Scale parameter

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.

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.

In the case where a parametrized family has a location parameter, a slightly different definition is often used as follows.

[1] This modification is necessary in order for the standard deviation of a non-central Gaussian to be a scale parameter, since otherwise the mean would change when we rescale

So for example the exponential distribution with scale parameter β and probability density could equivalently be written with rate parameter λ as A statistic can be used to estimate a scale parameter so long as it: Various measures of statistical dispersion satisfy these.

In order to make the statistic a consistent estimator for the scale parameter, one must in general multiply the statistic by a constant scale factor.

Note that the scale factor depends on the distribution in question.

For instance, in order to use the median absolute deviation (MAD) to estimate the standard deviation of the normal distribution, one must multiply it by the factor where Φ−1 is the quantile function (inverse of the cumulative distribution function) for the standard normal distribution.

Different factors would be required to estimate the standard deviation if the population did not follow a normal distribution.