A distribution that is symmetric about the mean with 68 percent of its data
within one standard deviation of the mean is a normal distribution.

\(\frac{68}{2} = 34\)% on each side of the mean

The mean plus one deviation (m + 1d) = (m + d) above the mean is (50% + 34%) = 84%
The percent of the distribution that is below (m + d) is 84%

Or: 50% of the data lies below the mean.
Another 34% lies within one standard deviation above the mean.
50% + 34% = 84% of the distribution lies below the mean.

The percent of the distribution that is less than (m+d) = 84%

Answer D


Memorizing these three numbers will make life easy in normal distribution questions:
34%, 14%, 2%
Standard deviation = SD = d

In normal distribution questions, the mean is 50%
34%, 14%, and 2% correspond with 1SD, 2SD, and 3SD in the distribution

Above the mean, add the percents (to 50%). Below the mean, subtract the percents (from 50%).

Knowing the percents that correlate with the distribution helps, too: 0-2-16-50-84-98-100

In a normal distribution:

• 68 percent of data lies within 1 standard deviation of the mean
(i.e., within one deviation on EACH side of the mean)
34% below the mean, 34% above the mean
See the diagram.
ABOVE the mean: 50% + 34% = 84%
BELOW the mean: 50% - 34% = 16%

• 95 percent of data lies within 2 standard deviations of the mean (95%-68%) = 27%. It is standard to approximate.
Roughly half of 27% is 14%, so after the first deviation
ABOVE the mean: (84% + 14%) = 98%
BELOW the mean (16% - 14%) = 2%

• \(99.7 \approx 100\)% of data lies within 3 standard deviations of the mean (100 - 95) = 5%
Roughly half of 5% is 2.5%. I use 2%
ABOVE the mean (98% + 2%) = 100%
BELOW the mean (2% - 2%) = 0%

Note: some distributions use 2.5% (3 SDs from mean) and 13.5% (2 SDs from mean) instead of 2% and 14%
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And above all, watch with glittering eyes the whole world around you
because the greatest secrets are always hidden in the most unlikely places.
Those who don't believe in magic will never find it. —Roald Dahl