Question

Can you think of a situation in which the population standard deviation remained constant while the mean changed? Describe such a situation as best you can, explaining why the standard deviation would not change although the mean would.

Answer 1

If we add(or subtract) the same amount to each of the observations then the mean of the observations will change but the standard deviation will not change.

consider a population variable X under study

if Y=X+a

then E(Y)=E(X)+a

and Var(Y)=Var(X)

where E() denotes expectation and Var() denotes variance.

standard deviation is denoted by sqrt(Var)

in this situation the satdard deviation will not change since standard deviation is a measure of dispersion whcih measures how scattered or dispersed the observations are from the mean.

[ we can proof this mathematically :

V(X)=E(X²)-[{E(X)}²]=E(X²)- mu², where mu=E(X)

E(Y)=E(X)+a=mu+a

V(Y)=V(X+a)=E{(X+a)²}-{E(X+a)}²=E{(X+a)²}- (mu+a)²=E(X²+2aX+a²)-(mu²+2a*mu+a²)

=E(X²)+2aE(X)+a²-mu²-2a*mu-a²= E(X²)+2a*mu+a²-mu²-2a*mu-a²= E(X²)-mu²= V(X)

we know that for a constant "a",E(a)=a,E(a²)=a²

^]

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