S is a set with at least two numbers. Is the range of S greater than

S is a set with at least two numbers. Is the range of S greater than its arithmetic mean?

(1) The median of the set S is negative
(2) The mean and the median are equal

The answer should be A.
If median of a set S is negative, then range is larger than the largest no. of the set and mean can never be greater than the largest no.(+ve or otherwise).

S is a set with at least two numbers. Is the range of S greater than its arithmetic mean?

(1) The median of the set S is negative
As range would always be positive, it would be greater than a negative median for all cases when number of elements in the set S is 2 or more.

Case I: -3, -2
Mean = \(\frac{-3 -2}{2} = \frac{-5}{2}\)
Range = -2 + 3 = 1
YES

Case II: -3, -2, 1
Mean = -2
Range = 1 + 3 = 4
YES

SUFFICIENT.

(2) The mean and the median are equal
Mean and Median of the set S can either be positive or negative independent of number of elements. As we have checked for negative median(YES cases), lets check for positive cases.

Case I: 2, 3
Mean = Median = \(\frac{3 + 2}{2} = \frac{5}{2}\)
Range = 3 - 2 = 1
NO

Since both YES and NO cases exist, INSUFFICIENT.

Answer A.

Given that S is a set with at least two members. We are to determine if the range of S is greater than the arithmetic mean.

(1) The median of the set S is negative
Statement 1 is sufficient. This is because we can have two different sets A: -7, -6, -5 and B: -7, -3, 100
Both A and B have medians that are negative, (i.e. -6, and -3) respectively. While the mean of A is -6, the mean of B is 30. The range of A is 2 while that of B is 107. Clearly the range of any set S that has a negative median will have a range greater than the mean. The mean of such a set that satisfies Statement 1 will have a mean that is either negative, in which case the range (always a positive number) is greater than the mean, or a positive number indicating that such a set has members that are widespread, and the range will be more than the mean.

Statement 2: The mean and the median are equal
Clearly statement 2 is insufficient. We can have two sets A:-7,-6,-5 and B: 5,6,7
While A has its median and mean =-6 and a range of 2, implying the range of A is greater than the mean, the median and mean of B =6 but the range of B is 2, which is less than the mean.

The answer is A.

(1) The median of the set S is negative sufic

range of a set is ≥0; if the median is a negative, then the range>mean.

(2) The mean and the median are equal insufic

set [0,0]: md,mean=0; rng=0
set [0,1,2]: md,mean=1; rng=2

Ans (A)