# Transformation to reduce standard deviation without changing median

Consider some negative skew and high kurtosis return time-series $X_t$. I do not know the functional form of the pdf of $X_t$ and have about 150,000 data points.

Suppose that I was to create an adjusted $X_t$ called $X_t^a$ where the mean, median, skew and kurtosis remain approximately the same as the original series, and the only main difference is that $\sigma (X_t) = 2*\sigma (X_t^a)$.

What's an appropriate transformation? I've tried:

$$X_t^a := (X_t - E[X_t])*0.5\frac{\sigma(X_t)}{\sigma(X_t)} + E[X_t] \\ = Z*0.5\sigma(X_t) + E[X_t] \sim N(E[X_t],(0.5\sigma(X_t))^2)$$ $$Z \sim N(0,1)$$

but this severely reduces the positive median towards zero which is not what I want (I want to keep $\text{median}(X_t)$ approximately equal to $\text{median}(X_t^a)$).