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s is the standard deviation of 3 real numbers x, y, and z

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by Max@Math Revolution » Wed Apr 08, 2020 5:41 pm
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[GMAT math practice question]

s is the standard deviation of 3 real numbers x, y, and z, where a = x - y, b = y – z and c = z - x. What is the expression of s^2 in terms of a, b, and c?

A. a^3 + b^3 + c^3
B.(abc)^2
C. 0
D. a^2 + b^2 + c^2
E. 1/9(a^2 + b^2 + c^2)
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Re: s is the standard deviation of 3 real numbers x, y, and z

by Max@Math Revolution » Fri Apr 10, 2020 4:06 pm
=>

Assume m = (1/3)(x + y + z), which is the average of x, y, and z.
x – m = x – (1/3)(x + y + z) = (1/3)(3x – x – y - z) = (1/3)(2x – y - z) = (1/3)(x – y - (z - x)) = (1/3)(a - c).
We have y – m = (1/3)(b - a) and z – m = (1/3)(c - b) similarly.

Then, we have a + b + c = (x - y) + (y - z) + (z - x) = 0.

s^2 = (1/3)[(x - m)^2 + (y - m)^2 + (z - m)^2]
= (1/3)(1/3)[(a - c)^2 + (b - a)^2 + (c - b)^2]
= (1/27)[(a - c)^2 + (b - a)^2 + (c - b)^2]
= (1/27)[(a – c)(a – c) + (b – a)(b -a ) + (c – b)(c – b)]
= (1/27)(a^2 – 2ac + c^2 + b^2 – 2ab + a^2 + c^2 – 2bc + b^2)
= (1/27)[2(a^2 + b^2 + c^2) - 2(ab + bc + ca)]
= (1/27)[3(a^2 + b^2 + c^2) - (a + b + c)^2]
= (1/27)[3(a^2 + b^2 + c^2)], since a + b + c = 0
= (1/9)(a^2 + b^2 + ^c2)

Therefore, E is the answer.
Answer: E
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