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OG 13, Problem Solving, Q112 - Just cant understand it

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OG 13, Problem Solving, Q112 - Just cant understand it

by rishjain » Wed May 22, 2013 4:57 am

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The problem goes like this:
A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 % of the distribution lies within one standard deviation d of the mean, what percentage of the distribution is less than m+d ?

A) 16%
B) 32%
C) 48%
D) 84%
E) 92%

Can somebody please help me on this. Just cant understand the whole thing. The solution given is equally confusing.
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by Atekihcan » Wed May 22, 2013 5:09 am
As the distribution is symmetric about mean (m), 50% of the distribution must be less than m.
As 68% of the distribution lies within one standard deviation (d) of the mean (m), 68% of the distribution lie between (m - d) and (m + d)
Again as the distribution is symmetric about mean (m), 68%/2 = 34% of the distribution must be greater than m but less than (m + d)

So, percentage of the distribution is less than (m + d)
= percentage of the distribution is less than m + percentage of the distribution greater than m but less than (m + d)
= 50% + 34%
= 84%

Answer : D

The following figure may help you to understand the situation
Image
The blue area is the part of the distribution that lies within one standard deviation (d) of the mean (m).
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by Jim@StratusPrep » Wed May 22, 2013 5:28 am
If the distribution is symmetric then 50% is above the mean and 50% is below the mean. Also, from the fact that 68% of the population is within 1 standard deviation means that 1/2 of this below the mean and 1/2 is above, or 34% is above the mean. So, 50% + 34% gives you the amount below the mean plus 1 standard deviation above the mean.

D) 84%
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Re: OG 13, Problem Solving, Q112 - Just cant understand it

by Brent@GMATPrepNow » Sun Jun 06, 2021 3:32 pm
rishjain wrote: ↑
Wed May 22, 2013 4:57 am
 The problem goes like this:
A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 % of the distribution lies within one standard deviation d of the mean, what percentage of the distribution is less than m+d ?

A) 16%
B) 32%
C) 48%
D) 84%
E) 92%

Can somebody please help me on this. Just cant understand the whole thing. The solution given is equally confusing.
Let's first sketch a distribution that is symmetric about the mean m.
Image
Notice that m+d represents 1 unit of standard deviation ABOVE the mean
Likewise, m-d represents 1 unit of standard deviation BELOW the mean
And m+2d represents 2 units of standard deviation ABOVE the mean, etc.
ASIDE: There are infinitely many distributions that are symmetric about the mean m. The above distribution is just one.

Our goal is to determine what portion of the population is LESS THAN than m+d
Image

First recognize that, since the distribution is symmetric about the mean m, 50% of the population is BELOW the mean, and 50% is ABOVE the mean.
Image

Next, we're told that 68% of the distribution lies within one standard deviation d of the mean
In other words, 68% the population is BETWEEN m-d and m+d
Image

Since the distribution is symmetric about the mean m, this 68%, is divided into two equal populations.
Image

When we COMBINE our two findings, we see that the percentage of the population that's below m+d = 50% + 34% = 84%
Image

Answer: D

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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