What is the Chi-Square (χ²) Distribution?
The Chi-Square distribution is a probability distribution used primarily for:
- Goodness of Fit Test - Testing if a sample comes from a population with a specific distribution
- Independence Test - Testing if there's a relationship between two categorical variables
- Homogeneity Test - Testing if several populations share the same distribution
Structure of the χ² Distribution Table
The χ² distribution table contains critical values of the χ² distribution for different significance levels (α) and degrees of freedom (df).
Degrees of Freedom (df)
The degrees of freedom in the χ² table depend on the test type:
- Goodness of Fit Test: df = k - 1 (where k is the number of categories)
- Independence Test: df = (r - 1) × (c - 1) (where r is the number of rows and c is the number of columns)
Significance Level (α)
The significance level represents the probability of rejecting the null hypothesis when it is true (Type I error).
Examples of Using the Table
Example 1: Goodness of Fit Test
Suppose we have 5 categories (df = 4) and a significance level of 0.05.
In the table, find the row corresponding to df=4 and the column corresponding to α=0.05.
The value in the table is 9.4877.
Example 2: Independence Test
Suppose we have a 3×3 table (df = 4) and a significance level of 0.01.
The critical value from the table is 13.2767.
If our calculated χ² value is greater than 13.2767, we reject the null hypothesis.
Example 3: Calculating Confidence Interval for Variance
To calculate a 95% confidence interval for variance, with df=9:
The critical values from the table are 2.7004 (α=0.975) and 19.0228 (α=0.025).
Confidence Interval = [(n-1)s²/χ²α/2, (n-1)s²/χ²1-α/2]
