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Empirical Rule

Empirical Rule : If the histogram of the data is approximately mound shaped then

1.: About 68% of the data fall in the interval $\bar{X} - s$ to $\bar{X} + s$ .
2.: About 95% of the data fall in the interval $\bar{X} - 2s$ to $\bar{X} + 2s$ .
3.: About 99.5% of the data fall in the interval $\bar{X} - 3s$ to $\bar{X} + 3s$ .

This is based on the normal distribution . Suppose X has a normal distribution with mean $\mu$ and standard deviation $\sigma$ . To determine the $P(\mu - \sigma < X < \mu + \sigma)$ , form the z-scores

Z₁	=	$\displaystyle \frac{\mu - \sigma - \mu}{\sigma} = -1 , \textrm{ and;}$
Z₂	=	$\displaystyle \frac{\mu + \sigma - \mu}{\sigma} = 1.$

Now call up the class code. The area under the curve to the left of Z₁ = -1 is .1586. The area to the left of Z₂ = 1 is .8413. Hence

$\begin{displaymath}P(\mu - \sigma < X < \mu + \sigma) = .8413 - .1586 = .6827 \end{displaymath}$

or about 68%. The other two parts of the rule are obtained in the problems.

2001-01-01