Problem Solving

mehaksal wrote: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

To begin, the SD formula looks like this:


The set {0, 2, 4, 6, 8} has a mean of 4.
So, first we need to find the difference between each value and the mean.

0 is 4 units away from the mean of 4.
2 is 2 units away from the mean of 4.
4 is 0 units away from the mean of 4.
6 is 2 units away from the mean of 4.
8 is 4 units away from the mean of 4.

So, the SD will equal the square root of (4^2 + 2^2 + 0^2 + 2^2 + 4^2)/5.
In other words, the SD = the square root of 40/5.
= the square root of 8

Okay, so which pair of new numbers, when added to the original 5 numbers will yield a new SD that is closest to the square root of 8?

Well, to begin, it's useful to notice that each pair consists of numbers that are equidistant from the original mean of 4.
For example, in answer choice A, -1 is 5 units less than 4, and 6 is 5 units more than 4.
As such, add the two values in each answer choice will yield a mean of 4.

Okay, let's see what happens if we add -1 and 9 (answer choice A).
Well, -1 is 5 units away from the mean of 4, and 9 is 5 units away from the mean of 4.
So, the new SD = the square root of (4^2 + 2^2 + 0^2 + 2^2 + 4^2 + 5^2 + 5^2)/7.
= the square root of 90/7.
= the square root of approximately 13
This is considerably larger than the original SD of sqrt(8)

Now let's skip a few answers and try answer choice D.
Here, 2 is 2 units away from the mean of 4, and 6 is 2 units away from the mean.
So, the new SD = the square root of (4^2 + 2^2 + 0^2 + 2^2 + 4^2 + 2^2 + 2^2)/7.
= the square root of 48/7.
= the square root of approximately 7
This one is pretty close to the original SD of sqrt(8).

In fact, if we check the other answer choices (lots of work!), we'll see that answer choice D is the best answer.

Cheers,
Brent

mehaksal wrote: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

Alternative approach.

Having said that . . .

For GMAT purposes, Standard Deviation (SD) can often be thought of as "the average distance the data points are away from the mean."

So, with {0, 2, 4, 6, 8}, the mean is 4.
0 is 4 units away from the mean of 4.
2 is 2 units away from the mean of 4.
4 is 0 units away from the mean of 4.
6 is 2 units away from the mean of 4.
8 is 4 units away from the mean of 4.

So, the SD can be thought of as the average of 4, 2, 0, 2, and 4. The average of these values is 2.4, so we'll say that the SD is about 2.4

Note: This, of course, isn't 100% accurate, but it's all you should really need for the GMAT.

Okay, so which pair of new numbers, when added to the original 5 numbers will yield a new SD that is closest to 2.4?

Well, to begin, it's useful to notice that each pair consists of numbers that are equidistant from the original mean of 4.
For example, in answer choice A, -1 is 5 units less than 4, and 6 is 5 units more than 4.
As such, add the two values in each answer choice will yield a mean of 4.

Okay, let's see what happens if we add -1 and 9 (answer choice A).
Well, -1 is 5 units away from the mean of 4, and 9 is 5 units away from the mean of 4. So, 5 and 5 will be added to 4, 2, 0, 2, and 4 to get a new SD. As you can see, this will result in a much larger SD.

Now, let's examine D (2 and 6)
Well, 2 is 2 units away from the mean of 4, and 6 is 2 units away from the mean. So, we'll be adding 2 and 2 to the five original differences of 4, 2, 0, 2, and 4. Since the average of 4, 2, 0, 2, and 4 is 2.4, adding differences of 2 and 2 should have the least effect on the original SD.

As such, the correct answer must be D

Cheers,
Brent