Question

Suppose a class of students have an average SAT math score of 720 and average verbal score of 640. The standard deviation for each part is 100. If possible, find the standard deviation of the composite score. If it is not possible, explain why.?

(Area of statistics: Combining and Transforming Random Variables)

Answer 1

`141`

Explanation:

If `X` = the math score and `Y` = the verbal score,

`E(X) = 720` and `SD(X) = 100`

`E(Y) = 640` and `SD(Y) = 100`

You cannot add these standard deviations to find the standard deviation for the composite score; however, we can add variances. Variance is the square of standard deviation.

`var(X+Y) = var(X) + var(Y)`

`= SD^2(X) + SD^2(Y)`

`= 100^2 + 100^2`

`= 20000`

`var(X+Y) = 20000`, but since we want the standard deviation, simply take the square root of this number.

`SD(X+Y) = sqrt(var(X+Y)) = sqrt20000 ~~ 141`

Thus, the standard deviation of the composite score for students in the class is `141`.