A chi-square goodness-of-fit test with three categories has how many degrees of freedom?

A chi-square goodness-of-fit test with three categories has how many degrees of freedom?

• A. Two
• B. One
• C. Three
• D. Four

Answer: (A)

In a chi-square goodness-of-fit test, the degrees of freedom are equal to the number of categories minus one. This reflects that the category counts must sum to the total, so once counts for all but one category are fixed, the last one is determined and cannot vary independently. With three categories, there are two independent pieces of information, so the degrees of freedom are two. If you had to estimate any category probabilities from the data, you would subtract those estimated parameters from this amount, further reducing the degrees of freedom; but for the standard setup with fixed probabilities and three categories, it’s two.

In a chi-square goodness-of-fit test, the degrees of freedom are equal to the number of categories minus one. This reflects that the category counts must sum to the total, so once counts for all but one category are fixed, the last one is determined and cannot vary independently. With three categories, there are two independent pieces of information, so the degrees of freedom are two. If you had to estimate any category probabilities from the data, you would subtract those estimated parameters from this amount, further reducing the degrees of freedom; but for the standard setup with fixed probabilities and three categories, it’s two.