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# PS questions about standard deviation.

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Manager
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19 Jun 2012, 07:53
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:@Ritesh Gupta: Let me try to explain this to you.

consider a straight line that represents the mean (m). and lets consider S.D d as one unit. so you have m+d above the line m and m-d below the line m. Now given that 68% lies within one standard deviation from mean, that means 68% = m+d and m-d, which means 34% each. Next, remaining 32% (100-68) is above m+d and below m-d, again equally distributed, hence, 16% each.

Thus, if you graphically visualize, the question is asking you, (m+d) + (m-d) + below (m-d) = 34 + 34 + 16 = 84.

I hope its clear!

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19 Jun 2012, 09:32
pavanpuneet wrote:
:@Ritesh Gupta: Let me try to explain this to you.

consider a straight line that represents the mean (m). and lets consider S.D d as one unit. so you have m+d above the line m and m-d below the line m. Now given that 68% lies within one standard deviation from mean, that means 68% = m+d and m-d, which means 34% each. Next, remaining 32% (100-68) is above m+d and below m-d, again equally distributed, hence, 16% each.

Thus, if you graphically visualize, the question is asking you, (m+d) + (m-d) + below (m-d) = 34 + 34 + 16 = 84.

I hope its clear!

thanks pavan, that was quite helpful
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26 Jun 2012, 02:12
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

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26 Jun 2012, 02:28
casanjiv 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

Question says greatest difference between any two integers in E is 4, how come the range is 4.

"The greatest difference between any two integers in E is 4" does not mean that if you pick any two integers their difference will be 4. It means that the greatest difference you can get from any pair of integers from E is 4. Since the range is the difference between the largest and the smallest elements of a set, then 4 must be the range of E.

Check this for more on this question: ps-questions-about-standard-deviation-85897-40.html#p810657 (or: e-is-a-collection-of-four-odd-integers-and-the-greatest-99774.html#p769204).

Hope it helps.
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30 Jun 2012, 15:10
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

the mean of the numbers is 4. and the SD is √8
now for each option mean remains the same
but when mean-4 =0 the the SD remains as the original one that is √8

so the answer is B i think !

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30 Jun 2012, 15:32
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

there is a competition in E and A because as we add squares of the difference to the previous sums, the denominator is increasing too. so we need to check for A and E , A is √7.4 almost and E is √7.2 thus A

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30 Jun 2012, 15:34
i went wrong in the previous reply as i forgot to consider the increasing denominator too.. option B will have SD much less than previous one.. question 2 is based on the same concept. please excuse me for the mistake

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10 Jul 2012, 23:13
Bunuel wrote:
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|}

How is II. {0, s-t, s-r} derived?

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10 Jul 2012, 23:39
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let me explain, it is 0,s-t,s-r, thus, ... if say we multiply this set by -....0,t-s,r-s....now add s in all the terms <adding/subtracting same number in the set does not change the SD>....the set becomes <s,t,r>...

for the third one, the set may |r|,|s|,|t|..now since...now since SD=sqrt {summation(x-mean)2}/n...u can see that SD will differ as x will differ...

I hope it is clear...

Pavan

I hope it clear.

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11 Jul 2012, 01:58
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Expert's post
dianamao wrote:
Bunuel wrote:
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|}

How is II. {0, s-t, s-r} derived?

Note that:
If we add or subtract a constant to each term in a set: SD will not change.

Changing the signs of the element of a set (multiplying by -1) has no effect on SD.

Now, multiply {s, r, t} by -1 to get {-s, -r, -t}. According to the above these two sets have the same standard deviation.

Next, add s to each term to get {0, s-r, s-t}, again according to the above {0, s-r, s-t} and {-s, -r, -t} have the same standard deviation.

So, {s, r, t}, {-s, -r, -t} and {0, s-r, s-t} have the same standard deviation.

Hope it's clear.
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11 Jul 2012, 05:19
Thanks Bunuel Makes sense.

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26 Jul 2012, 11:29
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 been given
Set A - Elements with Median > Mean.
Set B - Elements with Median = Mean.

I. Z > N : We cannot compare standard deviations of two alien sets - False. [ we don't know what is A, what is B and what is the relationship between them] - So I Out.
II. R > M : We are comparing mean of the combined sets with mean of set B. We have no knowledge of number of elements (cardinality) of A or B. Consider this as a case of weighted average. Unless we know the number of elements of A, number of elements of B and mean values of Both the sets, we can not evaluate this.
For Instance, If A has 4 elements and mean of 5, B has 10 elements and mean of 8 .Then M =8, R = (4*5 + 10*8)/(4+10) = 100/14 = approx 7. So R < M
now, suppose Set A has a mean of 20 instead of 5. M remains unchanged. R = (4*20 + 10*8)/(4+10) = 160/14 = approx. 11. So R > M
So II is Out.
III. Q > R : Mean > Median. Again as above, since we do not know anything about composition of A and B. This might be true or false depending on distribution of A and B.
As there is no co-relationship provided between A and B, we cannot derive any conclusion.

