The chi-squared test
The two uses of the chi-squared test: fit to a stated distribution and independence of two characteristics in a contingency table; expected counts, statistic, degrees of freedom, reading the table and the condition of counts of at least 5.
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Chi-squared test: what is it?
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A test that compares observed counts with the counts expected under the null hypothesis, through the sum of (observed − expected)² / expected.
The statistic is zero if everything matches and grows with the gap. It does not say which cell the gap is in: look at each cell's contribution.
In a contingency table of 200 individuals, the “mobile” row has 80 and the “purchase” column has 50. What is the expected count of the “mobile, purchase” cell?
- 20
- 40
- 65
- 130
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20
80 × 50 / 200 = 20. Multiply the two marginal totals, then divide by the grand total. 40 is half of 80, ignoring the column.
Expected counts must be at least …; otherwise … are merged or another test is used.
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Expected counts must be at least 5; otherwise classes are merged or another test is used.
With counts too small, the statistic no longer follows the chi-squared distribution well and the p-value becomes unreliable. The condition is on expected counts, not on observed ones.
Expected count: what is it?
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The count one would expect in a cell if the null hypothesis were true.
In a contingency table it equals row total × column total / grand total: what independence of the two characteristics would give.