Normal Distributions
Many real data sets — heights, test scores — pile up in a symmetric bell shape called a normal distribution. It is centered at the mean and described by its standard deviation.
The empirical rule, or 68-95-99.7 rule, tells you what fraction of the data lies within one, two, and three standard deviations of the mean.
The bell shape
A normal distribution is symmetric around its mean, with most values near the center and fewer in the tails. The mean, median, and mode all sit at the middle.
Its spread is set by the standard deviation: a larger one makes a wider, flatter bell.
The 68-95-99.7 rule
About 68% of the data lies within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.
So values beyond 2 standard deviations are rare — only about 5% total in both tails. This rule answers most normal-distribution questions quickly.
Worked examples
Example 1: within two deviations
About what percent of data lies within 2 standard deviations of the mean?
Answer: About 95%
Example 2: within one deviation
About what percent lies within 1 standard deviation?
Answer: About 68%
Example 3: beyond two deviations
About what percent lies more than 2 standard deviations from the mean?
Answer: About 5%
Try one yourself
Common questions
What is the 68-95-99.7 rule?
For a normal distribution, about 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
What shape is a normal distribution?
A symmetric bell curve centered at the mean, with most data near the center and thin tails.
Are values beyond 2 deviations common?
No — only about 5% of data lies beyond 2 standard deviations, split between the two tails.
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