Problem Solving Pack 3, Question 3 A set of 11 positive integers...

Hi All,

When dealing with Quant questions involving Statistical terms (mean, median, mode, range, standard deviation), it's important to understand the specific definitions of the terms involved.

We're told that a set of 11 POSITIVE integers has an average of 25. We're asked for the GREATEST possible MEDIAN of that set.

To find the median, we have to first put the numbers in order from least to greatest. One of the 'keys' to this question is that the prompt does NOT state that the 11 terms have to be distinct (re: different), so duplicates can be used. To maximize the median, we need to minimize ALL of the other numbers though...

For the first 5 numbers, the smallest we can make each of them is 1. Once we do that, we can make the median (and each of the 5 other numbers) all X...

1 1 1 1 1 X X X X X X

Since the average of the 11 numbers is 25, the SUM of those numbers is 11(25) = 275.

Subtracting the five 1's from the sum gives us 275 - 5(1) = 270. Thus, the remaining 6 values are all the SAME and sum to 270. The largest possible median would be....

270/6 = 45

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Don't you think that the question has an error ?
You have mentioned that 11 no.s are in a set. A no. never repeats in a set. Given that, we cannot take 2 nos to be same.

Hi Sejal16,

Unless the question specifically states that there are no 'duplicate' numbers (for example, by describing the numbers as "distinct" or "different"), then a number can appear more than once in a set of numbers. Here, the 11 numbers have an average of 25, meaning that all 11 numbers COULD be 25s - meaning that the median would also be 25. The question asks for the GREATEST possible median though; to find that value, we'll need two 'groups' of duplicate values (one for the smallest possible number - in this case, "1" - and one for the largest).

GMAT assassins aren't born, they're made,
Rich

We can solve this by another approach. ( Standard deviation approach)
Since median has to be greater lets start with option E,
If Median = 46. to get a avg 25, Right hand side numbers to the median shall be equal to 46
that is 21 more than the avg. And 6 numbers have 21 more than the avg so 21*6 = 126 more than avg
So below the median we need 5 numbers that will balance out this 126 i.e -126 additive.
126/6 > 25. So any numbers you take the average will be more than 25.

Same trial with 45 and you will give you the possibility.
IMO D

This is wrong. In mathematics, a set cannot contain repeated values, by the definition of a set. Every official GMAT question observes this correct definition of a set, and if a question only mentions a 'set', you will never get the wrong answer if you assume the elements are all distinct. If a GMAT question wants to allow repeated values in a collection of values, the question will talk about a "list", or about a "data set" (which is a different thing than a pure mathematical set), or will use similar language; it will never describe that collection as a simple 'set', as the question in this thread does.