Question

Consider three observations a, b, and c such that \(b=a+c\). If the standard deviation of \(a+2,\ b+2,\ c+2\) is \(d\), then which of the following is true?

• A. \(b^2=3(a^2+c^2)+9d^2\)
• B. \(b^2=3(a^2+c^2)-9d^2\)
• C. \(b^2=3(a^2+c^2+d^2)\)
• D. \(b^2=a^2+c^2+3d^2\)

Hint:

Adding 2 to every observation does not change the standard deviation, so \(d^2=\frac{a^2+b^2+c^2}{3}-\left(\frac{a+b+c}{3}\right)^2\) with \(a+c=b\).

Answer 1

The Correct Option is B

Standard deviation is unchanged by shifting, so the variance of \(a,b,c\) is \(d^2\).
Mean \(=\frac{a+b+c}{3}=\frac{2b}{3}\) (because \(a+c=b\)).
\(d^2=\frac{a^2+b^2+c^2}{3}-\frac{4b^2}{9}\Rightarrow9d^2=3(a^2+c^2)+3b^2-4b^2\).
So \(b^2=3(a^2+c^2)-9d^2\).
Answer: (2) \(b^2=3(a^2+c^2)-9d^2\)