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# PS questions about standard deviation.

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09 Apr 2013, 02:52
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doe007 wrote:
Bunuel wrote:
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.

For the original set 0, 2, 4, 6, 8, standard deviation = 3.16227766

A) For the set -1, 0, 2, 4, 6, 8, 9, standard deviation = 3.872983346
B) For the set 0, 2, 4, 4, 4, 6, 8, standard deviation = 2.581988897
C) For the set 0, 2, 3, 4, 5, 6, 8, standard deviation = 2.645751311
D) For the set 0, 2, 2, 4, 6, 6, 8, standard deviation = 2.828427125
E) For the set 0, 0, 2, 4, 6, 8, 8, standard deviation = 3.464101615

[As the sample size is very small, SD's are calculated using formula for sample.]

Difference between original stdev and stdev of option D is 0.333850535
Difference between original stdev and stdev of option E is 0.301823955

Though option D has a very close call, closest to the original standard deviation is found in option E.

I believe this is not a real GMAT question and GMAC would not ask for such lengthy calculations.

Bunuel, you said correctly that answer to this question can be found through calculation only. I am always impressed by your explanations. For this question, however, something is amiss. I wish there is a simple way to find correct answer for this question.

That's not correct.

SD of {0, 2, 4, 6, 8} = ~2.83 (3.16 is sample standard deviation and 2.83 is population standard deviation, which is tested on the GMAT).

C. SD of {0, 2, 4, 6, 8, 3, 5} = ~2.45 --> difference=0.38
D. SD of {0, 2, 4, 6, 8, 2, 6} = ~2.62 --> difference=0.21
E. SD of {0, 2, 4, 6, 8, 0, 8} = ~3.21 --> difference=0.38

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09 Apr 2013, 03:22
Bunuel wrote:
That's not correct.

SD of {0, 2, 4, 6, 8} = ~2.83 (3.16 is sample standard deviation and 2.83 is population standard deviation, which is tested on the GMAT).

C. SD of {0, 2, 4, 6, 8, 3, 5} = ~2.45 --> difference=0.38
D. SD of {0, 2, 4, 6, 8, 2, 6} = ~2.62 --> difference=0.21
E. SD of {0, 2, 4, 6, 8, 0, 8} = ~3.21 --> difference=0.38

Bunuel, it is true that Official Guide for GMAT Review mentions only the SD for population though Statistics encourages SD for sample for small set of numbers. In fact, during study, we were asked to use only SD for sample for calculations in quantitative finance. However, GMAT is the playground of GMAC and their rules will prevail!

I withdraw my stand here considering GMAT strategy.
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17 Jun 2013, 11:17
Could you please explain the Q8, Brunel ? Thank you so much!
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17 Jun 2013, 11:25
Expert's post
kloklo wrote:
Could you please explain the Q8, Brunel ? Thank you so much!

Hope it helps.
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14 Oct 2013, 09:01
Bunuel wrote:
santy wrote:
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?

Q#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

Let the smallest odd integer be 1, thus the largest one will be 5. We can have following 6 types of sets:

1. {1, 1, 1, 5} --> mean=2 --> |mean-x|=(1, 1, 1, 3);
2. {1, 1, 3, 5} --> mean=2.5 --> |mean-x|=(1.5, 1.5, 0.5, 2.5);
3. {1, 1, 5, 5} --> mean=3 --> |mean-x|=(2, 2, 2, 2);
4. {1, 3, 3, 5} --> mean=3 --> |mean-x|=(2, 0, 0, 2);
5. {1, 3, 5, 5} --> mean=3.5 --> |mean-x|=(2.5, 0.5, 1.5, 1.5);
6. {1, 5, 5, 5} --> mean=4 --> |mean-x|=(3, 1, 1, 1).

CALCULATING STANDARD DEVIATION OF A SET {x1, x2, ... xn}:
1. Find the mean, $$m$$, of the values.
2. For each value $$x_i$$ calculate its deviation ($$m-x_i$$) from the mean.
3. Calculate the squares of these deviations.
4. Find the mean of the squared deviations. This quantity is the variance.
5. Take the square root of the variance. The quantity is th SD.

Expressed by formula: $$standard \ deviation= \sqrt{variance} = \sqrt{\frac{\sum(m-x_i)^2}{N}}$$.

You can see that deviation from the mean for 2 pairs of the set is the same, which means that SD of 1 and 6 will be the same and SD of 2 and 5 also will be the same. So SD of such set can take only 4 values.

Solutions and OA's for other questions are on previous pages.

Hope it's clear.

Is there a way to solve 9 using logic, as opposed to calculations? I got every other question right, based on the logic involved in SD, or knowledge of what it is, but for #9 I can't wrap my head around it. I chose A
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19 Oct 2013, 10:21
Bunuel wrote:
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.

