DS questions about standard deviation

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Lately, many questions were asked about the standard deviation. So I'm posting here my collection of DS problems 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. What is the standard deviation of Company R’s earnings per month for this year?
(1) The standard deviation of Company R’s earnings per month in the first half of this year was $2.3 million.
(2) The standard deviation of Company R’s earnings per month in the second half of this year was $3.9 million.

2. What is the standard deviation of Q, a set of consecutive integers?
(1) Q has 21 members.
(2) The median value of set Q is 20.

3. Lifetime of all the batteries produced by certain companies have a distribution which is symmetric about mean m. If the distribution has a standard deviation of d , what percentage of distribution is greater than m+d?
(1) 68 % of the distribution in the interval from m-d to m+d, inclusive
(2) 16% of the distribution is less than m-d

4. Question deleted

5. List S and list T each contain 5 positive integers, and for each list the average of the integers in the list is 40. If the integers 30,40 and 50 are in both lists , is the standard deviation of the integers in list S greater than the standard deviation of the integers in list T?
(1)The integer 25 is in list S
(2)The integer 45 is in list T

6. Set T consists of odd integers divisible by 5. Is standard deviation of T positive?
(1) All members of T are positive
(2) T consists of only one member

7. Set X consists of 8 integers. Is the standard deviation of set X equal to zero?
(1) The range of set X is equal to 3
(2) The mean of set X is equal to 5

8. {x,y,z}
If the first term in the data set above is 3, what is the third term?
(1) The range of this data set is 0.
(2) The standard deviation of this data set is 0.

9. Question deleted

10. A scientist recorded the number of eggs in each of 10 birds' nests. What was the standard deviation of the numbers of eggs in the 10 nests?
(1) The average (arithmetic mean) number of eggs for the 10 nests was 4.
(2) Each of the 10 nests contained the same number of eggs.

11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?
(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.


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 PS questions of SD at: ps-questions-about-standard-deviation-85897.html

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.

Answer is A. But, the SD will decrease by 30%.

1.
Any set of 21 consecutive numbers will have exact same standard deviation, irrespective of its value.
Std dev for {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21} = 6.0553
Std dev for {11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31} = 6.0553
Believe this is true for all evenly spaced sets with equal numbers of elements and same common difference.
Sufficient.

2.
Set can be:
{19,20,21}. Median=20. Std Dev=0.8165
{15,16,17,18,19,20,21,22,23,24,25}. Median=20. Std Dev=3.16228
Not Sufficient.

Thus, the answer is A.

A. If range is not zero, then SD is not 0. Suff..
B. tells nothing..

My ans
1C
2C
3D
4A
5C
6D
7A
8D
9E
10B
11A

ANSWERS (OA):

1. What is the standard deviation of Company R’s earnings per month for this year?
(1) The standard deviation of Company R’s earnings per month in the first half of this year was $2.3 million.
(2) The standard deviation of Company R’s earnings per month in the second half of this year was $3.9 million.
Answer: E.

2. What is the standard deviation of Q, a set of consecutive integers?
(1) Q has 21 members.
(2) The median value of set Q is 20.
Answer: A.

3. Lifetime of all the batteries produced by certain companies have a distribution which is symmetric about mean m. If the distribution has a standard deviation of d , what percentage of distribution is greater than m+d?
(1) 68 % of the distribution in the interval from m-d to m+d, inclusive
(2) 16% of the distribution is less than m-d
Answer: D.

4. Question deleted

5. List S and list T each contain 5 positive integers, and for each list the average of the integers in the list is 40. If the integers 30,40 and 50 are in both lists , is the standard deviation of the integers in list S greater than the standard deviation of the integers in list T?
(1)The integer 25 is in list S
(2)The integer 45 is in list T
Answer: C.

6. Set T consists of odd integers divisible by 5. Is standard deviation of T positive?
(1) All members of T are positive
(2) T consists of only one member
Answer: B.

7. Set X consists of 8 integers. Is the standard deviation of set X equal to zero?
(1) The range of set X is equal to 3
(2) The mean of set X is equal to 5
Answer: A.

