Chi-Square p-Values and Conclusions

Chi-square statistics only come in one flavor of trouble: positive and large when observed counts stray far from expected counts. There's no direction to it like there is with a z or t statistic, no "too high" versus "too low" tail. Squaring every deviation before adding them up means all the evidence against the null piles up on the right side of the distribution. So the p-value is always the area to the right of your calculated chi-square value — one tail, always the upper one.

To find that area you need degrees of freedom, and for a two-way table those come from the table's shape, not its sample size. With r rows and c columns, df = (r minus 1) times (c minus 1). The logic: once you fix the row and column totals, only (r-1)(c-1) of the cell counts are free to vary — the rest are forced to make the totals work out. Different df values shift and reshape the chi-square curve, so the same statistic can mean different p-values depending on table size.

Once you have the p-value, the conclusion step matches every other test in this unit: compare it to your significance level and decide whether the evidence is strong enough to reject independence. But chi-square conclusions are coarser than other tests. A small p-value says the variables are associated somewhere in the table — it does not say where. Seeing which cells have observed counts far from expected (informally comparing them) is a separate, follow-up step, not something the p-value itself delivers.