Question

Is the standard deviation of a data set invariant to translation?

Answer 1

Yes it is. See explanation for a proof.

Explanation:

Let `S` be a data set:

`S={x_1,x_2,...,x_n}`
Its mean and standard deviation:

`bar(x)=1/nxxSigma_{i=1}^{i=n}(x_i)`

`sigma=sqrt(Sigma_{i=1}^{n}(x_i-bar(x))^2)`

Let `S_1` be a data set `S` translated by `a`:

`S_1={x_1+a,x_2+a,...,x_n+a}`

Its mean would equal:

`bar(x_1)=(x_1+a+x_2+a+...+x_n+a)/n=`
`=(x_1+x_2+...+x_n)/n+(na)/n=bar(x)+a`

The standard deviation would be:

`sigma_1=sqrt(Sigma_{i=1}^{n}(x_i+a-(bar(x)+a))^2)=`

`=sqrt(Sigma_{i=1}^{n}(x_i+a-bar(x)-a))^2)=`

`=sqrt(Sigma_{i=1}^{n}(x_icancel(+a)-bar(x) cancel(-a)))^2)=`

`=sqrt(Sigma_{i=1}^{n}(x_i-bar(x))^2)=sigma`

The standard deviation of the new set is equal to the deviation of the set before translation.

QED