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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?

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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]. $

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    @tarrasch: yes, it is the same since it also scales linearly2012-03-23