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.

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%

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%.