For the set {2, 2, 3, 3, 4, 4, 5, 5, x}, which of the following values

Question

For the set {2, 2, 3, 3, 4, 4, 5, 5, x}, which of the following values of x will most increase the standard deviation?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Kudos for a correct  solution .

Lucky2783 · Accepted Answer

in nutshell 
 To increase standard deviation add the element which has largest distance from Mean .
To decrease standard deviation add the element which has smallest distance from Mean .

mean is = 3.5

3.5 - 1 gives 2.5 
whereas 5-3.5 gives 1.5 only

Answer A.

BrentGMATPrepNow · Answer

-------------ASIDE----------------
For the purposes of the GMAT, it's sufficient to think of Standard Deviation as the  Average Distance from the Mean . Here's what I mean:

Consider these two sets: Set A {7,9,10,14} and set B {1,8,13,18}
The mean of set A = 10 and the mean of set B = 10
How do the Standard Deviations compare? Well, since the numbers in set B deviate the more from the mean than do the numbers in set A, we can see that the standard deviation of set B must be greater than the standard deviation of set A.

Alternatively, let's examine the Average Distance from the Mean for each set.

Set A {7,9,10,14}  
Mean =  10 
7 is a distance of 3 from the mean of  10 
9 is a distance of 1 from the mean of  10 
10 is a distance of 0 from the mean of  10 
14 is a distance of 4 from the mean of  10 
So, the average distance from the mean = (3+1+0+4)/4 =  2

B {1,8,13,18}  
Mean =  10 
1 is a distance of 9 from the mean of  10 
8 is a distance of 2 from the mean of  10 
13 is a distance of 3 from the mean of  10 
18 is a distance of 8 from the mean of  10 
So, the average distance from the mean = (9+2+3+8)/4 =  5.5

IMPORTANT: I'm  not  saying that the Standard Deviation of set A equals 2, and I'm  not  saying that the Standard Deviation of set B equals 5.5 (They are reasonably close however).

What I am saying is that the average distance from the mean can help us see that the standard deviation of set B must be greater than the standard deviation of set A. 
More importantly, the average distance from the mean is a useful way to think of standard deviation. This model is a convenient way to handle most standard deviation questions on the GMAT.
-----ONTO THE QUESTION!!!---------------------------

Remove x from the original set. 
The set {2, 2, 3, 3, 4, 4, 5, 5} has a mean of 3.5

In order to affect the greatest increase in the standard deviation, x must be the furthest from the mean (3.5)

Check the answer choices ..... answer choice A is furthest from the mean.

Answer: A

RELATED VIDEO
 https://www.youtube.com/watch?v=FbAlIjw2fwY

AmoyV · Answer

Without x, the average is 3.5. All the values are within +-1.5 units from 3.5. To increase the SD, we need something which is farther than 1.5 from 3.5.

1 is 2.5 units away from 3.5 and hence will increase the SD the most.

Answer: A

gauravmishra132 · Answer

Correct answer:A

Because if x=1 then mean will be greater than 3.......thus,it will create in maximum difference...than other four choices.

Vaibhav21 · Answer

Standard Deviation Step 1, as pointed out by others, is to find out the mean = 3.5
Step 2, For each number: subtract the Mean and square the result = 
(1-3.5)^2=(-2.5)^2 
(2-3.5)^2=(-1.5)^2 
(3-3.5)^2 =(.5)^2 
(4-3.5)^2=(.5)^2 
(5-3.5)^2=(1.5)^2
Clearly (1-3.5)^2=(-2.5)^2 will give you the greatest value among all the other options. 
Hence A

Step 3 Work out the mean of those squared differences.
Step 4 Take the square root of that and we are done!

SherLocked2018 · Answer

A for me too. 
when x = 1, the range of the set increases , (5-1) = 4. 
when x = 2/3/4/5, the range remains the same, (5-2) = 3. 
Thus A.

A set with higher the range and fewer the number of element has the higher Standard deviation. Inthis case, the no of elements remains the same. So we can just use Range. 
In case the Range comes out to be same, then we can calculate the average and see. But for this one, I think Range suffices.