So the answer in E. None.

The problem could be twisted, if instead of L-M=0, we would be given N=0.
In that case, Z is always >0. So answer would be A then.
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30 Jul 2012, 09:10
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.

Hi Bunuel .. why cant we have a set {1,1,1,1} .. the question says that the greatest difference between any 2 numbers is 4 but it doesnt say that the range has to be 4.

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26 Aug 2012, 11:53
can anyone help me to understand question 1? I pick B as 4 & 4 are the two values close to mean and therefore will not chand SD.

AA

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15 Oct 2012, 14:11
Hello,
Could someone kindly check if my explanation to Question 6 is correct?

According to the question, m+d=m-d=68%, therefore, 2d=68%, d=34%. Question is asking what is the percentage value of m+d, we already know the distribution is symmetric about the mean, therefore, mean is at 50%. Therefore, m+d=50%+34%=84%.

Cheers

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25 Jan 2013, 20:01
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

How is OA Option E - 6 and 6 -the average of the set in the second quetsion while OA is not the average (Option B - 4 and 4) in the first question?

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08 Apr 2013, 20:10
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.

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08 Apr 2013, 21:11
Apex231 wrote:
4. Which of the following distribution of numbers has the greatest standard deviation?
(A) {-3, 1, 2} - r = 5 , n =3
(B) {-2, -1, 1, 2} - r = 4 , n = 4
(C) {3, 5, 7} - r = 4 , n = 3
(D) {-1, 2, 3, 4} - r = 5 , n = 4
(E) {0, 2, 4} - r = 4 , n = 3

r = range
n = number of elements

r/n is max for
[Reveal] Spoiler:
A.

Is this the right way to solve? or any better method?

Often more is the range, more is the standard deviation. But, this may not be true always.

Case A: Observe the following two sets with same number of elements.

Set A1: 10, 20, 20, 20, 20, 30
Range = 20
Standard Deviation = 6.32455532

Set A2: 12, 12, 12, 28, 28, 28
Range = 16
Standard Deviation = 8.76356092

Here, stdev of A1 is < stdev of A2 though range of A1 > range of A2.

Case B: Observe the following two sets with different number of elements.

Set B1: 10, 20, 20, 20, 30
Range = 20
No. of elements = 5
Standard Deviation = 7.071067812

Set B2: 12, 12, 12, 28, 28, 28
Range = 16
No. of elements = 6
Standard Deviation = 8.76356092

Here, stdev of B1 is < stdev of B2 though range of B1 > range of B2 and no. element is more in B2.

Case C: Observe the following two sets with different number of elements.

Set C1: 10 20 20 20 20 30
Range = 20
No. of elements = 6
Standard Deviation = 6.32455532

Set C2: 12 12 12 28 28
Range = 16
No. of elements = 5
Standard Deviation = 8.76356092

Here, stdev of C1 is < stdev of C2 though range of C1 > range of C2.

Standard deviation is not directly related to range and number of elements. Stdev is how spread are the elements from the mean. As you can see here, in sets A1, B1, and C1, more elements are closer to mean and that's how those sets have lower stdev.

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08 Apr 2013, 21:28
GMAT TIGER wrote:
Bunuel wrote:
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}

Look for range and # of elements in the set.

A set with higher the range and fewer the number of element has the higher SD. i.e. A.

This logic may not be true always. The following post has examples where the result is different.

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09 Apr 2013, 00:21
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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}

(A) standard deviation = 1.58113883
(B) standard deviation = 0.707106781
(C) standard deviation = 1.414213562
(D) standard deviation = 2.236067977
(E) standard deviation = 3.16227766

To go by logic:

All the sets got same number of elements; so, not considering any effect from that.

Set E has highest range and elements are more distributed towards edge. --> set has the highest standard deviation.
Set D has next highest range and but little concentric towards mean. Still the distribution looks better than remaining sets. Thus, this set has the second highest standard deviation.
Set B has lowest range and elements are mainly centered towards mean. Thus, this set has the lowest standard deviation.
Now we are left with options A and C. Range of set A is more than range of set C. Elements of A are evenly distributed (more spread) and elements of C are concentrated towards one side. From these two observations, we can speculate that option A has higher standard deviation than that of option C.

Note: All these sets have small sample size, very small range, and the numbers are very close to each other. This scenario say little without calculation and, hopefully, real GMAT will not come up with this sort of questions.

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