Chi-squared worked example, and where it goes 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, with 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.