page_title: Scale parameter
section_title: Definition
hierarchical_section_title: Scale parameter / Definition
caption_attribution_description: English: Animation showing the effects of a scale parameter on a probability distribution supported on the positive real line. Note that, unlike the real-line supported distribution, the mean is modified by the scale. A higher scale spreads out the distribution, lowering its peaks and raising the tails and valleys.
context_page_description: In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution.
context_section_description: If a family of probability distributions is such that there is a parameter s (and other parameters θ) for which the cumulative distribution function satisfies
  then s is called a scale parameter, since its value determines the "scale" or statistical dispersion of the probability distribution.  If s is large, then the distribution will be more spread out; if s is small then it will be more concentrated.
  If the probability density exists for all values of the complete parameter set, then the density (as a function of the scale parameter only) satisfies
  where f is the density of a standardized version of the density, i.e. .
  An estimator of a scale parameter is called an estimator of scale.
caption: Animation showing the effects of a scale parameter on a probability distribution supported on the positive real line.