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Re: PS questions about standard deviation.
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19 Jun 2012, 06:53
:@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 md below the line m. Now given that 68% lies within one standard deviation from mean, that means 68% = m+d and md, which means 34% each. Next, remaining 32% (10068) is above m+d and below md, again equally distributed, hence, 16% each.
Thus, if you graphically visualize, the question is asking you, (m+d) + (md) + below (md) = 34 + 34 + 16 = 84.
I hope its clear!



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Re: PS questions about standard deviation.
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Re: PS questions about standard deviation.
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10 Jul 2012, 22:13
Bunuel wrote: 5. Which of the following has the same standard deviation as {s,r,t}?
I. {r2, s2, t2} II. {0, st, sr} III. {r, s, t}
How is II. {0, st, sr} derived?



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Re: PS questions about standard deviation.
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10 Jul 2012, 22:39
let me explain, it is 0,st,sr, thus, ... if say we multiply this set by ....0,ts,rs....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(xmean)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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Re: PS questions about standard deviation.
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11 Jul 2012, 00:58
dianamao wrote: Bunuel wrote: 5. Which of the following has the same standard deviation as {s,r,t}?
I. {r2, s2, t2} II. {0, st, sr} III. {r, s, t}
How is II. {0, st, sr} 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, sr, st}, again according to the above {0, sr, st} and {s, r, t} have the same standard deviation. So, {s, r, t}, {s, r, t} and {0, sr, st} have the same standard deviation. Hope it's clear.
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Re: PS questions about standard deviation.
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26 Jul 2012, 10:29
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 corelationship provided between A and B, we cannot derive any conclusion. So the answer in E. None. The problem could be twisted, if instead of LM=0, we would be given N=0. In that case, Z is always >0. So answer would be A then.
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Re: PS questions about standard deviation.
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08 Apr 2013, 19:10
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. Correct answer is 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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Re: PS questions about standard deviation.
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08 Apr 2013, 20:11
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 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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Re: PS questions about standard deviation.
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08 Apr 2013, 23:21
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
Answer is A.
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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Re: PS questions about standard deviation.
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09 Apr 2013, 01:52
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. Correct answer is 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 The correct answer is D.
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Re: PS questions about standard deviation.
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09 Apr 2013, 02:22
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
The correct answer is D. 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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Re: PS questions about standard deviation.
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29 Jun 2014, 17:32
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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Re: PS questions about standard deviation.
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IanStewart wrote: 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. Official answer from Manhattan Prep. If X – Y > 0, then X > Y and the median of A is greater than the mean of set A. If L – M = 0, then L = M and the median of set B is equal to the mean of set B. I. NOT NECESSARILY: According to the table, Z > N means that the standard deviation of set A is greater than that of set B. Standard deviation is a measure of how close the terms of a given set are to the mean of the set. If a set has a high standard deviation, its terms are relatively far from the mean. If a set has a low standard deviation, its terms are relatively close to the mean. Recall that a median separates the set into two as far as the number of terms. There is an equal number of terms both above and below the median. If the median of a set is greater than the mean, however, the terms below the median must collectively be farther from the median than the terms above the median. For example, in the set {1, 89, 90}, the median is 89 and the mean is 60. The median is much greater than the mean because 1 is much farther from 89 than 90 is. Knowing that the median of set A is greater than the mean of set A just tells us that the terms below set A’s median are further from the median than the terms above set A’s median. This does not necessarily imply that the terms, overall, are further away from the mean than in set B, where the terms below the median are the same distance from the median as the terms above it. In fact, a set in which the mean and median are equal can have a very high standard deviation if the terms are both far below the mean and far above it. II. NOT NECESSARILY: According to the table, R > M implies that the mean of set [A + B] is greater than the mean of set B. This is not necessarily true. When two sets are combined to form a composite set, the mean of the composite set must either be between the means of the individual sets or be equal to the mean of both of the individual sets. To prove this, consider the simple example of one member sets: A = [3], B = [5], A + B = [3, 5]. In this case the mean of A + B is greater than the mean of A and less than the mean of B. We could easily have reversed this result by reversing the members of sets A and B. III. NOT NECESSARILY: According to the table, Q > R implies that the median of the set [A + B] is greater than the mean of set [A + B]. We can extend the rule given in statement II to medians as well: when two sets are combined to form a composite set, the median of the composite set must either be between the medians of the individual sets or be equal to the median of one or both of the individual sets. While the median of set A is greater than the mean of set A and the median of set B is equal to the mean of set B, set [A + B] might have a median that is greater or less than the mean of set [A + B]. Therefore none of the statements are necessarily true and the correct answer is E.
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Re: PS questions about standard deviation.
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14 Oct 2017, 00:46
Bunuel wrote: 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: http://gmatclub.com/forum/psquestions ... ml#p810657 (or: http://gmatclub.com/forum/eisacollec ... ml#p769204). Hope it helps. Bunuel thank you for such a classified question,I could cover SD topic, but unfortuantely I was just wrong about this question. Why we don't assume negative numbers? such as {1,1,1,3} etc...??



