= Affine invariance of kurtosis
{title2=$\kappa(aX+b)=\kappa(X)\quad(a\ne0)$}
Changing units or shifting the origin preserves <kurtosis>. For real constants $a\ne0$ and $b$, the <mean> of $Y=aX+b$ is $a\mu+b$, its <variance> is $a^2\sigma^2$, and its fourth <central moment> is $a^4\mathbb E[(X-\mu)^4]$. The factors $a^4$ cancel in the ratio, even for negative $a$. Hence <excess kurtosis> is also unchanged. For example, $100X+7$ and $X$ have the same <kurtosis>, although their <means> and <variances> differ. The case $a=0$ produces zero <variance> and is excluded.
Back to article page
