Data Sufficiency

VyDinh wrote:x, 3, 1, 12,8

If x is an integer, is the median of the 5 numbers shown greater than the average (arithmetic mean) of the 5 numbers?

1. x > 6
2. x is greater than the median of the 5 numbers

Target question: Is the median greater than the average?

Given: We have the set {1, 3, 8, 12, x}
Notice that there are 3 possible scenarios we need to consider:
Scenario #1: x is less than 3, in which case the median is 3
Scenario #2: 3 < x < 8, in which case the median is x
Scenario #3: x is greater than 8, in which case the median is 8

The average of this set will be (24+x)/5.

Okay, now onto the statements

Statement 1: x > 6
This rules our scenario #1, but we must still consider scenarios #2 and #3
Here are two possible values of x that yield conflicting answers to the target question:
Case a: x = 7 (median = 7 and average = 31/5), in which case, the median is greater than the average
Case b: x = 21 (median = 8 and average = 9), in which case, the median is NOT greater than the average
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x is greater than the median of the 5 numbers
This rules our scenarios #1 and #2, which leaves scenario #3
Here are two possible values of x that yield conflicting answers to the target question:
Case a: x = 11 (median = 8 and average = 7), in which case, the median is greater than the average
Case b: x = 21 (median = 8 and average = 9), in which case, the median is NOT greater than the average
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that the x-values we used to show that statement 2 is not sufficient ALSO satisfy statement 1. So, we know immediately that the combined statements are NOT SUFFICIENT.
To see what I mean, here are the two conflicting cases:
Case a: x = 11 (median = 8 and average = 7), in which case, the median is greater than the average
Case b: x = 21 (median = 8 and average = 9), in which case, the median is NOT greater than the average
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT.

Answer = E

Cheers,
Brent