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Re: PS questions about standard deviation.
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14 Oct 2017, 00:53
soodia wrote: Bunuel wrote: 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: http://gmatclub.com/forum/psquestions ... ml#p810657 (or: http://gmatclub.com/forum/eisacollec ... ml#p769204). Hope it helps. Bunuel thank you for such a classified question,I could cover SD topic, but unfortuantely I was just wrong about this question. Why we don't assume negative numbers? such as {1,1,1,3} etc...?? {1, 1, 1, 3} > range = largest  smallest = 3  (1) = 4.
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Re: PS questions about standard deviation.
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14 Oct 2017, 02:12
Bunuel wrote: soodia wrote: so what's the problem?? I think I'm confused. What is your question? P.S. Check here: https://gmatclub.com/forum/psquestions ... ml#p810657 and here: https://gmatclub.com/forum/eisacolle ... ml#p769204 for more. The question says: E is a collection of four odd integers and the greatest difference between any two integers in E is 4. in set {1,1,1,3} or {1,1,1,3},..... the greatest difference is 4 so why we do not assume these sets to calculate the SD?



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Re: PS questions about standard deviation.
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14 Oct 2017, 02:33
soodia wrote: Bunuel wrote: soodia wrote: so what's the problem?? I think I'm confused. What is your question? P.S. Check here: https://gmatclub.com/forum/psquestions ... ml#p810657 and here: https://gmatclub.com/forum/eisacolle ... ml#p769204 for more. The question says: E is a collection of four odd integers and the greatest difference between any two integers in E is 4. in set {1,1,1,3} or {1,1,1,3},..... the greatest difference is 4 so why we do not assume these sets to calculate the SD? I believe I already answered this question on previous pages. {1,1,1,3} has the same SD as {1, 1, 3, 5}: by by adding 2 to each term. {1,1,1,3} has the same SD as {1, 3, 3, 5}: by by adding 2 to each term. If we add or subtract a constant to each term in a set SD will not change.
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07 Apr 2018, 12:58
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 OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/asetofdat ... 47858.html2. 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 OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/acertainli ... 59504.html3. 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 OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/foracertai ... 28661.html4. 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} OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/whichofthe ... 18777.html5. Which of the following has the same standard deviation as {s,r,t}? I. {r2, s2, t2} II. {0, st, sr} III. {r, s, t}(A) I only (B) II only (C) III only (D) I and II only (E) I and III only OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/whichofthe ... 62888.html6. 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% OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/acertainch ... 43982.html7. 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} OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/whichofthe ... 18778.html8. 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 SOLUTION IS HERE: https://gmatclub.com/forum/psquestions ... ml#p6643029. 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 OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/eisacolle ... 99774.html10. 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 OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/ifacertain ... 30542.html11. 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 OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/thearithmet ... 29117.htmlCALCULATING STANDARD DEVIATION OF A SET {x1, x2, ... xn}:1. Find the mean, m, of the values. 2. For each value xi calculate its deviation (xim) 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. TIPS: 1. MedianMean <= SD.
2. Variance is the square of the standard deviation.
3. If Range or SD of a list is 0, then the list will contain all identical elements. And vise versa: if a list contains all identical elements then the range and SD of a list is 0. If the list contains 1 element: Range is zero and SD is zero.
4. SD is always >=0. SD is 0 only when the list contains all identical elements (or which is same only 1 element).
5. Symmetric about the mean means that the shape of the distribution on the right and left side of the curve are mirrorimages of each other.
6. 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.
7. If we increase or decrease each term in a set by the same percent: Mean will increase or decrease by the same percent. SD will increase or decrease by the same percent.
8. Changing the signs of the element of a set (multiplying by 1) has no effect on SD.
9. The SD of any list is not dependent on the average, but on the deviation of the numbers from the average. So just by knowing that two lists having different averages doesn't say anything about their standard deviation  different averages can have the same SD.You can also check collection of DS questions of SD at: http://gmatclub.com/forum/dsquestions ... 85896.html20. Descriptive Statistics For more check: ALL YOU NEED FOR QUANT ! ! !Ultimate GMAT Quantitative Megathread
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Re: PS questions about standard deviation. &nbs
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07 Apr 2018, 12:58



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