If set X and set Y consist of more than 1 element, what is the ratio

Question

Competition Mode Question

If set X and set Y consist of more than one element, what is the ratio of the standard deviation of set X to the standard deviation of set Y?

(1) Set X consists of consecutive multiples of 5 and set Y consists of consecutive multiples of 3

(2) The number of elements in sets X and Y are equal

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M36-138

Bunuel · Accepted Answer

Official Solution:

If set X and set Y consist of more than one element, what is the ratio of the standard deviation of set X to the standard deviation of set Y?

(1) Set X consists of consecutive multiples of 5 and set Y consists of consecutive multiples of 3
 
If set X is say {5, 10} and set Y is say {3, 6, 9, ..., 300}, then the standard deviation of set X will be less than that of set Y (because variation of numbers from the mean in set X is less than the variation of numbers from the mean in set Y) and thus the ratio would be less than 1. But if set X is say {5, 10, ..., 500} and set Y is say {3, 6}, then the standard deviation of set X will be more than that of set Y and thus the ratio would be more than 1. Not sufficient.
 
(2) The number of elements in sets X and Y are equal
 
Clearly insufficient. We don't know anything about numbers in the sets.
 
(1)+(2) Two important properties:
 
If we multiply (or divide) all the numbers in a set by a non-zero constant, the standard deviation will be respectively multiplied or divided by the absolute value of that constant. For example, if the SD of  \{a, \ b, \ c\}  is  s , then the SD of  \{2a, \ 2b, \ 2c\}  will be  2s  and the SD of  \{-3a, \ -3b, \ -3c\}  will be  s*|-3|=3s .
 
If we add (or subtract) a constant to all the numbers in a set, the standard deviation will NOT change. For example, if the SD of  \{a, \ b, \ c\}  is  s , then the SD of  \{a+4, \ b+4, \ c+4\}  or of  \{a-1, \ b-1, \ c-1\}  will also be  s .
 
From the second property it follows, that any  n  consecutive multiples of  x  will have the same standard deviation. For example, the SD of  \{3, \ 6, \ 9\}  will be the same as the SD of  \{-12, \ -9, \ -6\}  because we can add 15 to each number in the second set to get the numbers in the first set. 
 
Back to the question, say there are  n  numbers in each set and X =  \{3, \ 6, \ 9, ..., 3n \}  and Y =  \{5, \ 10, \ 15, ..., 5n \}  (as discussed above, it does not matter which  n  multiples of 3 or 5 we consider). Notice that numbers in Y are  \frac{5}{3}  times the respective numbers in X. So, according to the first property above,  SD_A*\frac{5}{3}=SD_B , thus  \frac{SD_A}{SD_B}=\frac{5}{3} . Sufficient.

Answer: C

yashikaaggarwal · Answer

Standard Deviation = 1/4*(range)
Range = Highest value-Lowest Value

Statement 1: There can N no. Of integers in both set, making the standard deviation vary with the range. Hence Insufficient

Statement 2: Even if No. Of elements are same in sets. The range can differ so is the SD. Insufficient

Statement 1&2 together since the elements in set is same and the set consists of 3&5 consecutive multiple.
The consecutive set range of multiple of 5 is always greater than consecutive set range of 3 multiples.
Hence the ratio of the standard deviation of set X to the standard deviation of set Y. Will always be in the range of 5:3 (Sufficient)

IMO C

Posted from my mobile device

rajatchopra1994 · Answer

Explanation:

Statement 1: 
Standard deviation depends on the number of terms.
Since we have no idea about the number of terms in both sets.
So Insufficient.

Statement 2:
No information about Set X & Set Y
Insufficient.

Together:
Let Set A = {5, 10, 15, 20, 25}
Set B = {3, 6, 9, 12, 15 }
S.D A > S.D B
IMO-C

ArunSharma12 · Answer

statement 1: X = {5k, 5k+1, 5k+2, 5k+3...}, Y = {3p, 3p+1, 3p+2, 3p+3...} 
we do not know the number of elements in each set, it is required to calculate the standard deviation.
not sufficient.

statement 2: number of elements in X = number of elements in Y.
we do not know the anything about the numbers.
not sufficient

combining both statements,
we know X and Y are evenly space sets with equal number of elements. so the ratio of standard deviation will remain same.
consider X = {5,10,15} Y = {3,6,9} 
SD(X) = 5 + 0 + 5 / 3 = 10/3
SD(Y) = 3 + 0 + 3 / 3 = 6/3 
ratio = 5/3

consider X = {5,10,15,20,25} Y = {3,6,9,12,15} 
SD(X) = 10 + 5 + 0 + 5 + 10 / 5 = 30/5
SD(Y) = 6 + 3 + 0 + 3 + 6 / 5 = 18/5 
ratio = 5/3

Ans: C

MayankSingh · Answer

IMO C

(1)   Set X consists of consecutive multiples of 5 and set Y consists of consecutive multiples of 3

Since Standard deviation depends on n number of element, and here no. of elements are unknown.
So, this statement is not sufficient.

(2)   The number of elements in sets X and Y are equal

We only know about the no. of elements but no idea about the elements in the set.
So, this statement is not sufficient.

Together  
 Let Set X= { 5,10,15} & Y={ 3,6,9}
SD (X) = √(5^2+5^2)/3 & SD (Y) = √(3^2+3^2)/3
Req Ratio = √50/18 = √25/9

But , if Let Set X= { 5,10,15,20,25} & Y={ 3,6,9,12,15}
SD (X) = √(10^2+5^2+5^2+10^2)/5 & SD (Y) = √(6^2+3^2+3^2+6^2)/5
Req Ratio = √250/90 = √50/18 = √25/9

So ,irrespective of the number of terms, the ratio of SD will remain the same.

