# Scaling of Statistical Operators

Transforming distributions

Shifting and scaling distribution has an effect on the distribution operators, and on sample operators when the samples are from a specified distribution. Location and scale parameters find their origin in a common transformation

to shift and stretch a given distribution. By choosing \(l=0\) and \(s=1\) one obtains the standard form of a given distribution, and the relation of operators working on the general and standard form of distributions is as follows:

Transformation of the mean

The transformation formula for the `Mean`

holds for both the
`DistributionMean`

and the `Mean`

of a sample. However, for the
`GeometricMean`

, the `HarmonicMean`

and the `RootMeanSquare`

, only
the scale factor can be propagated easily during the transformation.
Thus, for a sample taken from a distribution \(X\) and a any mean
operator \(M\) from the `GeometricMean`

, `HarmonicMean`

or
`RootMeanSquare`

, it holds that

but in general

Transformation of the other moments

The transformation formula for the deviation is valid for the
`DistributionDeviation`

, the `SampleDeviation`

and
`PopulationDeviation`

, while the transformation formulae for the
`Skewness`

and `Kurtosis`

hold for both the distribution and sample
operators.