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.
The normal curve showing the empirical rule.
µ−2 σ µ−1 σ µ+1 σ µ−3 σ µ+3 σ µ µ+2 σ 68% 95% 99.7%
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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