Set P has n integers. What is the standard deviation of Set P?

Solution:

We need to determine the standard deviation of set P.

Statement One Alone:

The range of Set P is equal to zero

Since the range of set P is zero, all the integers in set P are the same. Thus, the standard deviation is zero. Statement one alone is sufficient.

Statement Two Alone:

The mean of Set P is equal to the median of Set P

We could have Set P = {5, 5, 5, 5, 5, } or {1, 1, 5, 9, 9}. Although the mean and median both equal 5 in each set, the standard deviation of the first example is 0, and the standard deviation of the second example is greater than 0. Thus, we cannot determine a unique value for the standard deviation of Set P. Statement two alone is not sufficient.

Answer: A

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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.

Condition 1) is sufficient for the following reason.
The five statements are equivalent each other.
1. The standard deviation of P is 0.
2. All data of P are equal.
3. The range of P is 0
4. The maximum of P is equal to the average of P.
5. The minimum of P is equal to the average of P.
Since condition 1) is the case 3, P has the standard deviation 0.

Condition 2)
{1,2,3} and {2,4,6} have different standard deviations, even thought each of them has its average equal to it median, respectively.
Since it does not yield a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.

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