PS questions about standard deviation.

ANSWERS (OA):

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
Answer: D.

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
Answer: E.

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
Answer: A.

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}
Answer: 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
Answer: D.

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%
Answer: 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}
Answer: A.

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
Answer: E

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
Answer: B

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
Answer: A.

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
Answer: A.

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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.

Good work.
1. I agree with point A.
2. I do not fully agree with point B. Agree with time saving part is ok but want to work on understanding the SD issues/questions in more detail as well. I cannot depend on praying for no SD questions, relying on chance.

Adding the value equivalant to mean lowers the SD.

E. 6 & 6.

GMAT TIGER point B no way means that one should't work on SD issues, not at all. The point B. means that taking into account the probability of getting SD on real GMAT one should spread the time wisely. Right the way you've mentioned: time saving is the key issue here.
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x - 2sd = 58
x + 3sd = 98

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

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.

(D) I and II only

Anything added/deducted to the set elements or the set elements deducted from anything results in no change in SD.

I. Deduct 2 from each of the elements in set result I. i.e. {r-2, s-2, t-2}
II. Deduct each set elements from s. The new set elements in II i.e. {0, s-t, s-r} result.
III. Taking the absolute value of the set elements is not the same as deducuting or adding the same. This act would not change the SD if all set elements have the same sign (+ve or -ve).

Suppose s = 5 and r = 6 and t = 7, {|r|, |s|, |t|} and {s, r, t} have same SD.
If s = -5 and r = -6 and t = -7, {|r|, |s|, |t|} and {s, r, t} have same SD.
If s = -5 and r = 6 and t = 7, {|r|, |s|, |t|} and {s, r, t} have different SD.

III is not always a true case.

Thats a real good question however I took more than 5 minuets to understand as I went in a wrong direction.

Since the greatest difference between any two elements in E is 4, different elements in E, lets say, could be: (3, x, x, 7) where x could be any of 3 or 5 or 7.

How many possibilities: {3,3,3,7}, {3,3,5,7}, {3,3,7,7}, {3,5,5,7}, {3,5,7,7}, {3,7,7,7}. So 6.

D.

Probably C only. III.

The order is:
1. (E) {-1, 1, 3, 5, 7}
2. (D) {0, 2, 3, 4, 6}
3. (A) {1, 2, 3, 4, 5}
4. (C) {2, 2, 2, 4, 5}
5. (B) {2, 3, 3, 3, 4}

i.e. A.

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)

Thanks Bunuel.



The mean of the set is 4.
sqrt[((0-4)^2+(2-4)^2+(6-4)^2+(8-4)^2)/5] = sqrt[(16+4+4+16)/5] = sqrt(40/5) = sqrt(8)

The mean of set 0,8 is 4. Std.dev. is sqrt[(0-4)^2+(8-4)^2/2]=sqrt(8). Answer is E.



E. If the set contained only 6 it's standard deviation would be 0. Since it is positive we can reduce the std.dev. by adding to integers equal to the mean - so answer is E.



2 eq in 2 unknowns. Let x denote mean and let y denote std.dev.

58 = x - 2y <=> x= 58 + 2y (1) AND
98 = x + 3y <=> 98 = 58+5y <=> y=8. Insert into (1) to get x=58+2*8= 74.
Answer is A.



A good way to go about these questions is to look for the range of the set compared to the number of elements in the set. Ceteris paribus it holds that the higher the range - the higher the std.dev and the higher the number of elements - the lower the std.dev. A is the only set with a range of 5 and only 3 numbers. Furthermore none of the numbers represent the average (0) and therefore all contribute to the std.dev. Answer is A.


The absolute value of the numbers doesn't matter since it is the differences to the mean that enters into the std.dev. Thus I is the same as in the Q. The set in 2 is the set (s,s,s) subtracted by (s,r,t). This gives the same std.dev. as in set {s,r,t} (If you're in doubt try plugging in numbers. In III there is clearly a difference between the set {-1,1,1} and {1,1,1} so this does not necessarily have the same std.dev as {s,r,t}. The answer is D.



I am little uncertain about the meaning of this question. I will assume that you mean that 68% of the distribution lies one standard deviation above the mean (alternative interpretation is that 68% of the distribution lies within 1 std.dev from the mean).

The total mass of the distribution is 100%. Just subtract 68% from the total mass to get this rest = 32%. Answers is B.



OK, gotta hit the sack now. Thanks again for the questions Bunuel.

Questions 1 and 9 are solved incorrectly. One of two answers for 6 is incorrect. 10 and 11 aren't solved yet, though they are relatively easy.

The hardest questions in this set are 8 and 9. Probably they are 750+ problems, so would be interesting to see the solutions for them. Also please note that I don't have the OA for 8!, only my own solution.

Good luck.
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Thats a real tricky question and is of 750+ level. Did not think that some of the SDs are of equal value.
Revised to 4 in B.

I'll attempt an easy one

10) A
Mean = 20.0
S.D = 3
More than 2.5 SD from mean means either less than 20- (2.5x3=7.5) = 12.5 or greater than 20+7.5= 27.5

By The same logic
11) A
13.5-3=10.5

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

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(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.

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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(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):
[1,5,1,1]
[1,5,5,5]
[1,5,3,3]
[1,5,1,3]
[1,5,1,5]
[1,5,3,5]

total of 6. 1st and 2nd have the same SD. Left with 5 other cases.