Expected Counts in Two-Way Tables

Expected counts in a table

• For a two-way table, each cell has its own expected count under the null of "no relationship."
• The formula:
• $$\text{expected} = \frac{(\text{row total})(\text{column total})}{\text{grand total}}$$
• Compute it separately for every cell.

Why that formula?

• Under the null, the two variables are unrelated — the column split is the same in every row.
• So a cell's expected count = (that row's total) $\times$ (that column's overall fraction).
• That column fraction is $\dfrac{\text{column total}}{\text{grand total}}$.
• Multiply out and you get the row$\times$column$\div$grand-total formula.

Expected count = row total × column total ÷ grand total

Check every cell

• The large counts condition applies to every cell's expected count $\ge 5$.
• Not the observed counts, and not just some cells — all of them.
• One small expected cell can invalidate the whole test.
• Compute the full table of expected counts and scan it.

Degrees of freedom

• For a two-way table, $df = (\text{rows} - 1)(\text{columns} - 1)$.
• A $2 \times 3$ table → $df = (2-1)(3-1) = 2$.
• It counts the "free" cells once the margins are fixed.
• It depends on the table's shape, not the sample size.

Write from an expected count of 12

• A row total is 40, a column total is 30, and the table total is 100.
• The lesson expected counts include 30. This cell is not the whole row.