Uniting Psychometric Modelling and Poisson Distributions: A Metrological Study of Elementary Counting

A study of elementary counting (of simple clouds of dots by the Munduruku indigenous people of Brazil) is reanalysed in order to compare and contrast three kinds of probability mass functions (PMFs): (i) quantitative response to a discrete range of counts; (ii) the classic Poisson distribution of miscounts; and (iii) psychometric (Rasch) distributions of counting task difficulty and person counting ability. This reanalysis highlights how best to handle PMFs which provide a means of defining—for discrete and qualitative data—the basic metrics, viz. location and dispersion, of metrology—quality-assured measurement, as increasingly required since the turn of the millennium in topical and challenging quality-assurance applications, amongst others, in the human sciences and in Artificial Intelligence. PMF-based metrics, useful in ‘clinical’ and other applications where meaning and value are sought, complement the traditionally dominating role played by the corresponding probability density functions (PDF) in ‘analytical’, quantitative and continuous Metrology in Physics. New insights are provided when benchmarking the Rasch Poisson Counts Model, which has received less attention in modern metrology, against full psychometric Rasch modelling.