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# PS questions about standard deviation.

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31 Oct 2010, 18:00
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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.
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17 Nov 2010, 00:36
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17 Nov 2010, 10:46
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 solve this x - 2sd = 58 and x + 3sd = 98 ?
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17 Nov 2010, 10:54
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shrive555 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 solve this x - 2sd = 58 and x + 3sd = 98 ?

A score of 58 was 2 standard deviations below the mean --> 58=Mean-2*sd
A score of 98 was 3 standard deviations above the mean --> 98=Mean+3*sd

Subtract (1) from (2) --> 98-58=Mean+3*sd-(Mean-2*sd) --> 40=5sd --> sd=8 --> Mean=58+2*sd=58+16=74.

Hope it's clear.
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17 Nov 2010, 20:56
This thread was really useful to me. Thanks a ton to everybody for their parts. I didn't understand the wording of question 9 but i'll check the links that you've posted for better explanations. Thanks again!
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22 Nov 2010, 07:14
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Interesting Questions on Mean,Median and Mode.
(Plucked from Dr.Math) I believe these type of conceptual questions would help us all. Very fundamental but somewhat tricky for a person like me. If this is not the right place to include these type of questions or that they do not qualify the post take action accordingly. Thanks.

Q1) Find a set of five data values with modes 0 and 2, median 2, and
mean 2.

Soln:
Mode = values appearing most often,
Median = value with as many other values above as below,
Mean = average (sum of the values divided by the number of values).

There are five values. If all were different, there would be five modes, but there are only two. The two modes must appear at least twice. They cannot appear three times each, because then you would have at least six values, not five. Thus four of the values must be 0, 0, 2, and 2.

For 2 to be the median, the remaining value, call it x, must be greater than 2. If 0 < x < 2, then x would be the median, and if
x < 0, 0 would be the median.

Then the mean is (0+0+2+2+x)/5 = 2, which you can solve for x.

Q2) We are told that the median of five numbers is 5, the mode is
1, and the mean is 4. Find the five numbers?

Soln: We can start by drawing a blank for each of the values. Then we'll try to fill them in, putting them in ascending order as we go. Here are the five blanks:

__ __ __ __ __

Now, what do we know about the numbers? We know that "the median of five numbers is 5." The median is the middle number when arranged in ascending order, so let's put it there:

__ __ 5 __ __

What else do we know? We're told that "the mode is 1." That means that 1 has to appear more than any other number. Since 1 is less than 5, all 1's will have to go to the left of the 5. We know we need at least two of them (otherwise, 5 would be a mode as well), so both blanks on the left will have to be 1's. Putting them in, we have:

1 1 5 __ __

Now the last clue is, "the mean is 4." The mean is the "average" of the numbers. It is computed by taking the sum of the numbers and dividing it by the number of numbers. Algebraically, if we call our values A, B, C, D and E, we'd write:

M = (A+B+C+D+E) / 5

Since we already know the first three numbers, let's plug them in for A, B, and C. We also know that the mean is 4, so we'll plug that in too:

4 = (1+1+5+D+E) / 5

4 = (7+D+E) / 5

Now let's solve for D+E, the two numbers we don't know:

4 = (7+D+E) / 5

4 * 5 = (7+D+E)

20 = 7+D+E

20 - 7 = D+E

D+E = 13

So the sum of the last two numbers must be 13. We also know that each of those two numbers must be greater than 5. What two numbers will work? Only 6 and 7 (can you think of WHY only 6 and 7 work?) So our five numbers must be:

1 1 5 6 7

To check; median = 5 (check), mode = 1 (check), mean is:

M = (1+1+5+6+7) / 5

= 20 / 5

= 4 (check)
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22 Nov 2010, 09:09
1.B
2.C
3.A
4.A
5.C
6.D
7.A
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22 Nov 2010, 09:10
I can see that I have some wrong answer probably i need to work the concept....
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03 Oct 2011, 01:39
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

m-2*s.d=58
m+3*s.d=98

subracting both the equations:
s.d=8
hence mean=74

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}

A) range -->2-(-3)=5
B) range -->2-(-2)=4
C)range -->7-3=4
D)range -->4-(-1)=5
E)range -->4-(0)=4

if we calacualte the SD for A and D option (as range is greatest) --> we get SD for A(option) >D option

Hence A

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

Option D
S.d of {s,r,t} is same as {r-2, s-2, t-2} and {0, s-t, s-r}

Example {s,r,t}={5,6,7}
{s-2, r-2, t-2}={5-2,6-2,7-2}={3,4,5}
{0, s-t, s-r}={0,5-6,7-5}={-2,-1,0}

hence S.d for all the three will be same.

for modulus we are not sure whether the value is positive or negative...hence not sure S.D will be the same.

