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PS-OG#112

by tapanmittal » Sun Jun 21, 2015 8:22 am
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 percent of the distribution
is less than,m+d

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

[spoiler]Ans-84%[/spoiler]

I am seeking for a reasonable solution unlike the OG one,which is far beyond my understanding.
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by GMATGuruNY » Sun Jun 21, 2015 11:38 am
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 percent of the distribution is less than m+d?

A) 16%
B) 32%
C) 48%
D) 84%
E) 92%
Let:
Mean m=10.
Standard deviation d=2.

One SD below the mean = m-d = 10-2 = 8.
One SD above the mean = m+d = 10+2 = 12.

The distribution must be symmetric about the mean of 10.
Since 68% of the distribution lies within one SD of the mean, 34% lies one SD below the mean of 10, and 34% lies one SD above the mean of 10.

The distribution looks like this:
-----16%-----8-----34%-----m=10-----34%-----12-----16%-----
Notice that 50% of the distribution is below the mean of 10, with the remaining 50% above the mean of 10, yielding the required symmetry about the mean.

Since m+d = 10+2 = 12, the portion in red is less than m+d:
16% + 34% + 34% = 84%.

The correct answer is D.
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by theCEO » Sun Jun 21, 2015 11:55 am
tapanmittal wrote: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 percent of the distribution
is less than,m+d

Ans-84%

I am seeking for a reasonable solution unlike the OG one,which is far beyond my understanding.
First we can use a number line to solve this
We then convert the line to -50% to 50%.

-E ---- -1d ------M------ +d ---- +E
-50%---- -1d ------0------ +d ---- +50%

68% of the distribution lies within one standard deviation d of the mean
this means 68% of the range of the line lies within one standard deviation d of the mean
this means 68% of the range of the line lies between -1d and +d
if the distribution is symmetric, it means 68%/2 of the range lies between -1d and 0
if the distribution is symmetric, it means 68%/2 of the range lies between 0 and 1d

the range = +50% - (-50%) = 100%
therefore 68%/2 of the range = 34%
therefore 34% of the range lines between 0 and 1d
therefore 34% of the range lines between 0 and -1d

-50%---- -1d ------0------ +d ---- +50%
-50%---- -34%d ------0------ +34% ---- +50%

Rephrasing the question.......
What is the % of the distribution that is less than M+d = 0+34% = 34%?

%less that (+34%) = +34% - (-50%) = +34% + 50% = 84%
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by [email protected] » Sun Jun 21, 2015 4:07 pm
Hi tapanmittal,

The broad concepts of 'normal distribution' and 'standard deviation' usually show up just 1 time on the GMAT. The "math" is based on a Bell Curve and is a standard concept in most Statistics classes.

With a Bell Curve, 50% of the data points are below the Mean and 50% are above it. Here, we're told that 68% of the data is within on Standard Deviation of the Mean - this means that 34% of THIS data is below the Mean and 34% is above the Mean.

From an organizational standpoint, the data is "spread out" like this:

16% - more than 1 SD below the Mean
34% - within 1 SD below the Mean
34% - within 1 SD above the Mean
16% - more than 1 SD above the Mean

The question asks for the percent of the data that is LESS than (Mean + 1 SD). Looking at the above information, that would be....

16 + 34 + 34 = 84%

Final Answer: D

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by Ian Stewart » Mon Jun 22, 2015 3:48 pm
[email protected] wrote:

The broad concepts of 'normal distribution' and 'standard deviation' usually show up just 1 time on the GMAT.
'Normal distributions' are never tested on the GMAT, and test takers should not spend any time studying them. The question in the post above is not about a normal distribution; it just happens to use a number (68%) that we also see when working with normal distributions. The question in the OP is about a 'symmetric distribution', and test takers should know what that means. It is also about how standard deviation can be used to measure how far elements are from the mean (sometimes called 'z-scores'), and questions about that are common.

I point this out because there are a few confusing (and mathematically incorrect) questions from one company that are regularly posted in this forum which are about 'normal distributions', and those questions are all irrelevant to people preparing for the GMAT.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by Brent@GMATPrepNow » Tue Jun 23, 2015 7:11 am
I second what Ian says.
From time to time, normal distribution questions show up on the BEG forum, and I think they end up scaring students who wonder whether they need to learn everything there is to know about normal distributions.
If you're interested, we have an article on how to spot substandard GMAT questions:
https://www.gmatprepnow.com/articles/questions-questions

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by Matt@VeritasPrep » Tue Jun 23, 2015 6:06 pm
Another approach:

Since 68% are within one SD, 32% are NOT. Of that 32% that are not, HALF of them are below, so 16%.

That gives us 68% that we already have + another 16% below, for 84%.
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