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# If Q is a set of consecutive integers, what is the standard

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If Q is a set of consecutive integers, what is the standard  [#permalink]

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Updated on: 24 Nov 2019, 01:36
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If Q is a set of consecutive integers, what is the standard deviation of Q?

(1) Set Q contains 21 terms.
(2) The median of set Q is 20.

Official explanation from the Manhattan GMAT

What is the standard deviation of Q, a set of consecutive integers?

(1) Q has 21 members.

(2) The median value of set Q is 20.

1) SUFFICIENT: From the question, we know that Q is a set of consecutive integers. Statement 1 tells us that there are 21 terms in the set. Since, in any consecutive set with an odd number of terms, the middle value is the mean of the set, we can represent the set as 10 terms on either side of the middle term x:

[x – 10, x – 9, x – 8, x – 7, x – 6, x – 5, x – 4, x – 3, x – 2, x – 1, x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7, x + 8, x + 9, x + 10]

Notice that the difference between the mean (x) and the first term in the set (x – 10) is 10. The difference between the mean (x) and the second term in the set (x – 9) is 9. As you can see, we can actually find the difference between each term in the set and the mean of the set without knowing the specific value of each term in the set!

(The only reason we are able to do this is because we know that the set abides by a specified consecutive pattern and because we are told the number of terms in this set.) Since we are able to find the "differences," we can use these to calculate the standard deviation of the set. Although you do not need to do this, here is the actual calculation:

Sum of the squared differences:
102 + 92 + 82 + 72 + 62 + 52 + 42 + 32 + 22 + 12 + 02 + (-1)2 + (-2)2(-3)2 + (-4)2 + (-5)2 + (-6)2(-7)2 + (-8)2 + (-9)2 + (-10)2 = 770
Average of the sum of the squared differences:

770/21 = 36 2/3

The square root of this average is the standard deviation: ≈ 6.06

Originally posted by CharuKapoor on 28 Apr 2013, 07:30.
Last edited by Bunuel on 24 Nov 2019, 01:36, edited 3 times in total.
Renamed the topic and edited the question.
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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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28 Apr 2013, 23:33
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Two very important properties of standard deviation:

If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

You can try it yourself:
SD of a set: {1,1,4} will be the same as that of {5,5,8} as second set is obtained by adding 4 to each term of the first set.

That's because Standard Deviation shows how much variation there is from the mean. And when adding or subtracting a constant to each term we are shifting the mean of the set by this constant (mean will increase or decrease by the same constant) but the variation from the mean remains the same as all terms are also shifted by the same constant.

Back to the original question:

If Q is a set of consecutive integers, what is the standard deviation of Q?

(1) Set Q contains 21 terms --> SD of ALL sets with 21 consecutive integers will be the same, as any set of 21 consecutive integers can be obtained by adding constant to another set of 21 consecutive integers. For example: set of 21 consecutive integers {4, 5, 6, ..., 24} can be obtained by adding 4 to each term of another set of 21 consecutive integers: {0, 1, 2, ..., 20}. So we can calculate SD of {0, 1, 2, ..., 20} and we'll know that no matter what our set actually is, its SD will be the same. Sufficient.

(2) The median of set Q is 20. Clearly insufficient.

Hope it's clear.
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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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28 Apr 2013, 07:37
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If Q is a set of consecutive integers, what is the standard deviation of Q?

(1) Set Q contains 21 terms.
The integers are consecutives, the first value has no importance in the stdDev. If we know that there are 21 integers, they will be in the form
x,x+1,x+2,...,x+20. We can locate the middle value and we know "how far" each value is.
Sufficient

(2) The median of set Q is 20.
Example: a set of {19 20 21} or a set of {18 19 20 21 22} both have median 20 but the stdDev in the latter is bigger.
Not sufficient

A
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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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28 Apr 2013, 07:46
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CharuKapoor wrote:
If Q is a set of consecutive integers, what is the standard deviation of Q?

(1) Set Q contains 21 terms.

(2) The median of set Q is 20.

Friends, I encountered this one on MGMAT test and initially got it wrong. Let's discuss.

From the F.S 1,consider the series [-10,-8,-9......0,1,2,3....9,10]. We have a fixed value for the S.D for the given series. Also, S.D. doesn't change if we add a constant across. Thus, we could add a constant 'a' to the above series --> The value of S.D will not change, irrespective the value of a.Sufficient.

From F.S 2, all we know is that the median is 20 and the series is of consecutive integers. For 19,20,21 the S.D would be different than that for 18,19,20,21,22.Insufficient.

A.
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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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29 Apr 2013, 00:24
Bunuel just one question what will be the SD of a set of (-10, -9 ,......0....9,10) and what will be the SD of (1,2,......20,21)..Also ranges of both sets.

Sorry for asking stupid questions but it confused me here.

Thanks,
Abhinav
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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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29 Apr 2013, 00:39
2
abhinav11 wrote:
Bunuel just one question what will be the SD of a set of (-10, -9 ,......0....9,10) and what will be the SD of (1,2,......20,21)..Also ranges of both sets.

