The chi-square distribution is a probability distribution used to describe sums of squared standard normal variables. It is important because many statistics used in hypothesis testing follow a chi-square distribution when the null hypothesis is true. Its shape depends on the degrees of freedom, so the curve changes as the number of independent pieces of information changes.
Students meet this distribution often in goodness-of-fit tests and tests of independence.
Key Facts
- If Z1, Z2, ..., Zk are independent standard normal variables, then X = Z1^2 + Z2^2 + ... + Zk^2 follows a chi-square distribution with k degrees of freedom.
- A chi-square random variable is always nonnegative, so x >= 0.
- The mean of a chi-square distribution is μ = k, where k is the degrees of freedom.
- The variance of a chi-square distribution is σ^2 = 2k.
- For a goodness-of-fit test, χ^2 = Σ((O - E)^2 / E), where O is observed count and E is expected count.
- For a test of independence in an r by c table, degrees of freedom are df = (r - 1)(c - 1).
Vocabulary
- Chi-square distribution
- A probability distribution for the sum of squares of independent standard normal random variables.
- Degrees of freedom
- The number of independent values that are free to vary after constraints have been applied.
- Goodness-of-fit test
- A chi-square test that checks whether observed category counts match a claimed distribution.
- Test of independence
- A chi-square test that checks whether two categorical variables are associated in a two-way table.
- Expected count
- The count predicted for a category or table cell if the null hypothesis is true.
Common Mistakes to Avoid
- Using negative x-values for a chi-square graph is wrong because chi-square values come from squared quantities and cannot be negative.
- Forgetting to check expected counts is wrong because the chi-square approximation can be unreliable when expected counts are too small.
- Using df = n - 1 for every chi-square test is wrong because goodness-of-fit and independence tests use different degrees of freedom formulas.
- Interpreting a small p-value as proof that the null hypothesis is false is wrong because a hypothesis test gives evidence, not absolute proof.
Practice Questions
- 1 A goodness-of-fit test has observed counts 18, 22, and 10 with expected counts 20, 20, and 10. Compute the chi-square statistic χ^2 = Σ((O - E)^2 / E).
- 2 A survey table has 3 rows and 4 columns. Find the degrees of freedom for a chi-square test of independence.
- 3 A chi-square curve with df = 2 is strongly right-skewed, while one with df = 20 is more spread out and more symmetric. Explain why increasing degrees of freedom changes the shape of the distribution.