The Scaling Law of Evaluation Failure: Why Simple Averaging Collapses Under Data Sparsity and Item Difficulty Gaps, and How Item Response Theory Recovers Ground Truth Across Domains

Abstract

Benchmark evaluation across AI and safety-critical domains overwhelmingly relies on simple averaging. We demonstrate that this practice produces substantially misleading rankings when two conditions co-occur: (1) the evaluation matrix is sparse and (2) items vary substantially in difficulty. Through controlled simulation experiments across four domains -- NLP (GLUE), clinical drug trials, autonomous vehicle safety, and cybersecurity -- we show that Spearman rank correlation $\rho$ between simple-average rankings and ground-truth rankings degrades from $\rho = 1.000$ at 100% coverage to $\rho = 0.809$ at 67% coverage with high difficulty heterogeneity (mean over 20 seeds). A standard two-parameter logistic (2PL) Item Response Theory (IRT) model maintains $\rho \geq 0.996$ across all conditions. A 150-condition grid sweep over sparsity $S \in [0, 0.70]$ and difficulty gap $D \in [0.5, 5.0]$ confirms that ranking error forms a failure surface with a strong $S \times D$ interaction ($\gamma_3 = +0.20$, $t = 13.05$), while IRT maintains $\rho \geq 0.993$ throughout. We discuss implications for Physical AI benchmarking, where evaluation matrices are often incomplete and difficulty gaps are extreme.