Expected counts
In a two-way table, the expected count for a cell is (row total x column total) / grand total. It is what you would see if the two variables were unrelated.
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Analogy
If 50 percent of everyone is in group A and 50 percent say yes, then half of half, or 25 percent, should be both.
Example
Group A: yes 30, no 20 (row 50)Group B: yes 20, no 30 (row 50)Column totals: 50 and 50, total 100Expected each cell = 50 x 50 / 100 = 25
In practice
Chi-square tests need every expected count to be at least 5 to be reliable.
Practice
Flashcards
- Expected count in a cell of a table?Row total x column total / grand total.
Interview questions
- In a 2 by 2 table the observed counts are 30, 20, 20 and 30. Test whether the two variables are independent at the 5% level.
- Expected count 25 in every cell
- Chi-square = 4.0
- df = 1, critical 3.841
- Reject independence
Questions people ask
Why do expected counts need to be at least 5?
The chi-square test relies on a smooth curve approximating discrete counts. When expected counts are tiny, that approximation breaks down and p-values become unreliable. If cells are too small, combine categories or use an exact test such as Fisher's, which does not rely on the curve approximation at all.
What is the difference between observed and expected counts?
Observed counts are what you actually recorded. Expected counts are what the model or the no-relationship assumption predicts. The test judges whether the gap between them is bigger than random sampling would plausibly cause. A small gap fits chance, while a large gap points to a real pattern or a poor model.