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The Normal Distribution Derived from Qualitative Conditions.

Reinhard Suck¹

¹University of Osnabrück

Journal of Mathematical Psychology

|April 17, 2001

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This study characterizes the normal distribution using measurement theory, deriving it from conjoint measurement axioms and showing error distribution as a consequence of these assumptions for statistical models.

Area of Science:

Measurement theory
Statistical modeling
Probability distributions

Background:

The normal distribution is fundamental in statistics.
Existing characterizations often rely on quantitative properties or limit theorems.
A measurement-theoretic framework offers a novel perspective.

Purpose of the Study:

To characterize the normal distribution within a measurement theoretic framework.
To derive the normal distribution from qualitative axioms.
To explore the implications for statistical models.

Main Methods:

Formulating properties of a linear model as a component structure.
Applying conjoint measurement axioms.
Utilizing qualitative conditions to define representations as random variables.

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Main Results:

Qualitative conditions guarantee representations as random variables.
Additional axioms yield the normal distribution.
Normal distribution of errors is a consequence of measurement theoretic assumptions.

Conclusions:

The normal distribution can be derived from fundamental measurement theoretic principles.
This approach provides a new foundation for understanding the normal distribution.
Results have potential impact on the formulation of statistical models.