IQ Tests And Scores Do Not Measure An Interval On A Continuum Of Intelligence
April 2, 2026 · Preprint
Psychometrics has alluded to implications of and represented IQ scores under a normal distribution, i.e. Gaussian or bell curve, without proof there is any quantitative data or sufficient statistics that are absolutely continuous with regard to a measure-theoretic continuum of presupposed latent g. We provide with statistical research elucidating the continuous formula for σ (standard deviation) has been used since at least the 1940's in lieu of the discrete version which is the appropriate one for the qualitative observations in IQ testing. Provided is a proof to show that the currently assumed continuity fails the Radon-Nikodym theorem which categorically renders the Gaussian for IQ invalid. A better interpretation of the Rasch model applicability is given with regard to its logit and probabilistic dichotomy with a classical example. Following is another proof supplying the argument that the test format fails the Kolmogorov axioms and definition of a σ-field. IQ is robustly shown to not be a measure-theoretic measure for the multiple item response test format (multiple choice) based on the current assumptions. This paper highlights historiographic connectionism errors between leaps about biometrics and twin studies with the misinterpretation of mathematical continuity as well as the absurdity of making analogies to Euclidian dimensional measurements for IQ. Its similarities to cult psychology are elucidated with passages and quotes.