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%

m+d and m-d =68%that to is divided into 2 parts 34 % and 34% on eithr side of the mean)
rest is 32% (that to is divided into 2 parts 16 % and 16% on eithr side of the mean)

hence 34+16+34=84% (less than m+d)

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) range -->5-1 =4
B) range -->4-2=2
C)range -->5-2=3
D)range -->6-0=6
E)range -->7-(-1)=8

hence the answer is C)range -->5-2=3

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

SD=2.5 hence 2.5*S.D=3*2.5=7.5

mean=20.0 hence 2.5*S.D from mean=12.5 or 27.5

hence greater than mean =12 will be the answer

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

SD=1.5 hence 2*S.D=1.5*2=3

mean =13.5 hence 2*S.D less than the mean=10.5

hence a
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06 Oct 2011, 06:55
gmattokyo wrote:
Trying the difficult one...

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

----------------------------------
(C) III only
I. SD of one set is greater than another. We cannot prove this as no information is given on the actual data of the individual sets
II. Mean of combined set may or may not be greater than one of the given sets.
Consider Set A [1, 3, 3, 3] Median-3, Mean-2.5
Set B [1, 4, 4, 4] Median-4, Mean-3.25
Set A+B [1, 1, 3, 3, 3, 4, 4, 4] Median-3, Mean-2.8
In this case R is not greater than M. But if you interchange set A & B, R>M.

The OA is E. Can somebody explain this?
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14 Jan 2012, 18:42
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
58 = m - 2 d
98 = m + 3 d

58 + 2d= 98 - 3d
5d = 40
d = 8

m = 58 + 2d = 58 + 2(8) = 74

Option
[Reveal] Spoiler:
A
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14 Jan 2012, 18:44
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

SD is deviation of elements of sets from mean.
Mean = 6
SD = d
To reduce SD further, add elements that have SD less than d.
6 and 6 have zero SD from mean = 6.
Option
[Reveal] Spoiler:
E
.
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14 Jan 2012, 18:47
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?
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14 Jan 2012, 19:09
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

m = 13.5
d = 1.5
V = m - 2d = 13.5 - 2(1.5) =
[Reveal] Spoiler:
10.5
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21 Mar 2012, 05:51
@ Bunuel

please explain the logical understanding/ difference between questions 10 and 11.
my doubt for q-11 is m getting

value=13.5+ 3= 16.5 and 10.5 so the value should be between 10.5 and 16.5,

however the options and the corresponding OA just confuses me.

pls explain
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21 Mar 2012, 06:15
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@ Bunuel

please explain the logical understanding/ difference between questions 10 and 11.
my doubt for q-11 is m getting

value=13.5+ 3= 16.5 and 10.5 so the value should be between 10.5 and 16.5,

however the options and the corresponding OA just confuses me.

pls explain

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

Value is more than 2.5SD from the mean means that the distance between the mean and the value must be more than 2.5*SD=7.5. So the value must be either less than 20-7.5=12.5 or more than 20+7.5=27.5.

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

The value which is exactly two SD less than the mean is: mean-2*SD=13.5-2*1.5=10.5.

Hope it's clear.
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21 Mar 2012, 06:39
+1
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20 Apr 2012, 11:28
With respect to the first question, we say that closer to the mean ... low variation in the SD. Then why is B incorrect....Or rather how to attack such problems?
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20 Apr 2012, 11:33
Expert's post
pavanpuneet wrote:
With respect to the first question, we say that closer to the mean ... low variation in the SD. Then why is B incorrect....Or rather how to attack such problems?

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16 Jun 2012, 10:51
rvthryet wrote:
Bunuel wrote:

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%

I m a little confused between 32% and 84% but i think 84% makes more sense.

If it is symmetric about the mean, then data less than m is and data greater than m is both 50%.

Then, each of the area from m to m+d and m to m-d would be 68/2=34%.

Thus, for data to be less than m+d, we have

Data from m+d to m = 34%
Data less than m = 50%
Thus, total 84% (D)

I am not clear with the solution provided above, can anyone elaborate, or submit another solution. In fact I am not clear about the question itself
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