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

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:
$$
\sigma[\lambda X] = |\lambda| \sigma[X].
$$
