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# PS questions about standard deviation.

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30 Oct 2009, 13:11
GMAT TIGER wrote:
Bunuel wrote:
Questions #1 and #9 are solved incorrectly. One of two answers for #6 is incorrect.

Are you talking seriously???

Yes. Your second solution for question #9 is correct. Indeed there are 6 possibilities and 2 pairs have the same SD, so there would be 4 different SD.

As for #1: if it would be E {0,8}, wouldn't these two numbers stretch the set making SD bigger? As {0,8} are too far from the the mean.

And for #6: 2 answers were given (by others) to this question and one of them is incorrect.

So yes, I'm serious. But maybe I didn't get your question right...
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30 Oct 2009, 16:23
Bunuel wrote:
GMAT TIGER wrote:
Bunuel wrote:
Questions #1 and #9 are solved incorrectly. One of two answers for #6 is incorrect.

Are you talking seriously???

Yes. Your second solution for question #9 is correct. Indeed there are 6 possibilities and 2 pairs have the same SD, so there would be 4 different SD.

As for #1: if it would be E {0,8}, wouldn't these two numbers stretch the set making SD bigger? As {0,8} are too far from the the mean.

Agree with your logic but did you check whether other choices could deviate more further than E?
So what is your OA and workout for #1.
Did you solve it or just guessing? If you solve it, you will find E as the closest.
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30 Oct 2009, 16:44
GMAT TIGER wrote:
Agree with your logic but did you check whether other choices could deviate more further than E?
So what is your OA and workout for #1.
Did you solve it or just guessing? If you solve it, you will find E as the closest.

Frankly speaking when I solved this for the first time I didn't calculated SD-s. Just trusted my logic, then compared my answer to the OA and as they matched I didn't double checked my own solution and OA.

But know I did it. And I can say it again E {0,8} is not correct.

Maybe there is some misunderstanding in stem?
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30 Oct 2009, 18:05
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GMAT TIGER wrote:
gmattokyo wrote:
9. 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
-----------------------------------

(C) 5
Not very sure of this one... would wait for an expert solution.
Given that the range is 4. So pick up any set of odd numbers (SD will be same for the scenarios, all the set of 4 odd integers with range of 4 will have 3 unique members).
Possible sets (each with a different SD):
1. [1,5,1,1]
2. [1,5,5,5]
3. [1,5,3,3]
4. [1,5,1,3]
5. [1,5,1,5]
6. [1,5,3,5]

total of 6. 1st and 2nd have the same SD. Left with 5 other cases.

1. [1,5,1,1] and 2. [1,5,5,5] have SD of 2.
3. [1,5,3,3] has a SD of 1.63.
4. [1,5,1,3] and 6. [1,5,3,5] have 1.91.
5. [1,5,1,5] has a SD of 2.31.

So there are altogather 4 different SDs.

Thanks to GMAT TIGER for the correction and to Bunuel for giving us kool practice.
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30 Oct 2009, 19:10
Bunuel wrote:
GMAT TIGER wrote:
Agree with your logic but did you check whether other choices could deviate more further than E?
So what is your OA and workout for #1.
Did you solve it or just guessing? If you solve it, you will find E as the closest.

Frankly speaking when I solved this for the first time I didn't calculated SD-s. Just trusted my logic, then compared my answer to the OA and as they matched I didn't double checked my own solution and OA.

But know I did it. And I can say it again E {0,8} is not correct.

Maybe there is some misunderstanding in stem?

lol.

Probably the difference is from how the SDs are calculated. I used (n-1) instead of n while dividing the sum of the sqare dev. If n is used, then it is (2, 6).
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1. A set of data consists of the following 5 numbers: 0,2,4,6, and 8. Which two numbers, if added to create a set of 7 numbers, will result in a new standard deviation that is close to the standard deviation for the original 5 numbers?
(A) -1 and 9
(B) 4 and 4
(C) 3 and 5
(D) 2 and 6
(E) 0 and 8

2. A certain list of 100 data has an average of 6 and standard deviation of d where d is positive. Which of the following pairs of data, when added to the list must result in a list of 102 data with the standard deviation less than d?
(A) 0 and 6
(B) 0 and 12
(C) 0 and 0
(D) -6 and 0
(E) 6 and 6

3. For a certain examination, a score of 58 was 2 standard deviations below the mean, and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination?
(A) 74
(B) 76
(C) 78
(D) 80
(E) 82

