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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