Sorry for asking stupid questions but it confused me here.

Thanks,
Abhinav

The two sets {-10, -9, ..., 10} and {1, 2, ..., 21} have the same standard deviation and the same range:

SD = $$\sqrt{\frac{110}{3}}$$;

RANGE = (largest)-(smallest) = 10-(-10) = 20 or 21-1=20.
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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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29 Apr 2013, 05:10
2
1
Thanks everyone who took initiative to explain.

I'm also posting an explanation from Manhattan GMAT and I hope none of us will get this or any other Standard Deviation Question wrong in future.

---------------------------------------------------------------------------------------

The procedure for finding the standard deviation for a set is as follows:

1) Find the difference between each term in the set and the mean of the set.

2) Average the squared "differences."

3) Take the square root of that average.

Notice that the standard deviation hinges on step 1: finding the difference between each term in the set and the mean of the set. Once this is done, the remaining steps are just calculations based on these "differences."

Thus, we can rephrase the question as follows: "What is the difference between each term in the set and the mean of the set?"

(1) SUFFICIENT: From the question, we know that Q is a set of consecutive integers. Statement 1 tells us that there are 21 terms in the set. Since, in any consecutive set with an odd number of terms, the middle value is the mean of the set, we can represent the set as 10 terms on either side of the middle term x:

[x – 10, x – 9, x – 8, x – 7, x – 6, x – 5, x – 4, x – 3, x – 2, x – 1, x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7, x + 8, x + 9, x + 10]

Notice that the difference between the mean (x) and the first term in the set (x – 10) is 10. The difference between the mean (x) and the second term in the set (x – 9) is 9. As you can see, we can actually find the difference between each term in the set and the mean of the set without knowing the specific value of each term in the set!

(The only reason we are able to do this is because we know that the set abides by a specified consecutive pattern and because we are told the number of terms in this set.) Since we are able to find the "differences," we can use these to calculate the standard deviation of the set. Although you do not need to do this, here is the actual calculation:

Sum of the squared differences:
10^2 + 9^2 + 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2 + 0^2 + (-1)^2 + (-2)^2 + (-3)^2 + (-4)^2 + (-5)^2 + (-6)^2 + (-7)^2 + (-8)^2 + (-9)^2 + (-10)^2 = 770

Average of the sum of the squared differences:
770/21 = 36 2/3

The square root of this average is the standard deviation: ≈ 6.06

(2) NOT SUFFICIENT: Since the set is consecutive, we know that the median is equal to the mean. Thus, we know that the mean is 20. However, we do not know how big the set is so we cannot identify the difference between each term and the mean.

Therefore, the correct answer is A.
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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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29 Apr 2013, 07:52
CharuKapoor wrote:
Thanks everyone who took initiative to explain.

I'm also posting an explanation from Manhattan GMAT and I hope none of us will get this or any other Standard Deviation Question wrong in future.

---------------------------------------------------------------------------------------

The procedure for finding the standard deviation for a set is as follows:

1) Find the difference between each term in the set and the mean of the set.

2) Average the squared "differences."

3) Take the square root of that average.

Notice that the standard deviation hinges on step 1: finding the difference between each term in the set and the mean of the set. Once this is done, the remaining steps are just calculations based on these "differences."

Thus, we can rephrase the question as follows: "What is the difference between each term in the set and the mean of the set?"

(1) SUFFICIENT: From the question, we know that Q is a set of consecutive integers. Statement 1 tells us that there are 21 terms in the set. Since, in any consecutive set with an odd number of terms, the middle value is the mean of the set, we can represent the set as 10 terms on either side of the middle term x:

[x – 10, x – 9, x – 8, x – 7, x – 6, x – 5, x – 4, x – 3, x – 2, x – 1, x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7, x + 8, x + 9, x + 10]

Notice that the difference between the mean (x) and the first term in the set (x – 10) is 10. The difference between the mean (x) and the second term in the set (x – 9) is 9. As you can see, we can actually find the difference between each term in the set and the mean of the set without knowing the specific value of each term in the set!

(The only reason we are able to do this is because we know that the set abides by a specified consecutive pattern and because we are told the number of terms in this set.) Since we are able to find the "differences," we can use these to calculate the standard deviation of the set. Although you do not need to do this, here is the actual calculation:

Sum of the squared differences:
10^2 + 9^2 + 8^2 + 7^2 + 6^2 + 5^2 + 4^2 + 3^2 + 2^2 + 1^2 + 0^2 + (-1)^2 + (-2)^2 + (-3)^2 + (-4)^2 + (-5)^2 + (-6)^2 + (-7)^2 + (-8)^2 + (-9)^2 + (-10)^2 = 770

Average of the sum of the squared differences:
770/21 = 36 2/3

The square root of this average is the standard deviation: ≈ 6.06

(2) NOT SUFFICIENT: Since the set is consecutive, we know that the median is equal to the mean. Thus, we know that the mean is 20. However, we do not know how big the set is so we cannot identify the difference between each term and the mean.

Therefore, the correct answer is A.

