Set S contains 16 integers whose range, mode, median, and

Question

A set S contains 16 integers whose range, mode, median, and mean are all equal to 16. What is the difference between the largest and smallest possible integers that could be members of S ?

A. 16
B. 18
C. 24
D. 28
E. 32

ankushsambare · Answer

Hi  kevincan  , 
Can you please check the answer?
I am getting (E) 32:

mean = 16 → total sum = 256
range = 16 → max = min + 16
median = 16 → 8th and 9th terms are 16
mode = 16 → 16 appears most often
to find extreme possible values across all valid sets:
make minimum as small as possible
make maximum = minimum + 16

and keep just enough 16s (at least 3) to maintain median and mode
balancing the sum, you can push: 
minimum down to 0 
→ then maximum = 16 above shifts across constructions up to 32
so across all valid sets:
smallest possible member = 0
largest possible member = 32

Difference = 32

tejas6 · Answer

If the range is 16, how can anything else be the answer?

chetan2u · Answer

Lots of restrictions but one of the important one is to keep the 8th and 9th number as 16.. 
 The median of 16 numbers will be average of 8th and 9th number.

Maximum possible value.  
Take all numbers except the largest as close to 16 as possible.
So, when placed in ascending order 8th number to 15th number will be 16. Now, the remaining 7 numbers below 16 should make up the distance of largest from 16. 
Therefore, taking all 7 numbers as 14 will require the largest to make up the difference of (16-14)×7 or 14 above 16. 
Algebraically, let the 7 numbers be x, so the largest will be x+16(for satisfying range criteria).
Thus 7*x+x+16 =16*8 as average of these 8 numbers will be 16.
This gives x=14, making the largest as 14+16 or 30.
Numbers will be  seven of them 14, eight of them 16 and largest as 30.

Smallest number 
Applying the above logic for minimum, we will get seven of them 18, eight of them 16 and the smallest as 2.
(The equation should come as x*7+8*16+x-16=16*16...Students can understand the above and solve for smallest number. )

Difference = 30-2 =28.

kevincan · Answer

Wonderful explanation!

chetan2u · Answer

That would not be correct. How can all numbers where smallest is 0 and largest are all 16 meet the criteria of 16 as mean?

Adit_ · Answer

If smallest possible number is 0, then the largest number of that set must be 16.
Is that possible? No.
Why? Because right side of median you are not increasing the value of mean at all.
Left side you have a -16 as compared to the mean and u need a +16 to compensate right side of median to match the average of 16.

The lowest possibility is 2:
Why?
When we keep 2, and every other value till median to be 16, 
then we have a -14 difference to the avg/median and the largest value here will be 18. 
Keeping every value right side of median to 18 we have a +14 right side, -14 left side because of 2.
Both deviations cancel out which means average matches as well. Mode matches too as there is one more 16 than 18.

Highest value is taken on similar principles. You need to do the exact opposite as the first case.
Take 2*7 = +14 deviation to the right of median = 16+14 = 30.
Lowest value becomes 14 now, and using deviation principles we again see that it is valid.
Symmeyrical to the first case except we aim for highest values here.

So difference between highest and lowest = 30-2 =  28 .

Hope it helps.

____________________________________

These stats problems are quite tricky to be fair, what do you think is the genuine difficulty of this  kevincan ? Definitely seems 700+ range.

kevincan · Answer

I knew it would be challenging , but I didn’t know whether it was too difficult for the GMAT. I wrote it for a student who wanted practice on hard questions about statistics. ChatGPT lost quite a bit of sleep over this one. Take this question with a grain of salt, and learn as much as you can from Chetan’s splendid explanation

Adit_ · Answer

I agree! And I also agree that Chetan's explanation is great, happy to have used the same method as himself!

Set S contains 16 integers whose range, mode, median, and

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Set S contains 16 integers whose range, mode, median, and

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