Problem Solving

A set of data consists of the following 5 numbers: 0,2,4,6, and 8. Which two numbers, if added to create a set of 7 numbers, will result in a new standard deviation that is close to the standard deviation for the original 5 numbers?

A). -1 and 9
B). 4 and 4
C). 3 and 5
D). 2 and 6
E). 0 and 8

I did this question keeping in mind that the GMAT will not ask you questions on how to CALCULATE standard deviation. Rather, it tests your knowledge of the concept, especially as it relates to the range.

The mean of the set is 4; (20 / 4).

This means that the range of the set is about 4 each way of the mean, i.e.- the difference between the mean [4] and the low [0] and high [8] numbers. Let's make sure our answer stays "close" to the original standard deviation, which is essentially keeping the numbers that are "close" to the mean in tact.

a) Range now moves to 5 away from mean. Eliminate.
b) Range stays 4 away from mean. However, with three "4s" in your answer your standard deviation moves closer to the mean [4], so it changes the standard deviation. Eliminate. (This is the "trap" answer)
c) Range stays 4 away from mean. However, since 3 and 5 are close to the mean [4], then the standard deviation again changes pretty dramatically. Eliminate.
d) Range stays 4 away from mean. Same justification as "C", but 2 & 6 move farther away from the mean [4]. Maybe.
e) Range stays 4 away from mean. So, this does not change the range or the mean and is the choice that is the furthest away from the mean and still does not change the range, this is the answer.

E is my answer.

What is the OA and where did this question come from?

fltingley wrote: b) Range stays 4 away from mean. However, with three "4s" in your answer your standard deviation moves closer to the mean [4], so it changes the standard deviation. Eliminate. (This is the "trap" answer)
c) Range stays 4 away from mean. However, since 3 and 5 are close to the mean [4], then the standard deviation again changes pretty dramatically. Eliminate.
e) Range stays 4 away from mean. So, this does not change the range or the mean and is the choice that is the furthest away from the mean and still does not change the range, this is the answer.

how do you know this without calculating? can you explain the concept

thanks in advance!

The range is nothing more than the lowest and the highest in a set of numbers. In B-E the high and low numbers did not change, vis-a-vis the range did not change.

OA is D. This question is from PS assignment passed on to me.

IMO, i too think answer is E. But i calculated it using the SD formula.

SD = Sqrt(Sum(X-x)^2/N) , where X is the mean

Here, X=4 ,N=5 ,Sum(X-x)^2 = 40 and Sum(X-x)^2/N = 8

Now with N=7, in order for SD to remain the same, Sum(X-x)^2/7 should be equal to 56.

This is only possible with choice E.

Coming back to your solution, what i understand is, for SD to remain constant while adding numbers, two conditions needs to be satisfied.

1) Mean and Range need to remain constant.
2)Numbers should be added as far away from the mean such that (1) is satisfied.

Is this a rule ? Can you please confirm ?