Standard poverty statistics treat households as either “poor” or “non-poor” and then attach a sampling variance after the fact. Fuzzy-set approaches, by contrast, allow partial deprivation but typically assume the survey data are error-free. We propose stochastic fuzzy indices, a single measure that embeds both partial membership and sampling randomness by integrating a fuzzy membership function \(\mu :X\!\rightarrow \![0,1]\) with respect to the survey probability measure \(P\) . An axiomatic analysis—non-negativity, dual monotonicity, decomposability, and continuity—shows that any index satisfying these properties admits a unique Choquet-type representation; the classical Foster–Greer–Thorbecke headcount, Cheli–Lemmi totally-fuzzy index, and Alkire–Foster \(M_0\) emerge as special cases. A stylised application to Tunisia illustrates how the index identifies borderline households that carry substantial sampling weight in interior regions, while a robustness check confirms stability under alternative membership functions and public aggregate data. Because the same object captures both partial deprivation and survey uncertainty, the stochastic fuzzy index offers an early-warning signal for vulnerable households and supports sharper geographic targeting without knife-edge jumps caused by minor data noise. We also benchmark SFI against crisp and fuzzy measures (Sect. 8) and illustrate its stability under small perturbations, together with more favourable budget-targeting properties in stylised policy exercises.
