If the highest observation in the set S is 4 standard deviations above

Question

If the highest observation in the set S is 4 standard deviations above the mean and the lowest observation in the set is 5 standard deviations below the mean, what is the standard deviation of the set?
 
(1) The mean of the set is 24.
(2) The range of the set S is 72.

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aditya1818 · Answer

As per the question:

H= M+4s ........(i)
L=M-5s ..........(ii)

(i)-(ii) ( subtracting (ii) from (i))

H-L=9s
s=(H-L)/9

Now, I will jump to the options:
Statement 1: M=24 
I still played with it and by substituting it in the equation (i) and (ii) and still it wasn't enough

Statement 2:H-L=72
This clearly gives s

Answer: Option (B)

egmat · Answer

The question stem gives us a hidden relationship that makes this problem easier than it looks.

Let's call the mean  μ  and the standard deviation  σ .

From the question stem:
• Highest observation = μ + 4σ
• Lowest observation = μ - 5σ

Principle: Range = Highest - Lowest

So: Range = (μ + 4σ) - (μ - 5σ) =  9σ

Notice that  the mean cancels out!  The range depends ONLY on σ.

Statement 1:  The mean is 24.
This tells us μ = 24. But we still have no information about σ.
• If σ = 5, then Highest = 44, Lowest = -1
• If σ = 10, then Highest = 64, Lowest = -26
Both scenarios are valid, but give different values of σ.
 → NOT SUFFICIENT

Statement 2:  The range is 72.
From our key insight: Range = 9σ
So: 9σ = 72
 σ = 8 
We found the standard deviation!
 → SUFFICIENT

Answer: B

Takeaway:  Before evaluating statements, always simplify relationships from the question stem. Here, connecting "highest" and "lowest" to the range eliminated the mean variable entirely, making Statement 2 directly sufficient.

Edskore · Answer

The egmat explanation above is excellent. I'd just add one thing that clicked for me during prep and might help others: the reason Statement 1 (mean = 24) fails is actually more interesting than it looks.

The question asks for standard deviation, and the stem tells you Range = 9s. Statement 1 gives you the mean — but the range formula has no mean in it. So knowing the mean = 24 is completely irrelevant to finding s. Students often fall for Statement 1 because "I know the mean, I know two boundary points in terms of the mean — surely I can figure out s?" The trap is that the mean and s are independent here; the equations H = mean + 4s and L = mean - 5s have two unknowns, and Statement 1 gives you one of them — but the one you actually need (s) can still be anything.

Statement 2 cuts right through by giving you Range = 72, so 9s = 72, giving s = 8. Clean and sufficient on its own.

The broader Statistics and Sets Problems lesson: always simplify the question stem first before touching the statements. Here, "Range = Highest - Lowest = (mean+4s) - (mean-5s) = 9s" does all the work.

Answer: B

ExpertsGlobal5 · Answer

Explanation Video:

https://www.youtube.com/watch?v=cECy8ZxBkPE

If the highest observation in the set S is 4 standard deviations above

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GMAT Data Sufficiency (DS) Questions

If the highest observation in the set S is 4 standard deviations above

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