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(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)`