Affine invariance of kurtosis ($\kappa (aX+b)=\kappa (X)(a\ne 0)$)

Changing units or shifting the origin preserves kurtosis. For real constants $a\ne 0$ 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.