# 3.3.4 - The Empirical Rule

3.3.4 - The Empirical Rule

The Empirical Rule is sometimes referred to as the 68-95-99.7% Rule. The rule is a statement about normal or bell-shaped distributions.

Empirical Rule

In any normal or bell-shaped distribution, roughly...

• 68% of the observations lie within one standard deviation to either side of the mean.
• 95% of the observations lie within two standard deviations to either side of the mean.
• 99.7% of the observations lie within three standard deviations to either side of the mean.
Note! Students tend to use these approximation instead of the more precise values found in the tables or by using software. The empirical rule should be used as a quick estimate. The more precise values should be used when possible.

## Try It!

Use the normal table to validate the empirical rule. In other words, find the exact probabilities $P(-1<Z<1)$, $P(-2<Z<2)$, and $P(-3<Z<3)$ using the normal table and compare the values to those from the empirical rule.

$P(-1<Z<1)= P(Z<1)-P(Z<-1) = .8413 - .1587 \approx .68$

$P(-2<Z<2)= P(Z<2)-P(Z<-2) = .9772 - .0228 \approx .95$

$P(-3<Z<3)= P(Z<3)-P(Z<-3) = .9987 - .0013 \approx .99.7$

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