Chi-Square Contribution of One Cell

The chi-square statistic is a sum of these cell terms, one per box in the table: square the gap between what you saw and what the model predicted, then divide by the expected count so that a discrepancy of 5 counts for more in a cell expecting 25 than in one expecting 500. Observing 30 where 25 were expected contributes (30 − 25)²/25 = 1.0. Karl Pearson introduced the statistic in 1900 and it remains the workhorse of categorical data analysis, from genetics ratios to A/B tests.

Two traps. Cell contributions are the most useful diagnostic in the whole procedure and are almost always ignored: a significant total tells you the model fails somewhere, and scanning the individual terms tells you where. And the approximation needs reasonably full cells: the standard rule is that every expected count should be at least 5, with small tables better handled by Fisher's exact test. Ronald Fisher built that test around a real 1920s Cambridge tea party, where Muriel Bristol claimed she could taste whether milk went into the cup before or after the tea; he designed eight cups, four each way, and worked out the exact probability of her guessing correctly by luck. Note that solving backwards for O returns the upper root: an observed count that overshoots the expectation.