B Why is V(1/5X) equal to 1/25*V(X)?

[Addez123]

TL;DR Summary

I don't understand how V(1/5X) can be turned into 1/25*V(X).
Shouldn't I just extract the 1/5 so:
V(1/5X) = 1/5*V(X) ?

V stands for variance.

In my book, when calculating the variance of X = (x_1 + x_2 + x_3 + x_4 + x_5)/5
in an example it says:

V(X) = V(1/5(X_1 + X_2 + X_3 + X_4 + X_5)) = 1/25*V(X_1) + 1/25*V(X_2) + 1/25*V(X_3) + 1/25*V(X_4) + 1/25*V(X_5) = 1/5Ф

I don't understand how V(1/5X) can be turned into 1/25*V(X), shouldn't I just extract the 1/5 so:
V(1/5X) = 1/5*V(X) ?

 

[Delta2]

According to the properties of the variance it is $V(aX)=a^2V(X)$. This follows from the definition of variance and the properties of the mean E(X).
It is $V(X)=E(X^2)-E(X)^2$
and $$V(aX)=E((aX)^2)-E(aX)^2=E(a^2X^2)-a^2E(X)^2=a^2E(X^2)-a^2E(X)^2=a^2(E(X^2)-E(X)^2)=a^2V(X)$$

 

[PeroK]

Summary:: I don't understand how V(1/5X) can be turned into 1/25*V(X).
Shouldn't I just extract the 1/5 so:
V(1/5X) = 1/5*V(X) ?

V stands for variance.

In my book, when calculating the variance of X = (x_1 + x_2 + x_3 + x_4 + x_5)/5
in an example it says:

V(X) = V(1/5(X_1 + X_2 + X_3 + X_4 + X_5)) = 1/25*V(X_1) + 1/25*V(X_2) + 1/25*V(X_3) + 1/25*V(X_4) + 1/25*V(X_5) = 1/5Ф

I don't understand how V(1/5X) can be turned into 1/25*V(X), shouldn't I just extract the 1/5 so:
V(1/5X) = 1/5*V(X) ?

What's definition of variance?

 

[FactChecker]

Variance is based on values of $X^2$ and $(1/5*X)^2 = 1/25*X^2$.