I. consider T (1, 2, 3) median=mean=2. Consider T (5, 9, 2, 19, 10) median=mean=9
II. Consider T (1, 2, 3) median=mean=2. Consider T (1, 1, 1, 1) median=mean=1
III. Consider T (1, 2, 3) mean=median=2 and there is no mode. Consider T (1, 1, 1) median=mean=1 and mode is 1.

Answer E.
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VERITAS PREP OFFICIAL SOLUTION

Correct Answer: E

None of the proposed statements must be true; to prove this, you might consider a hypothetical: Set T could be [1,1, 3, 3], in which the average and median are both 2. This set proves I, II, and III wrong. (Note: This set has two modes, 1 and 3)
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This is a very good 'trap question'.

Consider Set A= { 0,0,0,0 } Mean=Median=Mode=0- Eliminates 1,2 and 3 OR
Consider Set B= {1,2,2,3 } Mean=Median=Mode=2- Eliminates 1,2 and 3

If you think that only in 'consecutive integers' the average is equal to the median you might fall for 'D'.

Ans: E

This is an Excellent Question.
Here is my take on this one =>


Firstly,

Quote:

For any evenly spaced set => Mean = Median = Average of the first and the last term.

Also =>

Quote:

Sum of deviations around the mean = Zero

Now We are told that a Set T is such that its mean = median
We are asked which of the statements must be true.

Statement 1-->
T is evenly Spaced

Consider two sets => 5,6,7,8,9,10,11
Mean = Median=8
Now Removing 7 and 9 wont affect the mean as 8-7 will be balanced by 8-9 and the sum of deviations around 8 will still be zero.
Hence 5,6,8,10,11 => Mean = 8 and Median = 8
But the above set isn't every spaced.

Hence this statement is not always true.
Statement 2=>

Consider a set => 7,7,7,7
Here => Mean=Median=7
And the number of terms is even.
Hence this statement also isn't always true.

Statement 3-->
Consider a set => 7,7,7,7
Here Mode = 7
Hence Set T will can have a mode=> The statement isn't always true.
Statement 4--> None of the above => TRUE.


Hence E
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So, in conclusion we can takeway from this question that for an evenly spaced set mean=median does not necessarily mean that the vice versa is true. Just because for a set mean = median does not render the set as an evenly spaced one.

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Yes, that's true.
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Let’s analyze each Roman numeral statement.

I. Set T consists of evenly spaced numbers.

This is not necessarily true. For example, if set T = {1, 2, 4, 4, 9}, we see that average = median, but T is not an evenly spaced set.

II. Set T consists of an odd number of terms.

This is not necessarily true. For example, if set T = {3, 3}, we see that average = median, but T has an even number of terms.

III. Set T has no mode.

This is not necessarily true. For example, if we use the same set T as in Roman numeral I, we see that average = median, but T does have a mode.

Answer: E
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