The Empirical Rule for the Normal Distribution

For any normal distribution, the proportion of values that lies within a given number of standard deviations of the mean is fixed. The empirical rule (also known as the 68-95-99.7 rule) states that for a normal distribution with mean $\mu$ and standard deviation $\sigma$:

• About $\mathbf{68\%}$ of values lie within $1$ standard deviation of the mean: in $(\mu-\sigma,\,\mu+\sigma).$
• About $\mathbf{95\%}$ of values lie within $2$ standard deviations of the mean: in $(\mu-2\sigma,\,\mu+2\sigma).$
• About $\mathbf{99.7\%}$ of values lie within $3$ standard deviations of the mean: in $(\mu-3\sigma,\,\mu+3\sigma).$

For example, suppose adult resting heart rates are normally distributed with $\mu = 72$ bpm and $\sigma = 8$ bpm. Then the rates within $1\sigma$ of the mean fall in

$$(72-8,\;72+8) = (64,\;80),$$

so about $68\%$ of adults have a resting heart rate between $64$ and $80$ bpm. Similarly, about $95\%$ have a rate in $(56,\,88),$ and about $99.7\%$ have a rate in $(48,\,96).$