Set Q consists of the following five numbers

Question

Set Q consists of the following five numbers: Q = {5, 8, 13, 21, 34}. Which of the following sets has the same standard deviation as Set Q?

I. {35, 38, 43, 51, 64}
II. {10, 16, 26, 42, 68}
III. {46, 59, 67, 72, 75}

(A) I only
(B) I & II
(C) I & III
(D) II & III
(E) I, II, & III

For a detailed explanation of everything you need to understand about Standard Deviation for the GMAT, as well as a solution to this particular question, see
https://magoosh.com/gmat/2012/standard-d ... -the-gmat/

Mike :-)

mikemcgarry · Accepted Answer

Dear Keats , My friend, I'm happy to respond. I see you have requested the genius Bunuel, but I can help: I'm the author of this question. One way to recognize that III is correct is to see the pattern of subtraction from 80. That's one way to do it, but not the only way. Think about this. Look at at set Q = {5, 8, 13, 21, 34}. Subtract adjacent numbers to get the pattern of spacing between the numbers: 3-5-8-13. Those are the spaces between the individual data points. Now, look at the pattern of spacing in III {46, 59, 67, 72, 75}--it's 13-8-5-3. That's the same pattern in reverse. In other words, the basic pattern of spacing in Q and III is the same. If the data points are spaced the same, it doesn't matter where they are on the number line, they have the same standard deviation. This blog has two more practice problems on Standard Deviation: Standard Deviation on the GMAT Does this make sense? Mike :-)

Pushpinder · Answer

Facts about Standard Deviation:
(1) Multiplying all the numbers change Standard Deviation
(2) Adding and subtracting all the numbers with the same number keeps the standard deviation same.
If you observe Set I and III are added versions of Set Q .
Set I: 5 has been added to the Set Q
Set III: Subtract each element from 80 and you would find a number there in the Set Q. 
Set II: elements are multiplied by 2 and standard deviation changes.

Hence the answer is C.

keats · Answer

Bunuel  Can you help us with a way as to how do we tackle such questions?

Bunuel · Answer

Please try to be a little bit more specific. For example, what part of the following solution is not clear: set-q-consists-of-the-following-five-numbers-147917.html#p1187415

mikemcgarry · Answer

Dear Keats , I see that the genius Bunuel answered your request briefly. I will echo his concern. My friend, one of the habits of excellence is asking excellent questions. When you ask a vague one-line question, the presupposition behind that question is that responsibility for the process of education lies with the expert asked, in this case, Bunuel. In fact, 100% of the responsibility for your learning lies with you. One tangible way to demonstrate this responsibility is to ask thorough and detailed questions that make explicitly clear exactly what you understand and exactly what you don't understand. See: Asking Excellent Questions Wishing you excellence, my friend. Mike :-)

keats · Answer

Bunuel

Set Q consists of the following five numbers: Q = {5, 8, 13, 21, 34}. Which of the following sets has the same standard deviation as Set Q?

I. {35, 38, 43, 51, 64}
II. {10, 16, 26, 42, 68}
III. {46, 59, 67, 72, 75}

Can you please help in reaching out to why III is true?

S.D. remains same if we add/subtract a constant number from each element of the set. In Q = {5, 8, 13, 21, 34} , we are trying to subtract each of the element of Q from 80
80 - 5 = 75
80 - 8 = 72
80 - 13 = 67
80 - 21 = 59
80 - 34 = 46

Is it a right way to do this? What I know is that you add or subtract elements to set Q. Not the other way!

keats · Answer

Also,  Bunuel  Can you please share a link, if any, for such problems?Thanks.

Bunuel · Answer

Check below:
 
*  Statistics Made Easy - All in One Topic! 
*  Theory on Statistics and Sets problems 
*  Data Sufficiency Statistics Questions 
*  Problem Solving Statistics Questions 
*  Standard Deviation Questions  from our  Special Questions Directory .

Hope it helps.

keats · Answer

Wohoo!! This is awesome to learn. It just simplifies. Thank You  mikemcgarry . +1Kudos!

gabrielyaquinta · Answer

Another approach is to look at the sequence of differences between the numbers in the set and look for them in the other 3 sets, regardless of the order.

Q = {5, 8, 13, 21, 34}. Differences goes as 3 ,5 ,8, 13

I. {35, 38, 43, 51, 64} --> 3 ,5 ,8, 13 --> Same Std Dev.
II. {10, 16, 26, 42, 68} --> 6, 10, 16, 26 --> Different Std Dev.
III. {46, 59, 67, 72, 75} --> 13, 8, 5, 3 --> Same Std Dev.

Hence the answer is C.

adkikani · Answer

niks18

Can you explain why are we focusing on patterns by difference between two adjacent no?
As per my understanding the SD should be approx difference of each no from mean.

Also note that adding / subtracting same integer will not change SD, but here we are adding 
different numbers to the list. How can this be a conclusive factor to determine same SD?

niks18 · Answer

Hi  adkikani

At first I would suggest that you keep the property in mind as explained by  mikemcgarry  in his earlier post. it will save you from unnecessary calculation.

As you rightly mentioned that adding / subtracting same integer will not change SD. There is another property which says if you multiply each element of the set by  k , then you multiply the standard deviation by  |k|  (remember SD is always positive because it is square root of variance. Variance is the average of the squares of difference between each element and mean, so variance will always be positive)

Set Q {5, 8, 13, 21, 34}. let's assume it's SD is  d

Set I, I hope is clear to you as you have used the addition property.

Set III {46, 59, 67, 72, 75}

Now add  -80  to each element of the set, you will get {-34,-21,-13,-8,-5}. Re-arrange (if may like to do so to see a pattern, otherwise not required) {-5,-8,-13,-21,-34}

This is same as set Q multiplied by  -1 , so SD of this set will be  d*|-1|=d=   same as set Q

so we can also derive that if basic pattern of spacing between adjacent element of two sets are same then SD will be equal.

This question tests basically few properties of SD which you can easily deduce if you know the STEP TO CALCULATE SD of a given set

bumpbot · Answer

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Set Q consists of the following five numbers

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