ANOVA & Chi-Square Tests Cheat Sheet

Key Facts

• One-way ANOVA tests whether several population means are equal using hypotheses $H_0: \mu_1 = \mu_2 = \cdots = \mu_k$ and $H_a:$ at least one mean is different.
• The ANOVA test statistic is $F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$, where large values of $F$ give stronger evidence against $H_0$.
• The ANOVA sums of squares are $SS_{\text{between}} = \sum n_i(\bar{x}_i-\bar{x})^2$ and $SS_{\text{within}} = \sum\sum (x_{ij}-\bar{x}_i)^2$.
• For one-way ANOVA with $k$ groups and $N$ total observations, $df_{\text{between}} = k-1$, $df_{\text{within}} = N-k$, and $df_{\text{total}} = N-1$.
• Mean squares are found by $MS_{\text{between}} = \frac{SS_{\text{between}}}{df_{\text{between}}}$ and $MS_{\text{within}} = \frac{SS_{\text{within}}}{df_{\text{within}}}$.
• A chi-square goodness-of-fit test uses $\chi^2 = \sum \frac{(O-E)^2}{E}$ to compare observed category counts to expected category counts.
• A chi-square test of independence uses $E_{ij} = \frac{(\text{row total})(\text{column total})}{\text{grand total}}$ for each table cell.
• For a chi-square independence test with $r$ rows and $c$ columns, the degrees of freedom are $df = (r-1)(c-1)$.

Vocabulary

One-way ANOVA

A statistical test that compares the means of $k$ independent groups to see whether at least one population mean differs.

F statistic

The ANOVA test statistic $F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$ that compares between-group variation to within-group variation.

Chi-square statistic

The statistic $\chi^2 = \sum \frac{(O-E)^2}{E}$ that measures how far observed counts are from expected counts.

Expected count

The count predicted for a category or table cell if the null hypothesis is true.

Degrees of freedom

The number of independent pieces of information used to find the reference distribution for a test statistic.

p-value

The probability, assuming $H_0$ is true, of getting a test statistic as extreme as or more extreme than the observed result.