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Is the standard deviation of the numbers X, Y and Z

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by NandishSS » Fri May 19, 2017 7:36 am
Is the standard deviation of the numbers X, Y and Z equal to the standard deviation of 10,15 and 20?

(1) Z - X = 10
(2) Z - Y = 5

OA:C

Source: GMAT Prep QP1
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Source: — Data Sufficiency |
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by DavidG@VeritasPrep » Fri May 19, 2017 7:50 am
NandishSS wrote:Is the standard deviation of the numbers X, Y and Z equal to the standard deviation of 10,15 and 20?

(1) Z - X = 10
(2) Z - Y = 5

OA:C

Source: GMAT Prep QP1
You can quickly eliminate each statement individually. The first gives you nothing about y and the second gives you nothing about x.

Together, we know that z = x + 10 and z = y + 5. So long as you recognize that the standard deviation is unchanged if you were to add or subtract the same value from each element in a set, you can quickly show that the statements together are sufficient.

Case 1: x = 0, z = 10, and y = 5. A set of 0, 5, 10 would have the same SD as the set of 10, 15, 20. The answer is YES
Case 2: x = 10, z = 20, and y = 15. The numbers are identical - obviously YES.

No matter what we pick, the spacing of the set will be identical, as z will always be 10 greater than x and 5 greater than y. The answer is C
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by DavidG@VeritasPrep » Fri May 19, 2017 7:54 am
The concept is tested in a nearly identical way here: https://www.beatthegmat.com/standard-dev ... 23533.html
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by [email protected] » Fri May 19, 2017 10:28 am
Hi NandishSS,

While this question requires some specific knowledge about Standard Deviation, you don't actually have to do much math to solve it. To start, it's worth noting that the GMAT will NEVER ask you to calculate the Standard Deviation of a group using the S.D. formula, so that is NOT what this question is actually about.

We're asked if the S.D. of three numbers (X, Y and Z) is the SAME S.D. as the one for the numbers 10, 15 and 20. This is a YES/NO question.

The numbers 10, 15 and 20 are 'evenly spaced' numbers that differ by 5. To have the same S.D. as this group, another group must ALSO have evenly spaced numbers that differ by 5. For example, (0, 5, 10) and (1, 6, 11) would have the same S.D. as (10 15. 20).

1) Z - X = 10

This Fact fits part of the pattern that we're looking for, but we don't know the relative value of Y.
Fact 1 is INSUFFICIENT

2) Z - Y = 5

This Fact also fits part of the pattern that we're looking for, but we don't know the relative value of X.
Fact 2 is INSUFFICIENT

Combined, we know....
Z - X = 10
Z - Y = 5

This means that Z is the largest value, that Z is 5 greater than Y, and that Z is 10 greater than X. By extension, Y would then be 5 greater than X. This is an exact match for the 'spread' created by (10, 15, 20), so the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

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by Brent@GMATPrepNow » Thu Aug 31, 2017 10:46 am
NandishSS wrote:Is the standard deviation of the numbers X, Y and Z equal to the standard deviation of 10,15 and 20?

(1) Z - X = 10
(2) Z - Y = 5

OA:C

Source: GMAT Prep QP1
Target question: Is the standard deviation (SD) of the numbers X, Y and Z equal to the standard deviation of 10,15 and 20?

IMPORTANT CONCEPT: In order for the SD of X, Y and Z to be equal to the SD of {10, 15, 20}, X, Y and Z must have the same dispersion as {10, 15, 20}
That is, if X, Y and Z are arranged in ascending order, the 2nd value must be 5 greater than the 1st value, and the 3rd value must be 5 greater than the 2nd value.
So, for example, the following sets will have the same standard deviation as {10, 15, 20}:
{1, 6, 11}
{8.3, 13.3, 18.3}
{-8, -3, 2}
Etc

Statement 1: Z - X = 10
No information about Y
So, statement 1 is NOT SUFFICIENT

Statement 2: Z - Y = 5
No information about X
So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that Z - X = 10, which means Z is 10 greater than X
Statement 2 tells us that Z - Y = 5, which means Z is 5 greater than Y
So, we can conclude that Z is the biggest number.

If we take the equation Z - X = 10 and subtract the equation Z - Y = 5, we get -X + Y = 5, which is the same as Y - X = 5
This tells us that Y is 5 greater than X

So, we now know that Z is 5 greater than Y and Y is 5 greater than X
So, {X, Y, Z} has the exact same dispersion as {5, 10, 20}, which means {X, Y, Z} has the same standard deviation as {5, 10, 20}
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C
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by Matt@VeritasPrep » Thu Aug 31, 2017 4:49 pm
As a tip for test takers, this seems to be a favorite property of the GMAC's:

SD of (a, b, c, ...) = SD of (a + x, b + x, c + x, ...)

I've seen this on a few official questions, so keep it in mind on test day.
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