# 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

$x \mapsto \frac{x-l}{s}$

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:

\begin{split}\begin{align} X(l,s) &= l + sX(0,1)\\ \mathtt{DistributionDensity}(x;l,s) &= \frac{1}{s} \mathtt{DistributionDensity}(\frac{x-l}{s};0,1)\\ \mathtt{DistributionInversDensity}(\alpha;l,s) &= s \cdot \mathtt{DistributionInversDensity}(\alpha;0,1)\\ \mathtt{DistributionCumulative}(x;l,s) &= \mathtt{DistributionCumulative(\frac{x-l}{s};0,1)}\\ \mathtt{DistributionInverseCumulative}(\alpha;l,s) &= l + s \cdot \mathtt{DistributionInverseCumulative}(\alpha;0,1)\\ \mathtt{Mean}(X(l,s)) &= l + s \cdot \mathtt{Mean}(X(0,1))\\ \mathtt{Median}(X(l,s)) &= l + s \cdot \mathtt{Median}(X(0,1))\\ \mathtt{Deviation}(X(l,s)) &= s \cdot \mathtt{Deviation}(X(0,1))\\ \mathtt{DistributionVariance}(X(l,s)) &= s^2 \cdot \mathtt{DistributionVariance}(X(0,1))\\ \mathtt{Skewness}(X(l,s)) &= \mathtt{Skewness}(X(0,1))\\ \mathtt{Kurtosis}(X(l,s)) &= \mathtt{Kurtosis}(X(0,1))\\ \mathtt{(Rank)Correlation}(X(l,s),Y) &= \mathtt{(Rank)Correlation}(X(0,1),Y) \end{align}\end{split}

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

$M(X(l,s)) = s \cdot M(X(l,1))$

but in general

$M(X(l,s)) \pmb{\neq} l+ M(X(0,s))$

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.