Let's denote the original set of observations as
(X={x1,x2,...,x20}). The variance of these observations
(Var(X)) is given as 5 . Variance is a measure of how spread out the values in a dataset are, and it is calculated as the average of the squared differences from the mean:
Var(X)=‌where
µ represents the mean of the original observations.
Now, if each observation is multiplied by 2 , we get a new set of observations
Y={2x1,2x2,...,2x20}. The new variance
(Var(Y)) can be calculated as follows:
Var(Y)=‌‌(2xi−µ′)2where
µ′ is the mean of the new observations. Since the mean of the new observations would also be twice the mean of the original observations (because each term in the sum is doubled), we have
µ′=2µ.
Now let's substitute and note that
(2xi−2µ)2=4(xi−µ)2, which gives us:
‌Var(Y)=‌‌4(xi−µ)2‌Var(Y)=4×‌‌(xi−µ)2‌Var(Y)=4×Var(X)Since we know
Var(X)=5, we can now calculate
Var(Y) :
Var(Y)=4×5=20Therefore, the correct answer is
Option D : 20