Scaling normal distributions only using the data

Question

Suppose that you have the following data:

$$ X=\left\lbrack x_1,x_2,\ldots,x_n\right\rbrack $$ that is normal distributed N(0,1)

$$ X\sim N\left(0,1\right) $$

To change the mean you will have to add a number in every element:

$$ X+Y=Z $$

The mean-value will now be Y.

Question:

Is there any way to manipulate the data so that you will change the variance?

Answer 1

There are 2 important points in your question related to $X$, with $E(X)=0$ and $V(X)=1$, and $Y=\sigma X+\mu$:

1. irrespective to the distribution of $X$, $E(Y)=\mu$ and $V(Y)=\sigma^2$. This is valid for all distributions in which $E(X)=0$ and $V(X)=1$. And it is valid the other way around: if $Y$ is a random variable with any distribution in which $E(Y)=\mu$ and $V(Y)=\sigma^2$, and $X=\frac{Y-\mu}{\sigma}$, it is true that $E(X)=0$ and $V(X)=1$.
2. The probability distribution of $Y=\sigma X+\mu$ is Normal if $X$ is Normal. This is not an obvious fact. The preservation of the distribution under linear afine transformations is a property of the Normal that is not necessarily shared by other distributions.