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Set S contains 16 integers whose range, mode, median, and 26 Apr 2026, 16:19
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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
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D
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Set S contains 16 integers whose range, mode, median, and 28 Apr 2026, 15:23
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If the range is 16, how can anything else be the answer?
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Set S contains 16 integers whose range, mode, median, and 26 Apr 2026, 20:30
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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
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Set S contains 16 integers whose range, mode, median, and 29 Apr 2026, 05:56
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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
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
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Set S contains 16 integers whose range, mode, median, and 29 Apr 2026, 05:58
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Wonderful explanation!
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Set S contains 16 integers whose range, mode, median, and 29 Apr 2026, 05:59
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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?
ankushsambare
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
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Set S contains 16 integers whose range, mode, median, and 29 Apr 2026, 06:15
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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.
ankushsambare
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
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Set S contains 16 integers whose range, mode, median, and 29 Apr 2026, 06:35
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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
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Set S contains 16 integers whose range, mode, median, and 29 Apr 2026, 08:36
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I agree! And I also agree that Chetan's explanation is great, happy to have used the same method as himself!
kevincan
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
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Set S contains 16 integers whose range, mode, median, and 14 Jun 2026, 00:37
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Please correct me if I am wrong.. But you created 2 sets.. Set 1, Largest possible last number....Set 2 smallest possible first number.. and you came up with 28 by taking these two sets together. But the whole point is the two sets are different. With range being 16 these largest and smallest numbers cannot be in same set.
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Set S contains 16 integers whose range, mode, median, and 16 Jun 2026, 07:48
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Hi Sunsh,

You've spotted something real: those two sets are different, and no single set can contain both 30 and 2 (the range would be 28, not 16). But that's actually fine, because the question isn't asking for the range of one set.

Read the stem carefully: "the difference between the largest and smallest possible integers that could be members of S."

That's a question about possibilities across all valid sets, not about one specific set. It's really asking two separate questions:

- Over every set that satisfies all four conditions, what is the biggest value any member could ever take? - 30
- Over every set that satisfies all four conditions, what is the smallest value any member could ever take? - 2

So we deliberately build two different sets - one that pushes an element as high as possible, one that pushes an element as low as possible - and report those two ceiling/floor values. The answer 28 = 30 - 2 is the gap between the highest reachable member and the lowest reachable member, not the range inside any one set.

Think of it like asking, "What's the tallest and shortest a player in this league could be?" The tallest player and the shortest player don't have to be on the same team - and we'd never expect them to be. We just need each extreme to occur in some valid lineup.

So Chetan's two constructions are doing exactly the right thing: each one is a fully valid set on its own (range 16 in each), and we're harvesting one extreme value from each. Your instinct that they can't coexist in one set is correct - it just doesn't matter for what's being asked.

Answer: D

Sunsh
Please correct me if I am wrong.. But you created 2 sets.. Set 1, Largest possible last number....Set 2 smallest possible first number.. and you came up with 28 by taking these two sets together. But the whole point is the two sets are different. With range being 16 these largest and smallest numbers cannot be in same set.
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Set S contains 16 integers whose range, mode, median, and 14 Jul 2026, 08:54
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When range is 16 how can the difference between smallest and largest be greater than 16

chetan2u
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.

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Set S contains 16 integers whose range, mode, median, and 15 Jul 2026, 04:30
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Hi nightwalkeuv,

Your instinct about the range is completely correct: inside any single valid set, the biggest and smallest members can differ by exactly 16, never more. That constraint is real and it never gets broken. So you're not making an error about the range itself.

The trick is in what the question is actually asking. Look at the wording:

"the difference between the largest and smallest possible integers that could be members of S"

That word possible is doing everything. It's not asking for the range of one set. It's asking two separate questions:

- Across every set that fits all four conditions, what's the highest value any member could ever reach? - 30
- Across every set that fits all four conditions, what's the lowest value any member could ever reach? - 2

Those two extremes come from two different sets, and that's allowed. In Chetan's constructions, each set on its own has range 16:

- Set with the max: seven 14s, eight 16s, one 30 - range = 30 - 14 = 16 ✓
- Set with the min: one 2, eight 16s, seven 18s - range = 18 - 2 = 16 ✓

So neither set violates the range rule. We just harvest the ceiling (30) from one and the floor (2) from the other, and the answer reports the gap between those two: 30 - 2 = 28.

A simpler version to lock it in. Suppose I ask: "For a set of two integers with range 3, what's the largest and smallest possible member?" One valid set is {2, 5}; another is {100, 103}. No single set has a spread bigger than 3 - but the smallest member you could ever see is unbounded low, and the largest unbounded high. The "possible members" question ranges over all valid sets, not one.

Your range logic was right the whole time - it just answers a different question than the one asked.

Answer: D

nightwalkeuv
When range is 16 how can the difference between smallest and largest be greater than 16


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Set S contains 16 integers whose range, mode, median, and 05 Aug 2026, 00:43
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Great explanation. Thank you so much!
chetan2u
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.

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