Expected frequency is the count you would see in a cell if the two variables were unrelated.
Why it matters
It is the benchmark for chi-square, and it tells you whether the test is reliable.
Key ideas
- E = row total × column total ÷ grand total
- Compare O and E cell by cell
- Rule of thumb: all E ≥ 1 and at least 80% of cells E ≥ 5
- If violated: merge categories or use Fisher
- Large (O − E)² ÷ E marks the cells driving the result
Real-life research example
25 of 200 adults are smokers with symptoms. If smoking and symptoms were unrelated, E is smaller than 25, so there is an excess.
When to use
- Always inspect E before trusting a chi-square result
When NOT to use
- Don't read chi-square if many cells have E < 5
Mini example
Row total 50, column total 55, N = 200 → E = 50 × 55 ÷ 200 = 13.75
Quick takeaway
Chi-square is only as good as its expected counts.
Try it yourself
Enter a 2×2 table with a very small group and look at the expected table. Is the chi-square rule satisfied?
Model answer
Cells with E < 5 break the rule, so merge categories if sensible or use Fisher's exact test.