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

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Answer 1 · 2016-12-13T21:17:07

Yes it is. See explanation for a proof.

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