The Chi-Square Distribution

The chi-square statistic measures total disagreement between observed and expected counts, and it does this by squaring every deviation before adding them up. Squaring erases sign, so every term contributes a positive amount, and the sum can never dip below zero. That single fact shapes the whole distribution: it lives entirely on the positive number line, piles up near zero when a sample matches expectations closely, and stretches out in a long right tail when observed counts drift far from expected ones.

Degrees of freedom control the shape. With just one degree of freedom the curve is crammed against zero and sharply skewed right, because there is only one squared deviation driving the whole statistic. Add more categories or more comparisons, and you're summing more squared terms, so the typical total grows and the curve's peak slides rightward. By the time degrees of freedom reach 20 or 30, the distribution looks almost symmetric and bell-shaped, a result of the same averaging-out effect that makes sums of many pieces behave more predictably.

Because bigger deviations always push the statistic higher, and smaller or zero deviations push it near zero, there is no meaningful 'too small' side to worry about. A chi-square test is never two-sided, even when the question behind it sounds directionless, like testing whether a die is fair: any kind of unfairness, in any direction, shows up as a larger statistic. This is also where chi-square connects back to z: for one proportion with one degree of freedom, squaring the z statistic reproduces the chi-square statistic exactly, linking the two-sided z test to the one-tailed chi-square test.