Maßparameter

Maßparameter, often translated as scale parameter, is a parameter that governs the spread or size of a probability distribution or a measured quantity. It is distinguished from location parameters, which shift a distribution, and from shape parameters, which alter its form.

In many families of distributions a random variable X can be expressed as X = μ + σ Z, where μ is the location parameter and σ > 0 is the Maßparameter. Here σ controls dispersion: larger σ produces a wider spread.

Common examples include the normal distribution, where μ and σ are the mean and standard deviation (the scale parameter); the exponential distribution, where the scale is 1/λ; and the Gamma or Weibull distributions, which have explicit scale parameters named θ or λ.

Estimation of a Maßparameter is typically done by maximum likelihood or method of moments. Under affine transformations, the scale parameter behaves predictably: if Y = cX, then the scale parameter of Y is c times that of X. This property enables standardization and comparison across data sets.

Applications appear in statistics, measurement theory, and engineering, where the parameter quantifies dispersion, precision, or resolution of measurements, as well as in probabilistic modeling and data normalization.