On the distribution of measurements in units that are not arbitrary

Measurement achieved in the physical sciences as some kind of mapping on a continuum partitioned into equal units from an origin is understood readily. However, measurement is also an advanced concept, the achievement of which has been integral to the remarkable progress of physical science. This article specializes the probabilistic Rasch model for a finite number of ordered response categories to a response structure which is analogous to measurement - the parameters are expressed in the multiplicative metric giving a natural origin of O, the thresholds partitioning the continuum are made equidistant to correspond to a unit of measurement, and the number of thresholds is made unlimited. The resultant model is the Poisson whose parameter is the ratio of the size of the object of measurement to the size of the unit of measurement of the instrument. It is shown that a number of features of measurement are characterized and unified by this distribution, suggesting that it is a candidate for an idealization of the distribution of replicated measurements. However, this unification arises only if a distinction is made between an arbitrary unit which partitions a continuum, and a non-arbitrary unit which is integral to an instrument. The unit integral to the instrument characterizes the distribution of potential locations on the continuum, and different instruments give different potential locations. As this unit becomes smaller, the variance of replicated measurements relative to the arbitrary unit becomes smaller, showing that, as expected, the precision of measurement increases. However, the variance of measurement in the unit integral to the instrument then becomes greater, showing that, perhaps unexpectedly, there is less consistency in measurements in a smaller unit that gives greater precision than in a larger unit that gives less precision.