A certain list of 100 data has an average (arithmetic mean) : GMAT Problem Solving (PS)
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# A certain list of 100 data has an average (arithmetic mean)

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A certain list of 100 data has an average (arithmetic mean) [#permalink]

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08 Dec 2009, 02:54
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A certain list of 100 data has an average (arithmetic mean) 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 of 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

Thanks!
[Reveal] Spoiler: OA

Last edited by Bunuel on 20 Feb 2012, 15:33, edited 1 time in total.
Edited the question and added the OA
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08 Dec 2009, 04:33
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Gmatter111 wrote:
Hi,

Please take a look at the question from GMAT Prep:

A certain list of 100 data has an average (arithmetic mean) 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 of 102 data with standard deviation less than d?

-6 and 0
0 and 0
0 and 6
0 and 12
6 and 6

Thanks!

Logically, SD of a set will decrease if you add numbers which are equal to its mean. Thus the answer should be E.

I will however provide you with mathematical reasoning to justify the above statement.

S.D. for hundred numbers = d = $$\sqrt{\frac{S}{100}}$$ where 'S' is sum of the squares of the difference between each number and the mean.

Now let the two numbers added be 'x' and 'y'.

S.D after adding the two numbers will be = $$\sqrt{\frac{S}{102}+\frac{(x-6)^2 + (y-6)^2}{102}}$$

Now it is obvious that $$\frac{S}{102}$$ will be less than $$\frac{S}{100}$$. Also, the minimum value of $$\frac{(x-6)^2 + (y-6)^2}{102}$$ will be 0 when both 'x' and 'y' are equal to 6.

Thus if the two numbers added are equal to the mean, the SD of the set must decrease. (Unless of course SD of the set was 0 to start with (not in our case) and then in that case SD will remain constant).

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08 Dec 2009, 05:07
I agree with sriharimurthy.
In most cases Standard Deviation question can be solved without using SD formula.
The main idea is to think of SD as of average deviation from mean: $$average (|x - x_{av}|)$$.
So, 6 and 6 add two 0s in average formula and must reduce SD.
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10 Jan 2010, 06:41
Hi Sriharimurthy,
thats a great explanation,but i am stuck with one point.
when u r calculating the new SD.how can u subtract 6 hrom both x and y??6 is the mean of the old set...for examle if u add 0 and 0 the new mean will be 600/102.ie,will be less than 6. the mean will change as well..
Kindly explain this point...will really appriciate..
Thanx
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17 Nov 2010, 20:17
itsreallytime wrote:
Hi Sriharimurthy,
thats a great explanation,but i am stuck with one point.
when u r calculating the new SD.how can u subtract 6 hrom both x and y??6 is the mean of the old set...for examle if u add 0 and 0 the new mean will be 600/102.ie,will be less than 6. the mean will change as well..
Kindly explain this point...will really appriciate..
Thanx

I have the same confusion.
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18 Nov 2010, 00:14
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Werewolf wrote:
itsreallytime wrote:
Hi Sriharimurthy,
thats a great explanation,but i am stuck with one point.
when u r calculating the new SD.how can u subtract 6 hrom both x and y??6 is the mean of the old set...for examle if u add 0 and 0 the new mean will be 600/102.ie,will be less than 6. the mean will change as well..
Kindly explain this point...will really appriciate..
Thanx

I have the same confusion.

Yes, mean won't be the same for A, B, and C but as it will differ from 6 very little then the calculations above will still hold true, so it doesn't really matter.

As for the question, GMAT SD questions are fairly straightforward and don't require actual calculation of SD, they are about the general understanding of the concept.

So we have:
A certain list of 100 data has an average (arithmetic mean) 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 of 102 data with standard deviation less than d?
A. -6 and 0
B. 0 and 0
C. 0 and 6
C. 0 and 12
D. 6 and 6

"Standard deviation shows how much variation there is from the mean. A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values."

So when we add numbers, which are far from the mean we are stretching the set making SD bigger and when we add numbers which are close to the mean we are shrinking the set making SD smaller.

According to the above adding two numbers which are closest to the mean will shrink the set most, thus decreasing SD by the greatest amount.

Closest to the mean are 6 and 6 (actually these numbers equal to the mean) thus adding them will definitely shrink the set, thus decreasing SD.

For more on this issue please check Standard Deviation chapter of Math Book (link in my signature) and the following two topics for practice:

Hope it helps.
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Re: A certain list of 100 data has an average (arithmetic mean) [#permalink]

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29 Nov 2013, 05:09
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Re: A certain list of 100 data has an average (arithmetic mean) [#permalink]

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25 Mar 2015, 02:44
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Re: A certain list of 100 data has an average (arithmetic mean) [#permalink]

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07 Aug 2016, 02:33
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Re: A certain list of 100 data has an average (arithmetic mean)   [#permalink] 07 Aug 2016, 02:33
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