Q.If $SD(X) = 3$, find $SD(4X+5)$.

The standard deviation under a linear transformation scales by the absolute value of the multiplying constant only; the added constant has no effect. …

Since $Var(aX+b) = a^2 Var(X)$, taking square roots on both sides gives the corresponding property for standard deviation:

Here $a=4$, $b=5$, and $SD(X)=3$, so:

$SD(4X+5) = |4|\times3 = 4\times3 = 12$ …

The variance route shown above as the cross-check (square $SD(X)$ to get $Var(X)$, apply $Var(aX+b)=a^2Var(X)$, then take the square root again) is an equally valid primary me …

A frequent error is forgetting to square the multiplier when working through variance but then also forgetting to take the square root again to return to standard deviation — or, working directly with standard deviation, mistakenly squaring $a$ instead of usin …