PS questions about standard deviation.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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/a-set-of-dat ... 47858.html


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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/a-certain-li ... 59504.html


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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/for-a-certai ... 28661.html


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}

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/which-of-the ... 18777.html


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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/which-of-the ... 62888.html


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%

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/a-certain-ch ... 43982.html


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}

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/which-of-the ... 18778.html


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

SOLUTION IS HERE: https://gmatclub.com/forum/ps-questions ... ml#p664302


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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/e-is-a-colle ... 99774.html


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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-a-certain ... 30542.html


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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/the-arithmet ... 29117.html


CALCULATING STANDARD DEVIATION OF A SET {x1, x2, ... xn}:
1. Find the mean, m, of the values.
2. For each value xi calculate its deviation (xi-m) 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. |Median-Mean| <= 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 mirror-images 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: https://gmatclub.com/forum/ds-questions- ... 85896.html

20. Descriptive Statistics

• Theory
  Statistics Made Easy - All in One Topic!
  GMAT Statistics 101
  Math: Standard Deviation
  How to Quickly Solve Standard Deviation Questions on the GMAT
  
  • Questions
    Standard Deviation Problems
    DS Questions
    PS Questions

For more check:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread