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Standard Deviation in Statistics

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Standard Deviation in Statistics

by dddanny2006 » Fri Feb 14, 2014 1:41 am
Hi.

{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?

A.14,16
b.9,21
C.15,100

1)A only 2)C only 3)A and B 4)B and C 5)A,B and C

Answer is 4

My doubt is ,
1.Do I have to add only 14 to the whole set and then check,along with adding only 16 to the whole set and then check again?In both the cases the SD has remained the same.
2.I also tried another way ,that is I added 14 to two numbers of the set and 16 to the remaining three numbers of the set.The SD increased.

Please tell me how do I add numbers to the set,individually or scramble them like --I add 14 to two numbers,16 to the other 3 in the set?

I am not comfortable with checking their proximity to the mean as I'm more comfortable calculating them,although I don't mind if someone could explain that to me in detail along with the doubts 1 and 2 above..

Thanks

Daniel
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by GMATGuruNY » Fri Feb 14, 2014 2:40 am
(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.
My doubt is ,
1.Do I have to add only 14 to the whole set and then check,along with adding only 16 to the whole set and then check again?In both the cases the SD has remained the same.
2.I also tried another way ,that is I added 14 to two numbers of the set and 16 to the remaining three numbers of the set.The SD increased.
You would calculate the SD when both 14 and 16 are included (9,12,14,15,16,18,21) and compare it to the SD of the original set (9,12,15,18,21).
But there is no reason -- and not enough time -- to calculate the SD of each set.
On the GMAT, problems about SD are best solved conceptually.
They are also RARE.
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by dddanny2006 » Fri Feb 14, 2014 3:14 am
Thanks for that Mitch,I did get your explanation.Could to please explain this point below in detail.

1 If you include new numbers in the set - that is tricky, because adding in most numbers will change the mean of the entire set, which will change the deviation from the mean for each number on the list, which changes the standard deviation. If you include an additional number or a few additional numbers that are far away from the other numbers, this inclusion will wildly increase the standard deviation.

2 If you include two new numbers that are symmetrical around the mean, then that will not change the mean. If the distance of these two numbers from the mean is greater than the standard deviation, adding them will increase the standard deviation (there's a larger "average" distance from the mean). If the distance of these two numbers from the mean is less than the standard deviation, adding them will decrease the standard deviation (there's a smaller "average" distance from the mean).
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