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

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08 Dec 2009, 03:54

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A

B

C

D

E

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65% (01:27) correct
35% (00:41) wrong based on 248 sessions

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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

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).

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. _________________

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

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

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.

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

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29 Nov 2013, 06:09

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

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25 Mar 2015, 03:44

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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