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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