4. Which of the following distribution of numbers has the greatest standard deviation?
(A) {-3, 1, 2}
(B) {-2, -1, 1, 2}
(C) {3, 5, 7}
(D) {-1, 2, 3, 4}
(E) {0, 2, 4}

5. Which of the following has the same standard deviation as {s,r,t}?
I. {r-2, s-2, t-2}
II. {0, s-t, s-r}
III. {|r|, |s|, |t|}
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

6. A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68% of the distribution lies one standard deviation d of the mean, what percent of the distribution is less than m+d?
(A) 16%
(B) 32%
(C) 48%
(D) 84%
(E) 92%

7. Which of the following data sets has the third largest standard deviation?
(A) {1, 2, 3, 4, 5}
(B) {2, 3, 3, 3, 4}
(C) {2, 2, 2, 4, 5}
(D) {0, 2, 3, 4, 6}
(E) {-1, 1, 3, 5, 7}

8. The table below represents three sets of numbers with their respective medians, means and standard deviations. The third set, Set [A+B], denotes the set that is formed by combining Set A and Set B.

Median Mean StandardDeviation
Set A: X, Y, Z.
Set B: L, M, N.
Set [A + B]: Q, R, S.
If X – Y > 0 and L – M = 0, then which of the following must be true?
I. Z > N
II. R > M
III. Q > R
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) None

9. 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

10. If a certain sample of data has a mean of 20.0 and a standard deviation of 3.0, which of the following values is more than 2.5 standard deviations from the mean?
(A) 12.0
(B) 13.5
(C) 17.0
(D) 23.5
(E) 26.5

11. Arithmetic mean and standard deviation of a certain normal distribution are 13.5 and 1.5. What value is exactly 2 standard deviations less than the mean?
(A) 10.5
(B) 11
(C) 11.5
(D) 12
(E) 12.5

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18 Dec 2009, 04:05
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GMAT TIGER wrote:
Bunuel wrote:
1. A set of data consists of the following 5 numbers: 0,2,4,6, and 8. Which two numbers, if added to create a set of 7 numbers, will result in a new standard deviation that is close to the standard deviation for the original 5 numbers?
(A) -1 and 9
(B) 4 and 4
(C) 3 and 5
(D) 2 and 6
(E) 0 and 8

E. I got 0 and 8.

I don't think Answer will be E, here i'll prove it and you can also check with the calculations that E cannot be true.

With the given set the mean = 4 (as it is set of consecutive even integers therefore, 8+0/2 = 4 )

For all the options the mean remains 4 (as they add 8 to the total it will be divided by 7)

As the values are close to the mean then SD will be lower than the original SD and if the values are far from the mean than SD will be higher.

so D is the option which is not closer and not far from the mean.
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18 Dec 2009, 08:07
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Bunuel wrote:

8. The table below represents three sets of numbers with their respective medians, means and standard deviations. The third set, Set [A+B], denotes the set that is formed by combining Set A and Set B.

Median Mean StandardDeviation
Set A: X, Y, Z.
Set B: L, M, N.
Set [A + B]: Q, R, S.
If X – Y > 0 and L – M = 0, then which of the following must be true?
I. Z > N
II. R > M
III. Q > R
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) None

We have no information that might allow us to compare Z and N, so I need not be true. For II, without knowing the relationship between Y and M, we cannot decide whether R is larger than M. For III, if set A is {0, 3, 4}, then the median of A is larger than the mean. If set B is {13}, then the median of B is equal to the mean. So these sets agree with the conditions given. Combining the sets, we have {0, 3, 4, 13}, which has a median of 3.5 and a mean of 5; the median is not greater than the mean. So III need not be true and the answer is E.
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02 Jan 2010, 10:57
Bunuel wrote:
GMAT TIGER wrote:
Agree with your logic but did you check whether other choices could deviate more further than E?
So what is your OA and workout for #1.
Did you solve it or just guessing? If you solve it, you will find E as the closest.

Frankly speaking when I solved this for the first time I didn't calculated SD-s. Just trusted my logic, then compared my answer to the OA and as they matched I didn't double checked my own solution and OA.

But know I did it. And I can say it again E {0,8} is not correct.

Maybe there is some misunderstanding in stem?

Last edited by hamza on 02 Jan 2010, 12:06, edited 1 time in total.
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02 Jan 2010, 11:02
hamza wrote:
Bunuel wrote:
GMAT TIGER wrote:
Agree with your logic but did you check whether other choices could deviate more further than E?
So what is your OA and workout for #1.
Did you solve it or just guessing? If you solve it, you will find E as the closest.

