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In this paper we introduce randomized $t$-type statistics that will bereferred to as randomized pivots. We show that these randomized pivots yieldcentral limit theorems with a significantly smaller magnitude of error ascompared to that of their classical counterparts under the same conditions.This constitutes a desirable result when a relatively small number of data isavailable. When a data set is too big to be processed, we use our randomizedpivots to make inference about the mean based on significantly smallersub-samples. The approach taken is shown to relate naturally to estimatingdistributions of both small and big data sets.

Inference from Small and Big Data Sets with Error Rates