Question

A class has 3 sections. Here is a summary of the scores on the final exam.

Answer 1

The SD of scores of all the students in the class

a) is equal to 9.
b) is less than 9.
c) is greater than 9.
d) cannot be placed relative to 9 based on the information given.

Answer 2

@ZeHanz

Answer 3

Think of the score classes as stochasts \(X_1\), \(X_2\) and \(X_3\).
For stochasts, the following rules apply:

If \(Z=aX+bY\), then \(E(Z)=aE(X)+bE(Y)\),

\(VAR(Z)=aVAR(X)+bVAR(Y)\)

Here, \(VAR(X)=SD^2(X)\).

If we want to calculate the mean and the standard deviation of all the students, we do this:

\(Z=\frac{20}{77}X_1+\frac{25}{77}X_2+\frac{32}{77}X_3\).

Then \(E(Z)=\frac{20}{77}E(X_1)+\frac{25}{77}E(X_2)+\frac{32}{77}E(X_3)\)

So \(E(Z)=\frac{20}{77}\cdot68+\frac{25}{77}\cdot75+\frac{32}{77}\cdot60=66.948\)

And: \(VAR(Z)=\frac{20}{77}VAR(X_1)+\frac{25}{77}VAR(X_2)+\frac{32}{77}VAR(X_3)\).

Then \(VAR(Z)=\frac{20}{77}81+\frac{25}{77}81+\frac{32}{77}81=\frac{77}{77}\cdot 81=81\).

But then \(SD(Z)=\sqrt{VAR(Z)}=\sqrt{81}=9\).

The conlusion is: the standard deviation of the whole group is equal to 9.

Answer 4

I might be wrong, of course! But then I'd like to know why...