Chi-squared worked example — and every place the calculation can quietly go wrong

A classic genetics experiment predicts four types of plant should appear in the ratio 9 : 3 : 3 : 1. We grow 160 plants and count: 95, 31, 24, 10.

Step one: what should the counts be if the theory is right? Split 160 in the ratio 9:3:3:1 (which is sixteenths):

Honesty check 1: every expected count is at least 5 (the smallest is 10), so the test is safe to use. With rarer categories it wouldn't be.

For each type: gap between observed and expected, squared, divided by expected. Then add up:

Honesty check 2: the comparison table we look this up in depends on how many categories we have, minus one — here $4-1=3$, called "degrees of freedom". (If we'd used the data itself to estimate part of the theory, we'd have to subtract more — a correction people routinely forget.)

For 3 degrees of freedom, chance alone produces a total bigger than 7.8 only 5% of the time. Our total, 1.51, is nowhere near that. Verdict: the counts are entirely consistent with the 9:3:3:1 theory.

Honesty check 3 — what we may NOT say: we haven't proved the theory. We've shown 160 plants can't tell it apart from similar theories. Small studies often "pass" simply because they're too small to catch a modest difference.

Honesty check 4 — the too-perfect test: with 3 degrees of freedom, a typical honest result lands around 3. Ours, 1.51, is fine. Had it been 0.02 — four counts all landing almost exactly on prediction — the right question would be about the counting, not the plants.

The arithmetic takes two minutes. The four checks are what make it worth doing.