Sneakysam · Answer

I get the concept but when you add in x, won't the average be a value OTHER than 3.5? In your scenario you keep the average constant as if x does not affect the mean of the "new" set. Can someone explain if i'm thinking about this correct?

sudh · Answer

Add the two extreme values from the answer choice (1 & 5) and equate the mean

\frac{28+1}{9}  < mean <  \frac{28+5}{9} 
 \frac{29}{9}  < mean <  \frac{33}{9} 
3  \frac{2}{9}  < mean < 3  \frac{6}{9}

Distance between 1 and 3  \frac{2}{9}  = 2  \frac{2}{9}

Distance between 3  \frac{6}{9}  and 5 = 1  \frac{3}{9} ; So 1 will increase the SD more

Answer A

EMPOWERgmatRichC · Answer

Hi All,

This is a great "concept" question - one that doesn't really require any calculations if you recognize the patterns involved.

Notice how the set of numbers consists of TWO EACH of the integers 2, 3, 4 and 5.
Also notice how the answer choices are 1, 2, 3, 4 and 5.

Since the terms in the set are consecutive and have the same frequencies, adding in another 2 or another 5 would have the SAME IMPACT on the SD, so neither of those can have the GREATEST increase. In that same way, adding in another 3 or another 4 would have the SAME IMPACT on the SD, so neither of those can have the GREATEST increase either.

The correct answer is the only term that's left.

Final Answer:  A

GMAT assassins aren't born, they're made,
Rich

nphatak · Answer

An Excellent post by Mike from Magoosh, that I have found useful to conceptualize STD. Dev. https://magoosh.com/gmat/2012/standard-d ... -the-gmat/

Bunuel · Answer

VERITAS PREP OFFICIAL SOLUTION:

If you recall the steps to calculating the standard deviation, what we really need to do first is to calculate the mean. (i.e. how mean are you?) You can add the eight elements together and divide by eight, but the fact that these elements follow a fairly obvious pattern helps us as well. The numbers each appear twice, and they are evenly spaced. This means that the average will be the same as the median, and the median is 3.5. Even if you take the long way, it shouldn’t take you more than 20 seconds to find that the mean of this set is 3.5

The next step is to take each element and find the difference from the mean, but this is what we need to do if the goal is to actually calculate the standard deviation. All we’re being tasked to do here is to determine which number will increase the standard deviation the most. In this regard, all we need to do is figure out which answer choice is furthest from the mean. That number will produce the biggest distance, which will then be squared and in turn produce the biggest difference in standard deviation. So although you can spend a lot of time calculating every last detail of this question, what it actually comes down to is “which of these numbers is furthest from 3.5”.

Asking about distance from a specific number is much more straightforward, and probably an elementary school level question. Yet, if you understand the concept, you can turn a GMAT question into something a 5th grader could answer (Are you smarter than a 5th grader?).  The answer is thus obviously choice A , as 1 is as far from 3.5 as possible given only these five choices.

The important thing about the standard deviation is that you will never have to formally calculate it, but understanding the underlying concept will help you excel at the quantitative section of the GMAT. Most standard deviation questions hinge primarily on the distance from the mean, as everything else is just a rote division or addition. Much like taking five practice exams and getting wildly different scores, having a high variance is bad for knowing what to expect. Understanding the way standard deviations are tested on the GMAT will help you consistently get the questions right and reduce the variance of your results (hopefully with a very high mean).

GMATYoda · Answer

The standard deviation question where we don’t actually need to calculate standard deviation.

Without the number, x, it is easy to see that the average of the remaining set is equal to 3.5 (notice the pairing of the numbers; the average of every 2-5 pair is equal to 3.5 and the average of every 3-4 pair is also equal to 3.5.)

This problem then simplifies down to the question: “which answer choice is farthest away from 3.5?” The answer, without much math at all, is “A”. (Incidentally, “A” is also the only answer which would increase the range of the set. For sets equally distributed around the average – like this problem has – such answers would naturally increase the standard deviation the most because they would be the farthest away from the average, even if we didn’t know what the average was.)

ScottTargetTestPrep · Answer

Without including x, we see that the mean is (3 + 4)/2 = 3.5. To increase the standard deviation the most, we need a value that is furthermost from 3.5. Therefore, 1 will be the correct answer.

Answer: A

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For the set {2, 2, 3, 3, 4, 4, 5, 5, x}, which of the following values

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For the set {2, 2, 3, 3, 4, 4, 5, 5, x}, which of the following values

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