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PS questions about standard deviation. [#permalink]

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Lately, many questions were asked about the standard deviation. So I'm posting here my collection of PS on SD, plus some tips about it.

Please note the following: A. I was assured MANY TIMS, by various GMAT tutors, that GMAT won't ask you to actually calculate SD, but rather to understand the concept of it. Though KNOWING how it's calculated helps in understanding the concept. B. During the real GMAT it's highly unlikely to get more than one ot two question on SD (as on combinatorics), actually you may see none, so do not spend too much of your preparation time on it, it's better to concentrate on issues you'll definitely face on G-day.

Many questions below are easy, some are tough, but anyway they are good to master in solving SD problems. I'll post OA after some discussions. Please provide your way of thinking along with the answer. Thanks.

Here we go:

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

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}

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

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%

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}

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

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

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.

Re: PS questions about standard deviation. [#permalink]

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27 Oct 2009, 20:14

Bunuel wrote:

Please note the following: A. I was assured MANY TIMS, by various GMAT tutors, that GMAT won't ask you to actually calculate SD, but rather to understand the concept of it. Though KNOWING how it's calculated helps in understanding the concept.

B. During the real GMAT it's highly unlikely to get more than one ot two question on SD (as on combinatorics), actually you may see none, so do not spend too much of your preparation time on it, it's better to concentrate on issues you'll definitely face on G-day.

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. _________________

Re: PS questions about standard deviation. [#permalink]

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27 Oct 2009, 20:20

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Bunuel wrote:

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

Adding the value equivalant to mean lowers the SD.

Re: PS questions about standard deviation. [#permalink]

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27 Oct 2009, 20:23

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GMAT TIGER wrote:

Bunuel wrote:

Please note the following: A. I was assured MANY TIMS, by various GMAT tutors, that GMAT won't ask you to actually calculate SD, but rather to understand the concept of it. Though KNOWING how it's calculated helps in understanding the concept.

B. During the real GMAT it's highly unlikely to get more than one ot two question on SD (as on combinatorics), actually you may see none, so do not spend too much of your preparation time on it, it's better to concentrate on issues you'll definitely face on G-day.

Good work. 1. I agree with point A. 2. I do not fully agree with point B. Agree with time saving issue but still work on understanding the SD issues/questions too. I cannot depend on praying for no SD question as I do not want to take a chance.

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. _________________

Re: PS questions about standard deviation. [#permalink]

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27 Oct 2009, 20:25

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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

Re: PS questions about standard deviation. [#permalink]

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27 Oct 2009, 20:43

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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|}

(A) I only (B) II only (C) III only (D) I and II only (E) I and III only

(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.

Re: PS questions about standard deviation. [#permalink]

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27 Oct 2009, 21:09

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Bunuel 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

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.

Re: PS questions about standard deviation. [#permalink]

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27 Oct 2009, 21:30

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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

Re: PS questions about standard deviation. [#permalink]

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27 Oct 2009, 21:33

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Bunuel wrote:

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}

Re: PS questions about standard deviation. [#permalink]

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28 Oct 2009, 12:14

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)

Re: PS questions about standard deviation. [#permalink]

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28 Oct 2009, 14:35

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Thanks Bunuel.

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

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.

Bunuel wrote:

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

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.

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

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.

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}

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.

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|} (A) I only (B) II only (C) III only (D) I and II only (E) I and III only

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.

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 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.

Bunuel wrote:

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}

Use the same principle I described above. All sets have 5 numbers. E have the larges range, and D comes in second. A has the third largest range and greater dispersion among numbers so answer is A.

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

Re: PS questions about standard deviation. [#permalink]

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29 Oct 2009, 20:32

Expert's post

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.

Re: PS questions about standard deviation. [#permalink]

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30 Oct 2009, 00:09

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GMAT TIGER wrote:

Bunuel 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

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.

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. _________________

Re: PS questions about standard deviation. [#permalink]

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30 Oct 2009, 09:37

Bunuel wrote:

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.

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

Re: PS questions about standard deviation. [#permalink]

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30 Oct 2009, 09:39

gmattokyo wrote:

Bunuel wrote:

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.

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

Re: PS questions about standard deviation. [#permalink]

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30 Oct 2009, 09:59

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.

Re: PS questions about standard deviation. [#permalink]

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30 Oct 2009, 10:30

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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 -----------------------------------

(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.

Re: PS questions about standard deviation. [#permalink]

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30 Oct 2009, 13:33

gmattokyo 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 -----------------------------------

(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. [1,5,1,1] 2. [1,5,5,5] 3. [1,5,3,3] 4. [1,5,1,3] 5. [1,5,1,5] 6. [1,5,3,5]

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

1. [1,5,1,1] and 2. [1,5,5,5] have SD of 2. 3. [1,5,3,3] has a SD of 1.63. 4. [1,5,1,3] and 6. [1,5,3,5] have 1.91. 5. [1,5,1,5] has a SD of 2.31.

So there are altogather 4 different SDs. _________________

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