A layered framework for neutrosophic statistics: foundational distinctions, empirical validation, and operational implementation

Neutrosophic statistics encompasses several mathematically distinct frameworks that are frequently conflated, blurring its relation to interval and classical methods. We organize neutrosophic statistics as three layers: (i) neutrosophic arithmetic 𝑁 = 𝑎 + 𝑏𝐼 with indeterminate components beyond intervals; (ii) neutrosophic set statistics over (𝑇, 𝐼, 𝐹) triplets as structurally independent dimensions; and (iii) plithogenic statistics with degree of appurtenance. We provide six lines of validation: a formal analysis showing interval projection is a lossy compression of neutrosophic numbers; empirical evidence on 510 expertannotated triplets that (𝑇, 𝐼, 𝐹) data does not reduce to an interval-derived construction (T-I coupling 3.4× weaker than the algebraic counter-factual, I-F sign reversal, and 34.9% paraconsistency 𝑇 + 𝐹 > 1 which interval-derived data cannot reach); a head-to-head benchmark on ten causal hypotheses where neutrosophic zone classification reaches 90% accuracy against expert ground truth; a UCI benchmark on five datasets with Consensuszone accuracy 0.973–1.000; an extended evaluation across fourteen UCI datasets and five base classifiers with Friedman 𝜒2 = 44.55, 𝑝 < 10−8; and layer-selection diagnostics. The thirdanswer library provides an open-source implementation. The framework supplies principled tools for multi-criteria decision analysis and uncertainty-aware evaluation.