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Standard Deviation

by Uva@90 » Fri Nov 08, 2013 9:21 pm
E is a collection of four odd integers and the greatest difference between any two integers in E is 4. The standard deviation of E must be one of how many numbers?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

OA B

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by [email protected] » Fri Nov 08, 2013 11:51 pm
Hi Uva@90,

First off, this question is NOT about calculating the possible Standard Deviations; it's about understanding HOW the Standard Deviation COULD change.

We're told that we have 4 odd integers, BUT the greatest difference between any two of them is 4. That means there will be at least one duplicate entry. I'm going to TEST Values to prove it. With each of the following possibilities, we'd have a different Standard Deviation:

1,1,1,5 same as 1,5,5,5
1,3,3,5
1,1,3,5 same as 1,3,5,5
1,1,5,5

Four different possibilities means for different possible Standard Deviations.

Final Answer: B

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Last edited by [email protected] on Sat Nov 09, 2013 12:54 pm, edited 1 time in total.
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by Uva@90 » Sat Nov 09, 2013 12:45 am
[email protected] wrote:Hi Uva@90,

First off, this question is NOT about calculating the possible Standard Deviations; it's about understanding HOW the Standard Deviation COULD change.

We're told that we have 4 odd integers, BUT the greatest difference between any two of them is 4. That means there will be at least one duplicate entry. I'm going to TEST Values to prove it. With each of the following possibilities, we'd have a different Standard Deviation:

1,1,3,5
1,3,3,5
1,3,5,5
1,1,5,5

Four different possibilities means for different possible Standard Deviations.

Final Answer: B

GMAT assassins aren't born, they're made,
Rich
Rich,
Could you explain me why you eliminated below possibilities
1,1,1,5
1,5,5,5

Thanks in advance.

Regards,
Uva.
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by GMATGuruNY » Sat Nov 09, 2013 4:04 am
Uva@90 wrote:E is a collection of four odd integers and the greatest difference between any two integers in E is 4. The standard deviation of E must be one of how many numbers?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

OA B
Let SD = standard deviation.
SD indicates how the data are SPREAD OUT from the mean.
Test values in the following list:
{1, x, x, 5}.

Options:
Case 1: {1, 1, 1, 5}
Case 2: {1, 1, 3, 5}
Case 3: {1, 1, 5, 5)
Case 4: {1, 3, 3, 5}
Case 5: {1, 3, 5, 5}
Case 6: {1, 5, 5, 5}

In Cases 1 and 6, the data are spread out the SAME WAY.
In each case:
3 values are the same.
Between these 3 values and the 4th value, there is a difference of 4.
Since the data are spread out the same way in each case, Cases 1 and 6 have the SAME SD.

In Cases 2 and 5, the data are spread out the SAME WAY.
In each case:
2 values are the same.
Between these 2 values and the next smallest or biggest value, there is a difference of 2.
Between the 3rd value and the 4th value, there is a difference of 2.
Since the data are spread out the same way in each case, Cases 2 and 5 have the SAME SD.

In Cases 3 and 4, the data are spread out DIFFERENTLY.
Thus, Cases 3 and 4 have DIFFERENT SDs.

Total different SDs = 4.

The correct answer is B.
Last edited by GMATGuruNY on Sat Nov 09, 2013 5:32 pm, edited 3 times in total.
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by Uva@90 » Sat Nov 09, 2013 5:25 am
GMATGuruNY wrote:
Uva@90 wrote:E is a collection of four odd integers and the greatest difference between any two integers in E is 4. The standard deviation of E must be one of how many numbers?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

OA B
Let SD = standard deviation.
SD indicates how the data are spread from the mean.
Test values in the following list:
{1, x, x, 5}.

Options:
Case 1: {1, 1, 1, 5}
Case 2: {1, 1, 3, 5}
Case 3: {1, 1, 5, 5)
Case 4: {1, 3, 3, 5}
Case 5: {1, 3, 5, 5}
Case 6: {1, 5, 5, 5}

Cases 1 and 6 have the SAME spread:
3 values the same.
A difference of 4 between these 3 values and the 4th value.
Since Cases 1 and 6 have the same spread, they have the SAME SD.

Cases 2 and 5 have the SAME SPREAD:
2 values the same
A difference of 2 between these 2 values and the next smallest or biggest.
A difference of 2 between the 3rd value and the 4th value.
Since Cases 2 and 5 have the same spread, they have the SAME SD.

Cases 3 and 4 have DIFFERENT SPREADS.
Thus, Cases 3 and 4 have DIFFERENT SDs.

Total different SDs = 4.

The correct answer is B.
Mitch,
You made clear. Thanks a bundle.

Regards,
Uva.
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by dominhtri1995 » Sat Nov 09, 2013 5:43 am
Hi GuruNY,

Can you clarify "the same spread" for me? I just didn't really get it


Thanks,
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by Uva@90 » Sat Nov 09, 2013 6:28 am
Hi Dominhtri1995,

Spread is deviation from the mean,

1,1,3,5, mean=2.5 [1.5, 1.5, 0.5, 2.5]
1,3,5,5, mean=3.5 [2.5, 0.5, 1.5, 1.5]
1,3,3,5, mean=3 [2, 0, 0, 2]
3,3,3,5, mean=3.5 [0.5, 0.5, 0.5, 1.5]
1,1,1,5, mean=2 [1, 1, 1, 3]
5,5,5,1, mean=4 [1, 1, 1, 3]

Hope it helps you.

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by GMATGuruNY » Sat Nov 09, 2013 8:39 am
dominhtri1995 wrote:Hi GuruNY,

Can you clarify "the same spread" for me? I just didn't really get it


Thanks,
Please revisit my post above, in which I've clarified the reasoning.
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As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

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