Is the Range of Set S Greater than its Mean? GMAT Data Sufficiency

Question: Is the range of set S greater than its mean?

1. All elements of set are negative.
2. The median of set S is negative.

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“Is the range of set S greater than its mean”– is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken f0rom the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation

Approach Solution 1:

There is only one solution approach to this problem.

Notice that the range of ANY set is more than or equal to zero.

(1) All elements of the set S are negative. The mean of a set with all negative elements is certainly negative so less than its range (which as discussed is always non-negative)
Hence, this statement is sufficient.

(2) The median of set S is negative. So, there is atleast one negative term is the set. Now, consider two cases:

1. If all elements in the set S are negative then we have the same scenario as above
   So, Range > Mean
2. If not all elements in set S are negative then Range = Largest – Smallest, which will mean that Range > Largest elements (that’s because the smallest element in the set S is negative. For example consider the following set {-1,-1,2}: the range of that set is Range = 2 – (-1) = 3 > 2).
   For the same reason the mean will be less than the largest element, so Range > Largest > Mean.

So, in any case, Range > Mean
Hence this statement is sufficient.

Correct Answer: D

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