compute the Chi-Square statistic (χ²).
Understanding the Chi-Square (χ²) Test
The Chi-Square (χ²) test is a statistical hypothesis test that is widely used in research to determine if there is a significant difference between observed frequencies and expected frequencies in one or more categories. It's particularly useful for analyzing categorical data.
When to Use the Chi-Square Test
There are two primary types of Chi-Square tests:
- Chi-Square Goodness-of-Fit Test: This test determines if an observed frequency distribution differs significantly from an expected frequency distribution. For example, you might use it to see if the number of people choosing different colors of a product matches a theoretical distribution (e.g., equal preference for all colors). This calculator focuses on the goodness-of-fit test.
- Chi-Square Test of Independence: This test assesses whether two categorical variables are related or independent. For instance, you could use it to see if there's a relationship between gender and political party preference.
The Chi-Square Formula
The Chi-Square statistic (χ²) is calculated using the following formula:
χ² = Σ [(Oᵢ – Eᵢ)² / Eᵢ]
Where:
- Σ (Sigma) represents the sum across all categories.
- Oᵢ is the observed frequency (the actual count) for category i.
- Eᵢ is the expected frequency (the count you would expect based on a null hypothesis or theoretical distribution) for category i.
How to Interpret the Chi-Square Result
A larger Chi-Square value indicates a greater discrepancy between the observed and expected frequencies. To determine if this difference is statistically significant, you would typically compare your calculated χ² value to a critical value from a Chi-Square distribution table. This comparison also requires knowing the "degrees of freedom" (df), which is usually calculated as the number of categories minus one (df = k – 1, where k is the number of categories).
If your calculated χ² value exceeds the critical value for a chosen significance level (e.g., 0.05), you would reject the null hypothesis, suggesting that the observed frequencies are significantly different from the expected frequencies.
Example Scenario
Imagine a candy company claims that its bags contain an equal distribution of five different colors: Red, Green, Blue, Yellow, and Orange. You buy a bag and count the following:
- Observed: Red=20, Green=15, Blue=25, Yellow=18, Orange=22 (Total = 100 candies)
If the company's claim is true, you would expect an equal distribution. With 100 candies and 5 colors, the expected frequency for each color would be 100 / 5 = 20.
- Expected: Red=20, Green=20, Blue=20, Yellow=20, Orange=20
Using the calculator, you would input these observed and expected pairs. The calculator would then compute the χ² value, allowing you to assess if your observed candy distribution significantly deviates from the company's claimed equal distribution.