How do i scale my errorbars when i scale my data?

Question

I am plotting distributions of data with the standard deviation and median of my data. Now when i want to scale my median by a another variable, how do i need to modify the standart deviation?

Answer 1

Let $X$ be some real-value random variable and $m$ be its median: $$ \mathsf P\{X\leq m\} = \mathsf P\{X>m\}. $$ Clearly, to scale median by the factor $\lambda> 0$ you just scale $X$ by the same factor since $$ \mathsf P\{\lambda X\leq \lambda m\} = \mathsf P\{\lambda X>\lambda m\}. $$ Note that although for the variance we have$\mathsf V[\lambda X] = \lambda^2 \mathsf V[X]$, the standard deviation scales with the same factor $\lambda$ being the square root of the variance: $$ \sigma[\lambda X] = |\lambda| \sigma[X]. $$