Question

An unknown data set has a standard deviation of 10. What is the standard deviation when each value in the data set is multiplied by 5?

Answer 1

Let $\mathbf x=\{x_i~:~1\le i\le n\}$ be the sample, with $n$ the size of the sample.

The mean of the sample is

$\displaystyle\bar x=\frac1n\sum_{i=1}^nx_i$

Let $\mathbf y$ be the same data set but with each value scaled up by 5. Then the mean here is

$\bar y=\displaystyle\frac1n\sum_{i=1}^ny_i=\frac5n\sum_{i=1}^nx_i=5\bar x$

The standard deviation for $\mathbf x$ is given by

$s_{\mathbf x}=\sqrt{\displaystyle\frac1{n-1}\sum_{i=1}^n(\bar x-x_i)^2}$

For $\mathbf y$, you would have

$s_{\mathbf y}=\sqrt{\displaystyle\frac1{n-1}\sum_{i=1}^n(\bar y-y_i)^2}$
$s_{\mathbf y}=\sqrt{\displaystyle\frac1{n-1}\sum_{i=1}^n(5\bar x-5x_i)^2}$
$s_{\mathbf y}=\sqrt{\displaystyle\frac{25}{n-1}\sum_{i=1}^n(\bar x-x_i)^2}$
$s_{\mathbf y}=5\sqrt{\displaystyle\frac1{n-1}\sum_{i=1}^n(\bar x-x_i)^2}$
$s_{\mathbf y}=5s_{\mathbf x}$