Consider three observations a, b and c, such that b=a+c.

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?

Accepted Answer

With $x_1=a+2, x_2=b+2, x_3=c+2$, mean $\mu = \dfrac{2b}{3}+2$ (using $a+c=b$).

$$d^2 = \frac{(a-2b/3)^2+(b-2b/3)^2+(c-2b/3)^2}{3}$$

$$3d^2 = a^2+b^2+c^2+\frac{4b^2}{9}-\frac43[b(a+c)+b^2] = a^2+b^2+c^2+\frac43b^2-\frac43(2b^2)$$
$$3d^2 = a^2+c^2-\frac{b^2}{3}$$

$b^2 = 3a^2+3c^2-9d^2$

Solutions · 1