chi-squared-statistic

The chi-squared statistic was designed by Pearson specifically for the purpose of testing goodness-of-fit — measuring how well observed frequencies match expected frequencies under a null model. The test statistic SERVES the goodness-of-fit evaluation: remove goodness-of-fit as a goal and the chi-squared statistic loses its primary designed purpose. Servant (chi-squared-statistic) points at master (goodness-of-fit).

Law 8d servant points at master. Pearson's chi-squared statistic is the decision quantity of the chi-squared test: compute X^2 from observed vs expected counts, compare against its asymptotic chi-squared reference distribution, decide about H0. Its accepted scope names the purpose — 'the standard form of the chi-squared hypothesis test' — and the master's own accepted definition names the chi-squared test among its operationalized standard procedures, with 'test statistic' as an explicit parameter slot. The statistic was made for the test's decision; the test is not built for the statistic's sake.

Which-came-first (Law 7): Pearson's 1900 statistic X² = Σ(O−E)²/E is the antecedent object. The chi-squared divergence is that same functional form, D_χ²(P||Q) = Σ(p−q)²/q, read as a measure on distributions — the statistic is the sample-size-scaled empirical instance (X² = n·D_χ²(empirical||model)) — and the general divergence reading was absorbed into the f-divergence framework (Csiszár 1966, φ(t)=(t−1)²). The statistic (1900) came first; the divergence construct is derived from it.