Standard Deviation

[Uva@90]

A certain list of 100 data has an average of 6 and standard deviation of d where d is positive. Which of the following pairs of data, when added to the list must result in a list of 102 data with the standard deviation less than d?
(A) 0 and 6
(B) 0 and 12
(C) 0 and 0
(D) -6 and 0
(E) 6 and 6
Answer: E.

Regards,
Uva.

[theCodeToGMAT]

Average = 6
Elements = 100
SD = d

Standard Deviation is lowest when the elements are added near the Mean Value; the farther the value from mean.. the greater the SD.

Only Option {E} satisfies

[GMATGuruNY]

A certain list of 100 data has an average of 6 and a standard deviation of d, where d is positive. Which of the following pairs of data, when added to the list, must result in a list 0f 102 data with standard deviation less than d?

A. -6 and 0
B. 0 and 0
C. 0 and 6
D. 0 and 12
E. 6 and 6

SD describes how the data are SPREAD OUT from the mean.
A HIGH SD means that the data are spread far from the mean.
A LOW SD means that the data are positioned close to the mean.

To guarantee a smaller SD, we need an answer choice that will introduce data close to the mean.
Answer choice E leaves the mean unchanged -- the new average = (600+6+6)/102 = 612/102 = 6, same as the original average -- but adds two pieces of data equal to the mean.
The result:
A greater proportion of the data will be close to the mean of 6, yielding a smaller SD.

The correct answer is E.

[Uva@90]

Mitch/Rahul,
As you both mentioned that if we add numbers close to Mean(Average) the New SD will close to Old SD.

I have applied the same logic to this question, but I am getting wrong. Could you please explain me why ?

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
OA is E .

Here the mean is 4. If we add number close to 4(4 and 4) then SD will be close to OLD SD na?
But answer is not B.

I couldn't get it.

Please help me with this.

Thanks in advance.

Regards,
Uva.

[Brent@GMATPrepNow]

Mitch/Rahul,
As you both mentioned that if we add numbers close to Mean(Average) the New SD will close to Old SD.

I have applied the same logic to this question, but I am getting wrong. Could you please explain me why ?

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
OA is E .

Here the mean is 4. If we add number close to 4(4 and 4) then SD will be close to OLD SD na?
But answer is not B.

I couldn't get it.

Please help me with this.

Thanks in advance.

Regards,
Uva.

Hi Uva,

If we add the numbers 4 and 4, the new SD will be closer to zero than the old SD was (since we're adding two additional means, and means do not deviate from the means at all. Yeesh, weird sentence ).

This new question asks us to find a pair of values to add such that the resulting SD is close to the old SD.

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

HAVING SAID all of that, this question isn't really GMAT-worthy since it requires too many calculations. Ian Stewart makes some good points here regarding the legitimacy of this question: https://www.beatthegmat.com/shortcut-in- ... 10473.html

Cheers,
Brent