11.1 Facts About the Chi-Square Distribution

Key Characteristics of the Chi-Square Distribution

The chi-square distribution is a continuous probability distribution used to analyze categorical data. It shows up in two major hypothesis tests you'll encounter in this unit: the goodness-of-fit test and the test of independence. Getting a solid handle on its shape and properties now will make those tests much easier to work with.

Shape and Properties

The chi-square distribution is defined only for positive values, ranging from 0 to positive infinity. It can never be negative. Its shape depends entirely on a single parameter: degrees of freedom ($df$).

• With low $df$, the distribution is strongly skewed to the right (positively skewed)
• As $df$ increases, the distribution becomes more symmetrical and bell-shaped
• The distribution is always positively skewed, but that skewness shrinks as $df$ grows
• It's technically a special case of the gamma distribution

Mean, Variance, and Standard Deviation

The formulas here are unusually clean compared to most distributions:

• Mean: $\mu = df$
• Variance: $\sigma^2 = 2 \cdot df$
• Standard deviation: $\sigma = \sqrt{2 \cdot df}$

So if $df = 10$, the mean is 10, the variance is 20, and the standard deviation is $\sqrt{20} \approx 4.47$. Notice that the mean sits at the degrees of freedom, which means the center of the distribution shifts right as $df$ increases.

Chi-Square vs. Normal Distribution

As $df$ increases, the chi-square distribution starts to resemble a normal distribution. The usual threshold is $df > 30$: beyond that point, a normal approximation works reasonably well.

When you use this approximation, the normal distribution has:

• Mean: $\mu = df$
• Variance: $\sigma^2 = 2 \cdot df$
• Standard deviation: $\sigma = \sqrt{2 \cdot df}$

These are the same formulas from above. The only thing that changes is you're now treating the distribution as if it were normal, which lets you use z-scores and normal probability tables.

Applications in Statistical Analysis

The chi-square statistic is built on a specific idea: measuring how far observed data falls from what you'd expect. It sums up the squared, standardized differences between observed and expected frequencies.

• Goodness-of-fit test: Checks whether observed data matches a proposed distribution. For example, do the colors in a bag of candy appear in the proportions the company claims?
• Test of independence: Examines whether two categorical variables are related. For example, is there a relationship between gender and preferred study method?

The chi-square statistic was introduced by Karl Pearson, and these tests remain some of the most widely used tools for working with categorical data.