Set A = {22, 27, 11, 12, 2}, and set B: {14, 19, 3, 4, x}

Question

If x < 0, and the standard deviation of set A = the standard deviation of set B, what is the value of x?

A) -2
B) -3
C) -4
D) -5
E) -6

BrentGMATPrepNow · Accepted Answer

The test-makers like to test the following concept:   If we take a set of numbers with a Standard Deviation of X and add k to each value in the set, the resulting set will still have a Standard Deviation of X.   
This should make sense, since Standard Deviation is a  measurement of dispersion  (i.e., how dispersed the numbers are)

For example, if we take the set {1,2,3} and add 15 to each set member, we get {16, 17, 18}
In both sets, the second number is 1 greater than the first number, and the third number is 1 greater than the second number. 
As such, both sets have the same Standard Deviation.

Likewise, if we take the set {7,7,17} and add 11 to each set member, we get {18, 18, 28}
In both sets, the first two numbers are the same, and the third number is 10 greater than the other two numbers. 
As such, both sets have the same Standard Deviation.

If we arrange both sets in ascending order we get: 
Set A = {2, 11, 12, 22, 27}
Set B = {x, 3, 4, 14, 19}

Notice that each of the 4 biggest numbers in set A are   8 greater   than each of the 4 biggest numbers in set B
For example, 27 = 19  + 8 
And 27 = 19  + 8 
And 22 = 14  + 8 
And 12 = 4  + 8 
And 11 = 3  + 8

Since the two sets have the same Standard Deviation, it must be the case that 2 = x  + 8 
Solve for x to get: x = -6

Answer: E

Cheers,
Brent

Kinshook · Answer

Set A = {22, 27, 11, 12, 2} = {2,11,12,22,27}
Set B: {14, 19, 3, 4, x} ={x,3,4,14,19} = {2-8,11-8,12-8,22-8,27-8}
If Set A and Set B are compared, it is noticed that each element of set B is 8 less than set A. So mean is changed but standard deviation remains the same.
We get = x = 2-8 =-6

IMO E

Archit3110 · Answer

mean of set A ; A = {22, 27, 11, 12, 2} ; 74/5; 14.8
and SD along mean ; 12.8,2.8,3.8,0,7.2,12.2
similarly at x=-6 we get the mean of set B ; 40-6/5 ; 34/5 ; 6.8
SD along mean is 12.8,2.8,3.8,0,7.2,12.2
IMO E

SchruteDwight · Answer

Total range lower than mean and bigger than mean set A=12+13=25, set B= 9+10+x=25 additional 6 needed, hence -6

pk12344 · Answer

The easiest way to solve this question is to line up the numbers of the sets in increasing order. That is start set A from 2 and start set B from x. You will notice that when you line them up the difference between each number of the respective set lines up. 11 is one way from 12, same as 3 is one away from 4. 12 is 10 away from 22, same as 4 is 10 away from 14 and so on.

Now to make sure that the standard deviation is the same, all we need to do is ensure that the difference between x and 3 is the same the difference between 2 and 11. That is x is 9 less than 3. Every other position is already set so the only movement to standard deviation can come from the value of x. To standardise that we insert -6 therefore smoothing out the distribution to ensure the same standard deviation. It more logic than solving complex formulae.

MathRevolution · Answer

Mean gets affected if the data in the set are increased or decreased by adding or subtracting the same number respectively. But S.D. remains the same.

Set A= {22, 27, 11, 12, 2}
Set B= {14, 19, 3, 4, x}

There is a difference of '8' between each term of Set A and Set B. Therefore, 2- (x) = 8

=> x = -6

Answer E

bv8562 · Answer

I solved it this way. Let me know if this is the right approach to solve this.  Bunuel

Since, SD <= Range/2

Range of Set A = 27-2 = 25
so, SD of set A <= 12.5
Given, SD of set A = SD of set B = 12.5

Range of Set B = 2*SD = 19-(-x) <= 25 => x <= 25-19 <= 6

since, x is negative, therefore, x <= -6 (E)

Hoozan · Answer

BrentGMATPrepNow  I arranged the set in ascending order and instead of considering the mean I thought about the median. Could you tell me how does this thinking work out despite not considering the mean??

Set A = {22, 27, 11, 12, 2}, and set B: {14, 19, 3, 4, x}

A = (2,11, 12 ,22,27
B = (x,3, 4 ,14,19)

Now we know that SD is the distance from the mean. BUT I yet considered the median to get an idea.

Here I noticed that A has a net displacement of 36 (from 12) and that B has a net displacement of 26 (from4. So X needs to compensate for that extra 10 (since the SD of the two sets are the same)

Hence x = -6

BrentGMATPrepNow · Answer

For this particular question, that approach works perfectly.

bumpbot · Answer

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Set A = {22, 27, 11, 12, 2}, and set B: {14, 19, 3, 4, x}

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Set A = {22, 27, 11, 12, 2}, and set B: {14, 19, 3, 4, x}

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