Sorry but this is way too complex to do in timed environment
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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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29 Apr 2013, 11:04
2
Hi Abhinav,

One actually need not do all this calculation.
The reason I posted the explanation is because, understanding the concept of Standard Deviation will help us tackle all other questions on this topic.
GMAT tricks us by combining things such as - Standard Deviation and other concepts of statistics with Data Sufficiency.

In case you need further explanation, kindly let me know.
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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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20 Dec 2016, 15:11
Top Contributor
amandeep_k wrote:
If Q is a set of consecutive integers, what is the standard deviation of Q?

(1) Set Q contains 21 terms.

(2) The median of set Q is 20.

Target question: What is the standard deviation of Q?

Given: Q is a set of CONSECUTIVE integers

Statement 1: Set Q contains 21 terms.
NOTE: Standard Deviation measures dispersion (spread-apart-ness). As such, the actual values mean nothing compared to RELATIVE values.
For example, the set {1,2,3,4} has the SAME STANDARD DEVIATION as the set {6,7,8,9}

So, knowing that set Q consists of 21 CONSECUTIVE integers is SUFFICIENT.
The Standard Deviation of Q will be the same as the Standard Deviation of {1,2,3,4...20,21}

Statement 2: The median of set Q is 20.
There are several different sets that satisfy this condition.
For example, set Q could equal {19, 20, 21} or set Q could equal {18, 19, 20, 21, 22}
These two sets have DIFFERENT standard deviations.
So, statement 2 is NOT SUFFICIENT

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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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13 Jul 2017, 15:50
CharuKapoor wrote:
If Q is a set of consecutive integers, what is the standard deviation of Q?

(1) Set Q contains 21 terms.
(2) The median of set Q is 20.

Official explanation from the Manhattan GMAT

1) SUFFICIENT: From the question, we know that Q is a set of consecutive integers. Statement 1 tells us that there are 21 terms in the set. Since, in any consecutive set with an odd number of terms, the middle value is the mean of the set, we can represent the set as 10 terms on either side of the middle term x:

[x – 10, x – 9, x – 8, x – 7, x – 6, x – 5, x – 4, x – 3, x – 2, x – 1, x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7, x + 8, x + 9, x + 10]

Notice that the difference between the mean (x) and the first term in the set (x – 10) is 10. The difference between the mean (x) and the second term in the set (x – 9) is 9. As you can see, we can actually find the difference between each term in the set and the mean of the set without knowing the specific value of each term in the set!

(The only reason we are able to do this is because we know that the set abides by a specified consecutive pattern and because we are told the number of terms in this set.) Since we are able to find the "differences," we can use these to calculate the standard deviation of the set. Although you do not need to do this, here is the actual calculation:

Sum of the squared differences:
102 + 92 + 82 + 72 + 62 + 52 + 42 + 32 + 22 + 12 + 02 + (-1)2 + (-2)2(-3)2 + (-4)2 + (-5)2 + (-6)2(-7)2 + (-8)2 + (-9)2 + (-10)2 = 770
Average of the sum of the squared differences:

770/21 = 36 2/3

The square root of this average is the standard deviation: ≈ 6.06

You don't actually have to know what the median value of the set is because if you know the number of terms and the common difference- like in this sequence then you can know the s.d

x-2, x-1 ,x, x+1 .. etc
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Re: If Q is a set of consecutive integers, what is the standard  [#permalink]

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10 Sep 2017, 22:17
CharuKapoor wrote:
If Q is a set of consecutive integers, what is the standard deviation of Q?

(1) Set Q contains 21 terms.
(2) The median of set Q is 20.

Official explanation from the Manhattan GMAT

1) SUFFICIENT: From the question, we know that Q is a set of consecutive integers. Statement 1 tells us that there are 21 terms in the set. Since, in any consecutive set with an odd number of terms, the middle value is the mean of the set, we can represent the set as 10 terms on either side of the middle term x:

[x – 10, x – 9, x – 8, x – 7, x – 6, x – 5, x – 4, x – 3, x – 2, x – 1, x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7, x + 8, x + 9, x + 10]

Notice that the difference between the mean (x) and the first term in the set (x – 10) is 10. The difference between the mean (x) and the second term in the set (x – 9) is 9. As you can see, we can actually find the difference between each term in the set and the mean of the set without knowing the specific value of each term in the set!

(The only reason we are able to do this is because we know that the set abides by a specified consecutive pattern and because we are told the number of terms in this set.) Since we are able to find the "differences," we can use these to calculate the standard deviation of the set. Although you do not need to do this, here is the actual calculation:

Sum of the squared differences:
102 + 92 + 82 + 72 + 62 + 52 + 42 + 32 + 22 + 12 + 02 + (-1)2 + (-2)2(-3)2 + (-4)2 + (-5)2 + (-6)2(-7)2 + (-8)2 + (-9)2 + (-10)2 = 770
Average of the sum of the squared differences:

770/21 = 36 2/3

The square root of this average is the standard deviation: ≈ 6.06

The SD is only restricted by the number of terms so we just need to know the number of terms

St 1

gives us this exactly

suff

St 2

clearly insuff

A
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