Animated Solution for Mathematics - Statistics: 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?

We are given three observations: $a$, $b$, and $c$, subject to the constraint $b=a+c$. We are also given that the standard deviation of the set $\{a+2,b+2,c+2\}$ is $d$.

The first step is to recognize a fundamental property of statistics: standard deviation is a measure of dispersion, not location. Shifting every data point by a constant does not change the distance between the points.

Therefore, the standard deviation of $\{a+2,b+2,c+2\}$ is identical to the standard deviation of $\{a,b,c\}$. We have effectively simplified the problem to finding the relationship between $a,b,c$, and $d$ for the original set.

Let the mean of the set be $\bar{x}=\frac{a+b+c}{3}$. Using our constraint $b=a+c$, we substitute $a+c$ with $b$ in the numerator.

The mean simplifies to:

We now invoke the variance formula, where the variance is the square of the standard deviation ($d^2$):

Substituting our specific values into this formula, we obtain:

Expanding the square term, we get:

To clear the denominators, we multiply the entire equation by $9$:

Expanding the bracket yields:

Combining the $b^2$ terms, we arrive at:

Rearranging to solve for $b^2$, we find the final relationship:

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