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Manager  Joined: 02 Dec 2012
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Difficulty:   55% (hard)

Question Stats: 59% (01:24) correct 41% (01:33) wrong based on 2831 sessions

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A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent 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%
Math Expert V
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Posts: 59726
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A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent 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%

Given that 68% lies between m-d and m+d, thus 32% lies out of this range.

Now, since the distribution is symmetric about m, then half of the 32%, so 16%, lies to the right of m+d. Therefore, 16% lies to the right of m+d, and hence 84% lies to the left of m+d, which means that 84% is less than m+d.

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Hope it helps.
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A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent 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%

This is easiest to solve with a bell-curve histogram. m here is equal to µ in the Gaussian normal distribution and thus m = 50% of the total population.

So, if 68% is one st.Dev, then on either side of m we have 68/2 = 34%. So, 34% are to the right and left of m (= 50%). In other words, our value m + d = 50 + 34 = 84% going from the mean m, to the right of the distribution in the bell shaped histogram.. This means that 84% of the values are below m + d.

Like I said, doing it on a bell-curve histogram is much easier to fully "get" how this works, or you could apply GMAT percentile jargon/theory to it ##### General Discussion
Intern  Joined: 09 Sep 2013
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Not understanding this. Would someone be kind enough to provide a picture?

Thanks,
C
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Joined: 02 Sep 2009
Posts: 59726
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runningguy wrote:
Not understanding this. Would someone be kind enough to provide a picture?

Thanks,
C

Check below:
Attachment: Distribution.png [ 23.16 KiB | Viewed 59188 times ]
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Concentration: Strategy, Marketing
GMAT 1: 660 Q49 V31 GMAT 2: 680 Q48 V34 Re: A certain characteristic in a large population has a distribution that  [#permalink]

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Bunuel wrote:
runningguy wrote:
Not understanding this. Would someone be kind enough to provide a picture?

Thanks,
C

Check below:
Attachment:
Distribution.png

-------16-------34-------m-------34-------16-------

We can't say that the 68% is divided equally on both the sides of m, can we?
I'm asking this, because that's how I solved it. However, I got it right...
Math Expert V
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Posts: 59726
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b2bt wrote:
Bunuel wrote:
runningguy wrote:
Not understanding this. Would someone be kind enough to provide a picture?

Thanks,
C

Check below:
Attachment:
Distribution.png

-------16-------34-------m-------34-------16-------

We can't say that the 68% is divided equally on both the sides of m, can we?
I'm asking this, because that's how I solved it. However, I got it right...

You are not right. Check the diagram.

A distribution is symmetric about the mean m, implies that half of 68% is to the left of m and another half to the right of m.
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GMAT 1: 750 Q49 V43 Re: A certain characteristic in a large population has a distribution that  [#permalink]

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A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent 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%

Any large population has a characteristic graph known as the Normal distribution curve that looks like a bell.
SEE THE ATTACHED IMAGE
The unique property of Normal distribution is that 68% of samples lies with in the 1st SD. 95% of samples fall with in 2nd SD and about 99% of samples fall within the 3rd sd.
The Normal distraction curve is symmetric around the arithmetic mean and positive SD and negative SD are equidistant from the mean
Half of the samples lies on the left of the mean and half of the samples lies on the right on the mean.
The x axis has the following form
3SD...........2SD.........MEAN...........2SD............3SD
3d.............2d..........mean............d................3d
2%............14%<---------50%---------->84%..............99%
m+d=84%
therefore remaining samples = 100-84 = 16 _________________
Posting an answer without an explanation is "GOD COMPLEX". The world doesn't need any more gods. Please explain you answers properly.
FINAL GOODBYE :- 17th SEPTEMBER 2016. .. 16 March 2017 - I am back but for all purposes please consider me semi-retired.
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Re: A certain characteristic in a large population has a distribution that  [#permalink]

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question says that 68% of population lies within 1 standard deviation, so this means that (Mean+deviation)-(Mean-deviation|=68%
the rest is 32. 16 to the very left, 16 to the very right. We need to find from the very left to the m+d, this is 16+68=84

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Hi... Can you tell me how do you know that (m-d)<68%<(m+d)
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Re: A certain characteristic in a large population has a distribution that  [#permalink]

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Drawing a bell curve such as the one in Bunuel's diagram makes this question a lot easier to visualize.

68% of the population lies within one std dev of the mean (one std dev above and one std dev below the mean). This tells us that 32% of the population is outside of this range, or 16% on either side of the one std devs each way. Hence, the portion under m+d equals 68% + 16% = 84%.
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Re: A certain characteristic in a large population has a distribution that  [#permalink]

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A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent 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%

symmetric about mean that is 50% left and 50% right so less than m+d is 68+16=84
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Re: A certain characteristic in a large population has a distribution that  [#permalink]

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This is how much i understood from the question, Since a bell shaped curve is symmetric and the 68% of the population is within 1 standard deviation of the mean then 68/2 =34% of the population is m+d and 34% of the population is m-d, Also the rest 32%(100-68) which is 16% for m-2d and m-3d , Therefore 34+16=50% is less than m+d. After this is what I am not understanding. Can someone please explain? U Thank you in advance.
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A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68 percent 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's first sketch a distribution that is symmetric about the mean m. 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 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. 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 Since the distribution is symmetric about the mean m, this 68%, is divided into two equal populations. When we COMBINE our two findings, we see that the percentage of the population that's below m+d = 50% + 34% = 84% Cheers,
Brent
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Originally posted by GMATPrepNow on 10 Oct 2017, 10:32.
Last edited by GMATPrepNow on 16 Apr 2018, 12:28, edited 2 times in total.
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Re: A certain characteristic in a large population has a distribution that  [#permalink]

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Thanks, Brent Hanneson

This is an amazing explanation Took me a lot of time to understand but such graph is saviors.
Would you recommend to use such graphs every time in questions as above?
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Top Contributor
amitpandey25 wrote:
Thanks, Brent Hanneson

This is an amazing explanation Took me a lot of time to understand but such graph is saviors.
Would you recommend to use such graphs every time in questions as above?

Thanks for that!

I can't speak for everyone, but I certainly benefit from sketching the information (although I think I'm a visual learner).

Cheers,'
Brent
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A certain characteristic in a large population has a distribution that  [#permalink]

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Since it’s given that the $$68\%$$ of the population lies within one standard deviation, this means that one deviation above the mean and one deviation below the mean total$$68\%$$ of the total (meaning $$34\%$$ above and $$34\%$$ below).

The remaining $$100 – 68 = 32\%$$ of the population is beyond $$1$$ standard deviation away from the mean;

$$\frac{32\%}{2} = 16\%$$ above one standard deviation away and $$16\%$$ below one standard deviation away.

Thus, $$68 + 16 = 84\%$$ is less than $$m+d$$.

The best way to understand this is to create a bell curve as shown below: D

Attachment: u3ftgz5.png [ 4.88 KiB | Viewed 24811 times ]
.
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Re: A certain characteristic in a large population has a distribution that  [#permalink]

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68% of the distribution is within 1 standard deviation (d) of the mean (m).

This means that 100% - 68% = 32% is > or < 1 standard deviation from the mean. Since the distribution is equally distributed, half of the distribution will be greater than 1 standard deviation (0.5*32% = 16%) and half will be less than 1 standard deviation.

The question asks what percentage of the distribution is LESS THAN 1 standard deviation GREATER THAN the mean. We can find this by subtracting the part of the distribution greater than 1 standard deviation (16%) from 100%.

100% - 16% = 84%.

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