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

1 Answer
Dec 13, 2016

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