Test observed counts against expected ones, or check whether two categorical variables are related, with expected counts shown cell by cell.
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Significant at α = 0.05 with 2 degrees of freedom. The variables appear to be associated.
Cramér’s V = 0.265
| Col 1 | Col 2 | Col 3 | Row total | |
|---|---|---|---|---|
| Row 1 | 20 (E 21.43; 0.095) | 15 (E 21.43; 1.929) | 25 (E 17.14; 3.601) | 60 |
| Row 2 | 30 (E 28.57; 0.071) | 35 (E 28.57; 1.446) | 15 (E 22.86; 2.701) | 80 |
| Total | 50 | 50 | 40 | 140 |
Cells with the largest (O − E)²/E contribute most to the result and show where the pattern departs from expectation.
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χ² = Σ (O − E)² ÷ E
Each cell compares what you observed with what the null hypothesis predicts. Squaring and dividing by E makes a difference of 10 matter more in a cell expecting 20 than in one expecting 2,000. The total is compared with a chi-square distribution with the appropriate degrees of freedom to get the p-value; you can also convert any χ² value with the p-value calculator.
Goodness of fit compares one categorical variable’s counts with a set of expected proportions, such as checking whether a die is fair. A test of independence uses a two-way table to check whether two categorical variables are associated, such as whether preferred study method differs by year group.
For independence: expected = row total × column total ÷ grand total. For goodness of fit: expected = total count × the hypothesised proportion for that category.
The chi-square approximation becomes unreliable. The usual rule (from W. G. Cochran) is that no expected count should be below 1 and no more than 20% below 5. Combine sparse categories, collect more data, or use Fisher’s exact test for 2×2 tables.
Observed values must be raw counts, never percentages, because the test depends on sample size. Expected values for goodness of fit can be proportions or counts; they are rescaled to the observed total.
An effect size for a test of independence, from 0 (no association) to 1 (perfect association). A significant χ² in a huge sample can have a tiny V, meaning the association is real but weak.