The Rasch distribution: A discrete, general form of the Gauss distribution of uncertainty in scientific measurement. (March 2021) - Andrich, David
Graphical abstract: Figure 3. Equivalences among two discrete Rasch distributions ○, m = 16, Δ = 0.50, β - δ = ± 4 Δ = ± 2, E [ X ] = m / 2 ± 4 = 8 ± 4 = 4 or 12, V [ X ] = 1 / 0.5 = 2.0, two discrete Gauss distributions, and the interpolated Gauss density functions, μ = 4 or 12, σ 2 = 2.0 . Highlights: Is a function of the origin and unit of the instrument. Is a function of the range of the instrument. Is commensurate with the continuous Gauss distribution. Is a discrete general form of the Gauss distribution. Abstract: The derivation of a distribution of random errors of measurement culminated in the continuous quadratic exponential law of Gauss, who recognised that the distribution cannot represent a law of error in full rigor because it assigns probabilities greater than zero outside the range of the possible errors . This paper shows that the distribution of replicated measurements derived from Rasch's measurement theory of invariant comparisons also culminates in the quadratic exponential law, but one that is discrete and is a function of all relevant properties of the instrument, its origin, its unit and its finite range, and assigns probabilities only to possible measurements. This convergence of the Rasch and Gauss distributions suggests that the former is a general case of the latter, and that the Rasch distribution can represent a law of error in full rigor. An application of the distribution to equating two instruments from the social sciences is shown.