I am looking for a method to transform my dataset from its current mean and standard deviation to a target mean and a target standard deviation. Basically, I want to shrink/expand the dispersion and scale all numbers to a mean.
It doesn't work to do two separate linear transformations, one for standard deviation, and then one for mean. What method should I use?
Suppose you start $\{x_i\}$ with mean $m_1$ and non-zero standard deviation $s_1$ and you want to arrive at a similar set with mean $m_2$ and standard deviation $s_2$.
Then multiplying all your values by $\frac{s_2}{s_1}$ will give a set with mean $m_1 \times \frac{s_2}{s_1}$ and standard deviation $s_2$.
Now adding $m_2 - m_1 \times \frac{s_2}{s_1}$ will give a set with mean $m_2$ and standard deviation $s_2$.
So a new set $\{y_i\}$ with $$y_i= m_2+ (x_i- m_1) \times \frac{s_2}{s_1} $$ has mean $m_2$ and standard deviation $s_2$.
You would get the same result with the three steps: translate the mean to $0$, scale to the desired standard deviation; translate to the desired mean.
Let’s consider the z-score calculation of data $x_i$ with mean $\bar{x}$ and standard deviation $s_x$.
$$z_i = \dfrac{x_i-\bar{x}}{s_x}$$
This means that, given some data $(x_i)$, we can transform to data with a mean of $0$ and standard deviation of $1$.
Rearranging, we get:
$$x_i = z_i s_x+ \bar{x}$$
This gives us back our original data with the original mean $\bar{x}$ and standard deviation $s_x$. But we could’ve gone to data $y_i$ with any mean $\bar{y}$ and standard deviation $s_y$.
$$y_i = z_i s_y +\bar{y}$$
Now combine the two transformations, first to $z_i$ and then to $y_i$.
$$y_i = \dfrac{x_i-\bar{x}}{s_x}s_y + \bar{y}$$
This is the same as what Henry posted, but I do think it is helpful to see that we get there by first going to standardized data and then transforming to data with the mean and standard deviation values we desire.