8. {x,y,z}
If the first term in the data set above is 3, what is the third term?
(1) The range of this data set is 0.
(2) The standard deviation of this data set is 0.
Answer: D.

9. Question deleted

10. A scientist recorded the number of eggs in each of 10 birds' nests. What was the standard deviation of the numbers of eggs in the 10 nests?
(1) The average (arithmetic mean) number of eggs for the 10 nests was 4.
(2) Each of the 10 nests contained the same number of eggs.
Answer: B.

11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?
(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.
Answer: A.

EDITED by walker

This this the question I saw on my G-Day:

*deleted*

Sorry, you can't post questions you saw on exam.

I'll try to provide some explanation of the questions with reference to answers, please correct me if i'm wrong.

Thanks

ANSWERS (OA):

1. What is the standard deviation of Company R’s earnings per month for this year?
(1) The standard deviation of Company R’s earnings per month in the first half of this year was $2.3 million.
(2) The standard deviation of Company R’s earnings per month in the second half of this year was $3.9 million.
Answer: E.

Reason : As none of the information above is adequate to find the SD, therefore SD is not possible to calculate

2. What is the standard deviation of Q, a set of consecutive integers?
(1) Q has 21 members.
(2) The median value of set Q is 20.
Answer: A.

Reason: A is the right answer as we know from the Stmt 1 the total number of members in the set and consecutive integers shows that they're not identical so we can calculate the SD from this statement. Stmt 2 is not sufficient as with median we can't calculate SD.


3. Lifetime of all the batteries produced by certain companies have a distribution which is symmetric about mean m. If the distribution has a standard deviation of d , what percentage of distribution is greater than m+d?
(1) 68 % of the distribution in the interval from m-d to m+d, inclusive
(2) 16% of the distribution is less than m-d
Answer: D.

Reason: This is the same problem which is also asked in the PS problem. If the distribution is symmetric about the mean then it means that 50% of the population is above and below the mean.

Stmt 1: if 68% population is within one SD of the mean then 34% is either above or below the mean. Out of 100, 32% population is outside 1SD of the mean so 16% of the population is m+d as well as 16% as m-d.

Stmt2: also reflects the same thing that 16% population is below m-d so both of them are sufficient.

4. Set A and B are 2 sets of numbers. A has a standard deviation 3 and a mean 5. B has the same mean but standard deviation 4. Can we find the standard deviation of the set A U B ? If no, then is it possible with the following additional information?
(1) Both sets have the same number of members each (let’s say 4)
(2) Both sets have distinct members (no number is common to both sets).
Answer: A.

Reason: A is the option as with the help of the Stmt 1 we know the total members of the set however Stmt 2 means distinct members so the set wont be AUB then it will be AnB.

5. List S and list T each contain 5 positive integers, and for each list the average of the integers in the list is 40. If the integers 30,40 and 50 are in both lists , is the standard deviation of the integers in list S greater than the standard deviation of the integers in list T?
(1)The integer 25 is in list S
(2)The integer 45 is in list T
Answer: C.


Reason:
Stmt 1 S = {25,30,40,50,x}
Mean = 40
then x = 55

Stmt 2 T = {x,30,40,45,50}
Mean = 40
then x= 35

So with the help of both stmts Set S has greater SD than T, that is why C is the answer

6. Set T consists of odd integers divisible by 5. Is standard deviation of T positive?
(1) All members of T are positive
(2) T consists of only one member
Answer: B.

Stmt 1 is not sufficient as all members are +ive but they may be identical so SD will be zero.
Stmt 2 has only one member so SD is zero

so that is why B is the answer

7. Set X consists of 8 integers. Is the standard deviation of set X equal to zero?
(1) The range of set X is equal to 3
(2) The mean of set X is equal to 5
Answer: A.


Stmt 1: if the range of the set is not equal to zero then SD will not be zero it will be greater than zero.
Stmt 2: Mean can be 5 but set may contain identical elements.

So A is right but SD will not be zero.

8. {x,y,z}
If the first term in the data set above is 3, what is the third term?
(1) The range of this data set is 0.
(2) The standard deviation of this data set is 0.
Answer: D.


