Multiple choice

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?

  1. b2 = 3(a2 + c2) + 9d2

  2. b2 = a2 + c2 + 3d2

  3. b2 = 3(a2 + c2 + d2)

  4. b2 = 3(a2 + c2) - 9d2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The standard deviation d of (a+2, b+2, c+2) is the same as the standard deviation of (a, b, c). The variance d^2 = [(a-m)^2 + (b-m)^2 + (c-m)^2] / 3, where m = (a+b+c)/3. Since b = a+c, m = (a+a+c+c)/3 = (a+c)/3 = b/3. Substituting m = b/3 into the variance formula: 3d^2 = (a - b/3)^2 + (b - b/3)^2 + (c - b/3)^2 = (a - b/3)^2 + (2b/3)^2 + (c - b/3)^2. Expanding this leads to the relation b^2 = 3(a^2 + c^2) - 9d^2.

AI explanation

Let the new observations be x, y, and z, representing a + 2, b + 2, and c + 2 respectively, which have a standard deviation of d. Using the direct variance formula, the variance of these three values is x squared plus y squared plus z squared, all divided by 3, minus the square of their mean. Because the standard deviation is unaffected by adding 2, the standard deviation of a, b, and c is also d, leading to an algebraic expansion where b squared equals 3 times the quantity a squared plus c squared minus 9 times d squared.

Keep practicing — related questions