A certain characteristic in a large population has a distribution that

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Not understanding this. Would someone be kind enough to provide a picture?

Thanks,
C

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

-------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.

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%


Answer: D

Cheers,
Brent

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

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:



The final answer is .

Solution:

Since 68% of the distribution is between the values of m - d and m + d, then 68/2 = 34% of the distribution is between m (the mean) and m + d (and the other 34% is between m - d and m). Since in a symmetric distribution, 50% of the distribution is less than m (the mean), then 50 + 34 = 84% of the distribution is less than m + d.

Answer: D

Answer: Option D

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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 ?

Gaussian distribution below:


without considering the figure (because I have never seen a gaussian distribution on the gmat)
with respect to the mean 50% will be greater and 50% smaller
we know that 50-68/2=16% will be greater than m + d.
thus we can conclude that 100%-16%=84% will be less than the m+d

A. 16
B. 32
C. 48
D. 84
E. 92

Since the distribution is symmetric about the mean m and 68 percent of the distribution lies within one standard deviation d of the mean, we can infer that 34 percent of the distribution lies between m and m + d, and another 34 percent lies between m - d and m.

To find the percent of the distribution that is less than m + d, we need to consider the 34 percent between m and m + d, as well as the 50 percent to the left of m - d.

Therefore, the percent of the distribution that is less than m + d is 34 percent + 50 percent = 84 percent.

So, the answer is (D) 84%.

I got this wrong because I thought that the region between m+d and m would also not count towards it.