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A certain list of 100 data has a mean of 6 and a standard

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SVP
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A certain list of 100 data has a mean of 6 and a standard [#permalink] New post 16 Jul 2008, 05:07
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A certain list of 100 data has a 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 explain your answer.
thanks
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Re: PS: Standard Deviation [#permalink] New post 16 Jul 2008, 05:45
tarek99 wrote:
A certain list of 100 data has a 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 explain your answer.
thanks



I think E 6,6 should be the answer.

sum of 100 numbers = 100 * 6 = 600

D = SQRT( 1/100 * summation of [(x[i] - 6) ^2 ]

with addition of 2 terms the total number of terms will be 102, so the bracketed part of SQRT above is already smaller,

My approach involves not increasing the summation part, and 6, 6 is ideal option for this

adding 6 + 6 to 600 = 612

mean will be 612/102 = 6..unchanged
since the added terms are 6,6 , the difference (6-6) = 0 and wont add to the summation part of SQRT above

thus SD will be smaller.
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Re: PS: Standard Deviation [#permalink] New post 16 Jul 2008, 07:38
Expert's post
My fast approach to SD. (10-20sec)

Think of SD as average deviations of data from a mean.
the less deviations we add the more we reduce SD. 6 and 6 add the least deviations: 0=(6-6) and 0=(6-6). So, E
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Re: PS: Standard Deviation [#permalink] New post 16 Jul 2008, 07:47
Mindful that, mean=6 and also keep formula for SD in mind.

Without adding anything in numerator, 6 & 6 add +2 in the denominator; So E.
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Re: PS: Standard Deviation [#permalink] New post 16 Jul 2008, 12:34
OA is E.

According to the OE, it says that when you add at least 2 numbers that are the same as the average, then the standard deviation becomes smaller. My question here is that aren't you reducing the standard deviation anyways when you add just one number that is the same as the average? do you really need at least 2 numbers??
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Re: PS: Standard Deviation [#permalink] New post 16 Jul 2008, 14:45
tarek99,

Yes. Bue, the question stem asks for addition of two numbers.
Re: PS: Standard Deviation   [#permalink] 16 Jul 2008, 14:45
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