Hi, there. I'm happy to help with this.
Prompt: Is the median of the four different positive integers a, b, c, and d, more than d?
Keep in mind, the median of a four-item list is going to be the average of the 2nd and 3rd numbers when the numbers are in numerical order. For example, the median of {2,3,5,8} is (3+5)/2 = 4. Also, incidentally, the median of {8,3,5,2} is also 4 --- changing the order doesn't change the median. To calculate the median you must put them all in numerical order.
Statement #1: The average (arithmetic mean) of a, b, c, and d, is d.
Well, in general, there's no guarantee what the size of the mean will be compared to the median. For sets in which all items are the same, or sets in which items are symmetrically distributed, the mean = median. If there are outliers one way or the other,
outliers pull the mean away from the median.
So, in set {3, 3, 3, 3}, d = 3, the mean = 3, and the median = 3, so the median is not greater than d.
But, in set {4, 13, 13, 10}, d = 10, mean = 10, median = 11.5, median is greater than d.
(NOTE: The low outlier of 4 pulls the mean below the median)
So, consistent with the conditions, we can answer the question either way. Statement #1 is
insufficient.
Statement #2: The average (arithmetic mean) of a, b, and c, is d.
So, now, the average of {a, b, c} is d, so when we add d to make a four-element set, that also will have an average of d. When you add the mean of a set as a new element in the set, the mean doesn't change. So, curiously, Statement #2 contains the information in Statement #1 --- it's not a "different" statement, mathematically.
Again, take set ={3,3,3}, mean = 3, add d = 3 to make {3,3,3,3}, median = 3 is not greater than d = 3
But take set = {4,13,13}, mean = 10, add d = 10 to make [4,13,13,10}, median = 11.5 is greater than d = 10.
Two answers possible, statement #2 is
insufficient.
Since Statement #2 contains Statement #1, nothing new is added when we combine the statements. Combined statements are also
insufficient.
Answer =
E
I'm not familiar with this problem source, but I will say that I've never seen a real GMAT DS question in which one statement can be completely deduced from the other statement. For that reason, I am a bit suspicious of this problem source.
Did everything I said in the solution make sense? Please let me know if I can clarify anything.
Here's a more typical GMAT DS question concerning median:
https://gmat.magoosh.com/questions/938
The question at that link should be followed by a video explanation of the answer.
Mike
