The Chi-Square Statistic and Its Interpretation

Once every cell has an observed frequency $O$ and an expected-under-independence frequency $E$, a single number is needed to summarise how far apart the whole observed table is from what independence would predict. Simply adding the differences $O - E$ is useless — they always cancel to zero, because $E$ reproduces the same totals as $O$. Squaring each difference removes the cancellation, and dividing by $E$ scales each squared gap relative to how large that cell was expected to be. Adding these up over every cell gives the chi-square statistic.

The symbol $\chi$ is the Greek letter chi (pronounced 'kai'), and $\chi^2$ is read 'chi-square'. Its value is interpreted by size:

• $\chi^2 = 0$ occurs only when every $O$ equals its $E$. This is exact agreement with independence, so the two attributes are taken to be independent (unassociated).
• $\chi^2 > 0$, and the larger it is, the further the observed table lies from independence — that is, the stronger the association between the two attributes. A small positive $\chi^2$ points to a weak association; a large $\chi^2$ points to a strong one.

Because $\chi^2$ is a sum of terms $(O-E)^2/E$ each of which is zero or positive, the statistic can never be negative — a negative $\chi^2$ is always an arithmetic error. Its size also grows with the size of the table (more cells means more non-negative terms added), which is why comparing a $\chi^2$ value against a formal threshold to decide significance uses the table's degrees of freedom, $(r-1)(c-1)$ for an $r \times c$ table.