What is the most efficient approach to choose between C and D here? They look pretty tentative both of them

Thanks
Cheers!

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07 May 2014, 07:05
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.

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07 May 2014, 08:04
Expert's post
himanshujovi wrote:
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.

Solutions are on pages 2, 3, 4 and 5.
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08 May 2014, 13:21
Hi Bunuel,

Do we have the OAs for all the questions?
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09 May 2014, 02:09
Expert's post
ravishankar1788 wrote:
Hi Bunuel,

Do we have the OAs for all the questions?

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29 Jun 2014, 18:32
GMAT TIGER wrote:
Bunuel wrote:
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

x - 2sd = 58
x + 3sd = 98

SD = 8 and Mean (x) = 74 in A.

How did you get 8 and 74 exactly?
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29 Jun 2014, 18:36
I checked and I don't think there's a step by step explanation for #5 please let me know or show the step by step answer
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30 Jun 2014, 02:53
Expert's post
sagnik2422 wrote:
GMAT TIGER wrote:
Bunuel wrote:
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

x - 2sd = 58
x + 3sd = 98

SD = 8 and Mean (x) = 74 in A.

How did you get 8 and 74 exactly?

By solving the system of equations:

x - 2*sd = 58
x + 3*sd = 98

Subtract (i) from (ii): 5*sd = 40 --> sd = 8 --> substitute this value in either one of the equations to get x = 74.

Hope it's clear.
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30 Jun 2014, 03:00
Expert's post
sagnik2422 wrote:
I checked and I don't think there's a step by step explanation for #5 please let me know or show the step by step answer

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03 Jul 2014, 03:13
Bunuel wrote:
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.

We do not need to calculate the deviation, as when we add a new number to a set, it will affect the original SD of the set, in the following way:
[y is the new number added, xav is the arithmatic mean]
|y-xav| < SD -> Decrease SD
|y-xav| > SD -> Increase SD
|y - xav| = SD -> SD remains constant.
So, the nearer the |y-xav| to SD is, the less the SD changes.

Make some comparation, we got the result is D
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20 Mar 2015, 07:58
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

ANS:mean of original set is 4. on adding any of the numbers from options mean remain same as 4. SD for given set is sqrt(40/5) =sqrt(8) , by adding 2 & 6 to the set deviation will be sqrt(48/7) which is close to original value . so ANS is D
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06 Apr 2015, 00:54
Quote:
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

Sorry for bringing this up again.

I am learning some concepts around SD and I take it, that for this Question, the ans is B. So I am trying to understand how to approach such problems.
For this Q, I was able to find out 6 different possibilities for the set E :
Since the max difference between any 2 elts of the set can be 4, thus if I take 1,5 as the min and max elts(or I can take 3,7 or any others too), we have range = 4.
now to get max difference between any 2 elts <=4, I take 3 as another element in the set(since only odd elts are there in the set). So I can make a total of 6 different sets of 4 elements each, using the nos 1,3 and 5. (using permutations. I did not actually find out these sets till now)

But Im not sure how to proceed further. Should I calculate the SD of each of these sets or maybe the variance and see if its same? But I'll also have to list down these sets , since I have just calculated the no till now, using permutation.
Bunuel,
Is this the correct approach? Or am I missing anything?
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06 Apr 2015, 05:17
Expert's post
SherLocked2018 wrote:
Quote:
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

Sorry for bringing this up again.

I am learning some concepts around SD and I take it, that for this Question, the ans is B. So I am trying to understand how to approach such problems.
For this Q, I was able to find out 6 different possibilities for the set E :
Since the max difference between any 2 elts of the set can be 4, thus if I take 1,5 as the min and max elts(or I can take 3,7 or any others too), we have range = 4.
now to get max difference between any 2 elts <=4, I take 3 as another element in the set(since only odd elts are there in the set). So I can make a total of 6 different sets of 4 elements each, using the nos 1,3 and 5. (using permutations. I did not actually find out these sets till now)

But Im not sure how to proceed further. Should I calculate the SD of each of these sets or maybe the variance and see if its same? But I'll also have to list down these sets , since I have just calculated the no till now, using permutation.
Bunuel,
Is this the correct approach? Or am I missing anything?

Check here: ps-questions-about-standard-deviation-85897-40.html#p810657 or here: e-is-a-collection-of-four-odd-integers-and-the-greatest-99774.html#p769204.

Hope it helps.
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10 Aug 2015, 23:24

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}

quote]

(A) & (C) has same value for distance from mean.
(A) 2+1+1+2 = 6
(c) 1+1+1+1+2 = 6
then why A is third largest.
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17 Oct 2015, 09:53
Hard to agree to point B.. Statistics and SD are on of the more frequently appearing questions. GMAC loves this topic. OG also consists handful of questions on Stats I believe.

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