Sufficient.

exc4libur · Answer

n of y and x > 1
sdx/y = ?

(1) insufic

x={ie.5,10,15} avg=5
sd=sqrt /3]

y={ie.3,6,9} avg=3
sd=sqrt /3]

sd of x and y depends on number of elemnts

(2) insufic

n of x and y are equal

(1/2) sufic

since we know the number elements is equal
then we can find their ratios:
sqrt /3] divided by
sqrt /3]

Ans (C)

freedom128 · Answer

(1) Number of elements for each set is unknown 
NOT SUFFICIENT

(2) 
Case 1: X={3,6}, Y={3,6}
Standard deviation of X = Standard deviation of Y
Case 2: X={3,6}, Y={5,10}
Standard deviation of X < Standard deviation of Y
NOT SUFFICIENT

(1)+(2)
Case 1: X={3,6,9}, Y={5,10,15}
Standard deviation of X < Standard deviation of Y
Case 2: X={3,6}, Y={5,10}
Standard deviation of X < Standard deviation of Y
SUFFICIENT

ANS. (C)

minustark · Answer

Competition Mode Question

If set X and set Y consist of more than one element, what is the ratio of the standard deviation of set X to the standard deviation of set Y?

(1) Set X consists of consecutive multiples of 5 and set Y consists of consecutive multiples of 3

(2) The number of elements in sets X and Y are equal

1) When set X = {5,10}, the SD is  /2 or, 12.5/2 = 6.25. When set Y = {0,3}, SD =  /2 = 2.25. When Set Y = {3,6,9}, SD is [3^2+0+3^2}/3 =6. So, for different number of elements the ratio will vary. not sufficient.

2) SD depends on the value of each element in the set. Not sufficient

Together, from the first example in statement 1, we can find the ratio when there are 2 elements in each set. The ratio is : 6.25 : 2.25 or, 1.25: .45. Now, lets examine when there are 3 elements. X= {0,5,10}, SD= (25+25)/3 = 50/3 and Y ={3,6,9}, SD= 6. Ratio is 50 : 18 or, 25 : 9. or, 1.25 : .45 (divided by 20). So, if know the nature of the set and the number of elements, we can find the ratio of the SDs. Sufficient.

C is the answer.

Poojita · Answer

Standard Deviation = 1/4*(range)
Range = Highest value-Lowest Value
I was not aware of this . Is it a rule or something?
and can you explain the logic behind this please?

yashikaaggarwal · Answer

It's a formula for getting SD using range.
I don't know the logic behind it, too lazy to draw all logic given this simple formula is easily memorable, but you can Google "SD formula using Range".
There is a whole definition and logic behind it.

Bunuel · Answer

I think this question did not get precise solution, so bumping up for more discussion.

IanStewart  do you think this one is out of scope for the GMAT? Thank you!

IanStewart · Answer

First, because some of the discussion in this thread might be misleading: it is certainly  not  true that "standard deviation = (1/4)*range". That's almost never going to be true, in fact. It might produce a decent approximation of standard deviation for certain types of sets, but that's all it might do.

I've seen all of the relevant concepts tested in one official question or another, so I wouldn't say it's out of scope, but I'd be surprised to see an official question precisely like this one, just because official standard deviation questions are usually pretty straightforward.

In this question, knowing only that a set consists, say, of consecutive multiples of 5 doesn't tell us much about its standard deviation. The standard deviation of {5, 10}, for example, is just 2.5 (because every element is 2.5 away from the mean) but the standard deviation of {5, 10, 15, ..., 990, 995, 1000} for example is much bigger than 2.5, because most values in the set are very far from the mean. So we have no hope of answering this question without using Statement 2, but Statement 2 alone is clearly insufficient (we know nothing about what is in the sets), so the answer is C or E.

Using both Statements, if each set contains n elements, then we can first imagine we have a set of the first n consecutive positive integers. Say that set has a standard deviation of s. If we multiply every value in that set by 3, to get a set of n consecutive multiples of 3, we'll be 'stretching out' the distances in the set by a factor of 3, which will multiply the standard deviation by 3. So that set of n consecutive multiples of 3 will have a standard deviation of 3s. Similarly, if we multiply all of our n consecutive integers by 5 to get a set of n consecutive multiples of 5, our standard deviation will become 5s. The ratio of these standard deviations is 5s/3s = 5/3. So the number of values in the sets doesn't matter, and the two statements are sufficient together, and the answer is C.

Note that it doesn't matter how big our multiples are; the set {5, 10, 15, 20} has the same standard deviation as {90, 95, 100, 105}, because they're spaced the same way, so we can answer the question just by imagining our two sets consist of the first n positive multiples of 3 and of 5 respectively.

Bunuel · Answer

Thank you. This one is my question and I was worried it might be too hard for the GMAT.

IN2MBB2PE · Answer

Hi  IanStewart   Bunuel  - will that ratio always be 5:3 in that case? As per the last statement of Ian ....  "Note that it doesn't matter how big our multiples are; the set {5, 10, 15, 20} has the same standard deviation as {90, 95, 100, 105}, because they're spaced the same way, so we can answer the question just by imagining our two sets consist of the first n positive multiples of 3 and of 5 respectively."

IN2MBB2PE · Answer

Tagging GMATNinja as well, given you asked on the video to be tagged in this question

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If set X and set Y consist of more than 1 element, what is the ratio

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If set X and set Y consist of more than 1 element, what is the ratio

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