Why the chi-squared test is right-tailed — and what a too-perfect result means

Remember what the test adds up: the gaps between what you observed and what your theory expected, each one squared. Squaring makes everything positive. Too many on the left, too many on the right — either way, the total goes up. All disagreement with the theory piles up on the high side. That's the whole reason: there is no "low side" of disagreement to test.

So the question the test asks is simply: is my total surprisingly big? If yes, the theory and the data are quarrelling.

Here's the bonus lesson the textbooks skip. A chi-squared total close to zero means your data matched the theory almost perfectly — every count landing nearly on its predicted value. Sounds great? Real randomness doesn't behave like that. Chance always adds some scatter, so an honest experiment should miss its predictions by a modest, predictable amount.

There's a famous historical example: the statistician Ronald Fisher looked back at Gregor Mendel's pea-breeding results — the foundation of genetics — and noticed the counts fit the theory too well, more perfectly than chance plausibly allows. The conclusion wasn't that the theory was wrong. It was that somewhere along the way, the numbers had probably been tidied up.

Test on the high side, always — that's where disagreement lives. But glance at the low side as a health check on your own bookkeeping. A big total questions your theory. An impossibly perfect one questions your data — massaged figures, double-counted records, or results quietly nudged toward the target.