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What is the standard deviation of a1, a2, a3, …, a100?

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What is the standard deviation of a1, a2, a3, …, a100?

by Max@Math Revolution » Thu Mar 12, 2020 12:10 am

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[GMAT math practice question]

What is the standard deviation of a1, a2, a3, …, a100?

1) The minimum of (x - a1)^2 + (x - a2)^2 + (x - a3)^2+…+(x - a100)^2 is 16.
2) The average of a1, a2, a3, …, a100 is 0.
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Source: — Data Sufficiency |
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Re: What is the standard deviation of a1, a2, a3, …, a100?

by Max@Math Revolution » Mon Mar 16, 2020 2:51 am

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Forget conventional ways of solving math questions. F or DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

The expression (x - a1)^2 + (x - a2)^2 + (x - a3)^2+…+(x - a100)^2 has a minimum when x is the average of a1, a2, …, a100 from condition 1).
The standard deviation of a1, a2, …, a100 is the square root of {(x - a1)^2 + (x - a2)^2 + (x - a3)^2+…+(x - a100)^2} / 100 where x is the average.
Thus, their standard deviation is √(16/100) = 4/10.

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
Since we don’t know their distribution, condition 2) does not yield a unique solution, and it is not sufficient.

Therefore, A is the answer.
Answer: A
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