Both stmt are sufficient as with Stmt 1 we know that range is zero so the third term will be 3 and with the stmt 2 it shows that SD = 0 so third term will be 3.

9. Question deleted


10. A scientist recorded the number of eggs in each of 10 birds' nests. What was the standard deviation of the numbers of eggs in the 10 nests?
(1) The average (arithmetic mean) number of eggs for the 10 nests was 4.
(2) Each of the 10 nests contained the same number of eggs.
Answer: B.



May be i don't understand the question properly, B is the right choice as each of the nest have identical eggs therefore SD = 0 or i missed something


11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?
(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.
Answer: A.


As 30% of the volume is removed from all the tanks so SD remains same. therefore stmt 1 is sufficient.

hi Bullet,
really nice explanation by u..

Q10) totally there are 10 nests with some number of eggs. SD=??
1) avg. number of eggs for 10 nests is 4, i.e total of 40 eggs. But, this does not provide us with the number of eggs in each nest. So, SD can vary. Hence, data insuff.
2) each nest contained same number of eggs. when the elements in a set are identical SD=0. so, in this case SD=0. data suff.

Bullet,ur explanation is correct.

Answer:B

hi Bunuel...
can you help me to understand the concept of SD little clearly...

1. ex: if set A={ 2 4 5 6 8}
here mean=5, and 2 & 8 are equidistant and farthest from the mean
so can we say SD= 8-5=3 or SD=5-2=3??

if set B={4 8 10 12 14}
here mean= 9.6 and 14 is the value that is spread farthest from the mean.
so can we say SD=14-9.6=4.4??

2. consider the two sets,

A={ 2 4 6 8 10}
B={2 6 8 12 14}

since the values of set B are widely spaced from the set's mean, it has a higher SD than set A, i.e., SD(B)>SD(A)
is it correct??

i hope i m not sounding stupid...

Also can u post the detailed explanations for Q4 and Q9...

4. Set A and B are 2 sets of numbers. A has a standard deviation 3 and a mean 5. B has the same mean but standard deviation 4. Can we find the standard deviation of the set A U B ? If no, then is it possible with the following additional information?
(1) Both sets have the same number of members each (let’s say 4)
(2) Both sets have distinct members (no number is common to both sets).

Can we have the explaination for this problem?

Don't worry about this question, it's not from the official source. Deleted form the set.

11. During an experiment, some water was removed from each of the 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?
(1) For each tank, 30% of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.
Answer: A.


As 30% of the volume is removed from all the tanks so SD remains same. therefore stmt 1 is sufficient.

The Std Dev does NOT remain the same but ALso decreases by 30% as All subjects decrease by 30% hence ans is A

can't we reject statement 2 for the same reason as 1. Set consist of One number means SD is zero, Zero is neither positive nor negative.

I'm not sure understood your question but here is a solution for this problem:

Question asks "is standard deviation of T positive?" Now, SD\geq{0}: the only case when standard deviation of a set equals to zero is when the set contains all identical elements (or which is same only 1 element). So, basically the question ask whether all elements of T are identical.


(1) All members of T are positive --> not sufficient, as we can have set {5, 5, 5} and SD will be zero or {5, 15, 25} and SD will be some positive number.
(2) T consists of only one member --> directly gives us the answer, as T consists of only one element then its SD will be 0. Sufficient.

Answer: B.

I agree. The new Sd will be 0.7*original Sd or in other terms it drops by 30%.
Sqrt(1/n*sum(0.7x-0.7*mean)^2), you can take 0.7 out of the sqrt and hence the new sd is 0.7 times the old.

Hi Bunuel,

I'd like to discuss question 9 a bit further. If the standard deviation of a set is 4, then the range within that SD is always 8, no matter you go the the left or right. Both (1) and (2) are redundant to answer the question right?

I see your point. The question is quite ambiguous and different interpretations are possible. Actually depending on how you interpret the question the answer can be B, D or even E. So, I'm deleting this question from the set.

not sure about answer of q2,a) says that there are 21 members in the set,but since we expect a unique value for std deviation we must choose option c

thanks a lot fluke,didnt know about this property,thanks a lot