Problem Solving

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 actually the describtion of normal distribution, and in normal distribution 68% ofthe population lies within one standard deviation, 14% within m-d and m-2d and 2% are less than m-2d. Adding these percents, we have:

68%+14%+2%=84%

Abhijit K wrote: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:
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.