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The average and standard deviation of

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The average and standard deviation of

by Max@Math Revolution » Fri Mar 27, 2020 12:59 am

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[GMAT math practice question]

The average and standard deviation of x1, x2, x3, x4 is 5 and 2, respectively. What is the average of x1^2, x2^2, x3^2, x4^2?

A. 11
B. 17
C. 22
D. 27
E. 33
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Re: The average and standard deviation of

by Max@Math Revolution » Sun Mar 29, 2020 9:41 pm
=>

We have (x1+x2+x3+x4)/4 = 5 or x1+x2+x3+x4 = 20.
Since we know the standard deviation of x1, x2, x3, and x4 is 2, we have
((x1 - 5)^2 + (x2 - 5)^2 + (x3 - 5)^2 + (x4 - 5)^2)/4 = ( \(\sqrt{2}\) )^2
=> (x1 - 5)^2 + (x2 - 5)^2 + (x3 - 5)^2 + (x4 - 5)^2 = 8
=> (x1^2 - 10x1 + 25) + (x2^2 - 10x2 + 25) + (x3^2 - 10x3 + 25) + (x4^2 - 10x4 + 25) = 8
=> (x1^2 + x2^2 + x3^2 + x4^2) - 10(x1 + x2 + x3 + x4) + 25 + 25 + 25 + 25 = 8
=> (x1^2 + x2^2 + x3^2 + x4^2) - 10(20) + 100 = 8
=> (x1^2 + x2^2 + x3^2 + x4^2) = 108
=> (x1^2 + x2^2 + x3^2 + x4^2)/4 = 27

Therefore, D is the answer.
Answer: D
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