Frankly speaking when I solved this for the first time I didn't calculated SD-s. Just trusted my logic, then compared my answer to the OA and as they matched I didn't double checked my own solution and OA.

But know I did it. And I can say it again E {0,8} is not correct.

Maybe there is some misunderstanding in stem?

Answer E for Q1 is not correct, OA is D. Pleas refer to my post with OA-s.
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02 Jan 2010, 12:37
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hamza wrote:

1. A set of data consists of the following 5 numbers: 0,2,4,6, and 8. Which two numbers, if added to create a set of 7 numbers, will result in a new standard deviation that is close to the standard deviation for the original 5 numbers?
(A) -1 and 9
(B) 4 and 4
(C) 3 and 5
(D) 2 and 6
(E) 0 and 8

I guess this is not real GMAT question as to answer this question with 100% certainty you should calculate SD for two sets and GMAT usually do not require actual calculation of SD. Though it's possible to eliminate 3 wrong answers at the beginning.

Mean is 4 and so are the means of all 5 pairs from answers choices.

A. (-1, 9) These two numbers are farthest from the mean and they will stretch the set making SD bigger

B. (4, 4) These two numbers are closest to the mean and the will shrink the set making SD smaller

C. (3, 5) Suitable option so far

D. (2, 6) Suitable option so far

E. (0, 8) These two numbers are also far from mean and they will also stretch the set making SD bigger.

So, when I looked at the options C and D I assumed that C is also too close to the mean and it will affect it more than D. So I ended with D and was correct. But still my logic eliminating C was not sure thing, without the calculations.
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02 Jan 2010, 22:23
Thanks a lot Bunuel for all the SD questions, I think SD, by means is covered,

Thanks again...
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15 May 2010, 20:18
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Convincing explanation for question 1.
Bunuel wrote:
1. A set of data consists of the following 5 numbers: 0,2,4,6, and 8. Which two numbers, if added to create a set of 7 numbers, will result in a new standard deviation that is close to the standard deviation for the original 5 numbers?
(A) -1 and 9
(B) 4 and 4
(C) 3 and 5
(D) 2 and 6
(E) 0 and 8

Mean = 4
Var = (16+4+0+4+16)/5 = $$40/5 = 8$$
After addition of 2 numbers, New Var = $$(40+x)/7$$
The question is What x will pitch New Var closest to 8 so that 56/7 = 8
OR Which of the options will give a value of x that is closest to 16
So from the 5 options find out which (deviation^2) from 4 is closest to 16
Naked eye will tell you that (A), (B) are a long shot.
(C) 1^2 + 1^2 = 2
(D) 2^2 + 2^2 = 8 ==> |16-8| = 8
(E) 4^2 + 4^2 = 32 ==> |16-32| = 16
So option (D) gives an SD that is closest to the original SD.
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06 Jun 2010, 11:39
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Really good collection of questions....I think SD is more or less covered now
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08 Jun 2010, 06:17
Bunuel,

I think the statement in the Q10 is incorrect. Some typo error, IMO:

10. If a certain sample of data has a mean of 20.0 and a standard deviation of 3.0, which of the following values is more than 2.5 standard deviations AWAY from the mean?
(A) 12.0
(B) 13.5
(C) 17.0
(D) 23.5
(E) 26.5

Bunuel wrote:
10. If a certain sample of data has a mean of 20.0 and a standard deviation of 3.0, which of the following values is more than 2.5 standard deviations from the mean?

(A) 12.0
(B) 13.5
(C) 17.0
(D) 23.5
(E) 26.5

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08 Jun 2010, 07:13
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ykaiim wrote:
Bunuel,

I think the statement in the Q10 is incorrect. Some typo error, IMO:

10. If a certain sample of data has a mean of 20.0 and a standard deviation of 3.0, which of the following values is more than 2.5 standard deviations AWAY from the mean?
(A) 12.0
(B) 13.5
(C) 17.0
(D) 23.5
(E) 26.5

Bunuel wrote:
10. If a certain sample of data has a mean of 20.0 and a standard deviation of 3.0, which of the following values is more than 2.5 standard deviations from the mean?

