Standard Deviation

[cazubuine]

Not sure why the answer is D, shouldn't it be B?

(9,12,15,18,21)

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only (B) III only (C) I and II (D) II and III (E) I, II, and III

[Uva@90]

Not sure why the answer is D, shouldn't it be B?

(9,12,15,18,21)

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only (B) III only (C) I and II (D) II and III (E) I, II, and III

Hi Cazubuine,

Even I Think Answer should be B.

Standard Deviation Increases or decrease Only if Outer Value Increase or decline.
By adding 9 or 21 does not affect the SD.

Experts,
Pls Correct me if I am wrong .

Thanks in advance.

Regards,
Uva.

[ani781]

Hi ,
The standard deviation is the dispersion of numbers about the arithmetic 'mean'.
This means how much the numbers contribute towards shifting or tilting from the mean.

Clearly , with the example set (9,12,15,18,21) :
when the following are added :
I. 14, 16 -- The mean remains the same. But the distribution gets closer to the mean 15. You can consider/picturize this to be putting more weights closer to the mean thereby dragging the numbers closer to a central number ( the mean). Hence the SD decreases.

II. 9, 21 -- The arithmentic mean remains the same. But since the extreme corner numbers are added/ repeated (again imagine putting weights on the extreme sides thereby straightening the line more than before), the numbers about the mean get dispersed. Hence the SD increases.

III. 15, 100 -- Here clearly the SD increases as 100 is too heavy.So SD increases.

So clearly , the answer is D.

[vinay1983]

Hi ,
The standard deviation is the dispersion of numbers about the arithmetic 'mean'.
This means how much the numbers contribute towards shifting or tilting from the mean.

Clearly , with the example set (9,12,15,18,21) :
when the following are added :
I. 14, 16 -- The mean remains the same. But the distribution gets closer to the mean 15. You can consider/picturize this to be putting more weights closer to the mean thereby dragging the numbers closer to a central number ( the mean). Hence the SD decreases.

II. 9, 21 -- The arithmentic mean remains the same. But since the extreme corner numbers are added/ repeated (again imagine putting weights on the extreme sides thereby straightening the line more than before), the numbers about the mean get dispersed. Hence the SD increases.

III. 15, 100 -- Here clearly the SD increases as 100 is too heavy.So SD increases.

So clearly , the answer is D.

Excellent work. D is the choice.I think this an OG question.

[Brent@GMATPrepNow]

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.

So, if you apply this concept to the original question, you'll find that the answer is, indeed, D

Cheers,
Brent

[GMATGuruNY]

Not sure why the answer is D, shouldn't it be B?

(9,12,15,18,21)

Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?

I. 14, 16
II. 9, 21
III. 15, 100

(A) II only (B) III only (C) I and II (D) II and III (E) I, II, and III

SD serves to describe how a set of data DEVIATES from the mean.
The MORE a set of data deviates from the mean, the GREATER the SD.
The LESS a set of data deviates from the mean, the LOWER the SD.

Note that the values in the set above are SYMMETRICAL about the median of 15.
When the values in a set are symmetrical about the median, the mean = the median.
Thus, the mean of the set above = 15.

I: 14, 16
Since these values are symmetrical about 15, adding them to the set will not change the mean.
14 and 16 are CLOSER to the mean of 15 than are 9, 12, 18, and 21.
Thus, adding 14 and 16 to the set will DECREASE the standard deviation.
Eliminate any answer choice that includes statement I (C and E).

II: 9, 21
Since these values are symmetrical about 15, adding them to the set will not change the mean.
9 and 21 are FURTHER from the mean of 15 than are 12, 15, and 18.
Thus, adding 9 and 21 to the set will INCREASE the standard deviation.
Eliminate any answer choice that does not include statement II (B).

III: 15, 100
15 and 100 are not symmetrical about 15.
Adding 100 to the set will increase the mean.
But 100 is MUCH, MUCH FURTHER from the current mean than are 9, 12, 15, 18 and 21.
Thus, adding 100 to the set will INCREASE the standard deviation.
Eliminate any answer choice that does not include statement III (A).

The correct answer is D.

[nikhilgmat31]

just visualizing the sets A,B,C gives that only set S will impact SD.

but actually it both set the sets - B, C

so simplify the calculation we can just calculate till Variance & compare variance of sets.

variance of original set = 36 + 9 + 0 + 9 + 36 / 5 = 18

variance of original set + A = 1 + 36 + 9 + 0 + 9 + 36 + 1 / 7 = 92/7 = 15......

variance of original set + B = 36 + 36 + 9 + 0 + 9 + 36 + 36 / 7 = 162/7 = 23......

variance of original set + C = 0 + 36 + 9 + 0 + 9 + 36 + 85^2 / 7 = clearly more than 18

so both sets B & C increases the SD.

Answer is D