Problem Solving

Dear GMAT aspirants,

I will design and post a Question frequently for all of you to solve. These questions will test important concepts and I hope they will help you in your preparation. Good luck!

Dear Experts/Others,

Please let me know if you have any feedback regarding the construction of the question and/or the options.
Thanks.
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PS # 1:

Which of the following must be true?
I. For a set of evenly spaced numbers, the mean is equal to the median.
II. A set of numbers for which the mean is equal to the median must be evenly spaced.
III. A set of numbers for which mean is equal to the median is uniformly distributed about the mean.

(A) I only
(B) I and II
(C) I and III
(D) I, II and III
(E) None of the above statements

Solution:

Lets start with the number '10'. Currently Mean = Median = 10
If two numbers are added to the set, such that
New set: {8, 10, 12} which is evenly spaced and has Median = 10
The sum of 8 and 12 is 20, which is same as adding two 10s to the set, so the Mean also does not change.
Mean = Median = 10
If two more numbers are included, such that
New set: {6, 8, 10, 12, 14}
The sum of the new numbers is again 20, so the mean will not change.
Mean = Median = 10

(I) must be true always.

Let's evaluate (II).

Let's take the evenly spaced numbers' set:
2, 4, 6, 8, 10
Without changing the median, if I add +1 to 2, and subtract -1 from 10, the Sum does not change. Since the number of terms is also the same as before, Mean = Median = 6
New set: 3, 4, 6, 8, 9
It is not evenly spaced anymore.

(II) is not true always.

Let's evaluate (III).
Let's take the same evenly spaced numbers' set:
2, 4, 6, 8, 10
Without changing the median, if I add +1 to 2, and subtract -1 from 8, the Sum does not change. Since the number of terms is also the same as before, Mean = Median = 6
New Set: 3, 4, 6, 7, 10
This set is not uniformly distributed because the first term and the last term has unequal distances from the mean.

(III) is not true always.

[spoiler](A)[/spoiler] is the correct option.

Please let me know if you have any doubts.