(A) 12.0
(B) 13.5
(C) 17.0
(D) 23.5
(E) 26.5

No typo there: the question is from GmatPrep but basically it's the same as you wrote.
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08 Jun 2010, 07:22
I found the Q11 in GMATPrep and based on the explaination for it, I asked you.

As per the question:
10. If a certain sample of data has a mean of 20.0 and a standard deviation of 3.0, which of the following values is more than 2.5 standard deviations AWAY from the mean?
(A) 12.0
(B) 13.5
(C) 17.0
(D) 23.5
(E) 26.5

So, the answer would be > 20+2.5x3 or >27.5, while none of the options say this.
But, if we are given that the required value is 2.5 SD more away then we can find the new value = 12.
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08 Jun 2010, 07:48
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ykaiim wrote:
I found the Q11 in GMATPrep and based on the explaination for it, I asked you.

As per the question:
10. If a certain sample of data has a mean of 20.0 and a standard deviation of 3.0, which of the following values is more than 2.5 standard deviations AWAY from the mean?
(A) 12.0
(B) 13.5
(C) 17.0
(D) 23.5
(E) 26.5

So, the answer would be > 20+2.5x3 or >27.5, while none of the options say this.
But, if we are given that the required value is 2.5 SD more away then we can find the new value = 12.

I don't quite understand your question... Again the question is from GmarPrep and there is no word "away" in it (at least in the version I have).

Value is more than 2.5SD from the mean means that the distance between the mean and the value must be more than 2.5SD=7.5. So the value either < 12.5 or > 27.5.

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16 Aug 2010, 07:12
good practice material.
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31 Oct 2010, 15:50
Bunuel,
First of all thanks for all the wonderful material that you compile and post here on this forum. I have been following lot of your math related posts for past few days. Your posts are great help in the gmat prep.

I was wondering if you have solutions for these PS SD questions? - specially to Q#8 & 9?

Bunuel wrote:

1. A set of data consists of the following 5 numbers: 0,2,4,6, and 8. Which two numbers, if added to create a set of 7 numbers, will result in a new standard deviation that is close to the standard deviation for the original 5 numbers?
(A) -1 and 9
(B) 4 and 4
(C) 3 and 5
(D) 2 and 6
(E) 0 and 8

2. A certain list of 100 data has an average of 6 and standard deviation of d where d is positive. Which of the following pairs of data, when added to the list must result in a list of 102 data with the standard deviation less than d?
(A) 0 and 6
(B) 0 and 12
(C) 0 and 0
(D) -6 and 0
(E) 6 and 6

3. For a certain examination, a score of 58 was 2 standard deviations below the mean, and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination?
(A) 74
(B) 76
(C) 78
(D) 80
(E) 82

4. Which of the following distribution of numbers has the greatest standard deviation?
(A) {-3, 1, 2}
(B) {-2, -1, 1, 2}
(C) {3, 5, 7}
(D) {-1, 2, 3, 4}
(E) {0, 2, 4}

5. Which of the following has the same standard deviation as {s,r,t}?
I. {r-2, s-2, t-2}
II. {0, s-t, s-r}
III. {|r|, |s|, |t|}
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

6. A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68% of the distribution lies one standard deviation d of the mean, what percent of the distribution is less than m+d?
(A) 16%
(B) 32%
(C) 48%
(D) 84%
(E) 92%

7. Which of the following data sets has the third largest standard deviation?
(A) {1, 2, 3, 4, 5}
(B) {2, 3, 3, 3, 4}
(C) {2, 2, 2, 4, 5}
(D) {0, 2, 3, 4, 6}
(E) {-1, 1, 3, 5, 7}

8. The table below represents three sets of numbers with their respective medians, means and standard deviations. The third set, Set [A+B], denotes the set that is formed by combining Set A and Set B.

Median Mean StandardDeviation
Set A: X, Y, Z.
Set B: L, M, N.
Set [A + B]: Q, R, S.
If X – Y > 0 and L – M = 0, then which of the following must be true?
I. Z > N
II. R > M
III. Q > R
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) None

9. 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

10. If a certain sample of data has a mean of 20.0 and a standard deviation of 3.0, which of the following values is more than 2.5 standard deviations from the mean?
(A) 12.0
(B) 13.5
(C) 17.0
(D) 23.5
(E) 26.5

11. Arithmetic mean and standard deviation of a certain normal distribution are 13.5 and 1.5. What value is exactly 2 standard deviations less than the mean?
(A) 10.5
(B) 11
(C) 11.5
(D) 12
(E) 12.5

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