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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The only thing more dangerous than ignorance is arrogance.
-- Albert Einstein