The Standard Normal or Z-Curve

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The Standard Normal or Z-Curve

Figure 4.5 contains plots of the normal curve with mean 36 and SD 8 (TV sales), compared with plots of the normal curve with mean 70 and SD 3 (adult men's heights, in inches).

**Figure 4.5:** Areas within One and Two SD's of the Mean for N(36, 8) and N(70, 3)
$\begin{figure} \begin{center} \epsfig{file=annie368_1SD.ps, height=3.2in, width=... ...ile=annie703_2SD.ps, height=3.2in, width=3in, angle=-90}\end{center}\end{figure}$

The areas within one and two SD's of the mean, respectively, are shaded. The area under the normal curve within one SD of the mean will always be .6827, or approximately 68%. The area within two SD's of the mean will always be .9545, or approximately 95%.

All the plots in Figure 4.5 contain a second horizontal axis, labeled Z . Note that the Z=1 whenever X is one SD above the mean. Similarly, Z=-1 whenever X is one SD below the mean .

$\fbox{ \parbox{5.5in}{ \vspace*{1ex} $X$\space is how many SD's above the mean? This number is $Z$ . \vspace*{1ex} }}$

If we replace the original X-axis by the Z-axis, then all normal curves will look like Figure 4.6. This is the Z-curve, also called the standard normal curve .

**Figure 4.6:** The Standard Normal Curve and Areas within 1, 2, 3 SD's of the Mean
$\begin{figure}\begin{center} \epsfig{file=emp123.ps, width=4in, angle=-90}\end{center}\end{figure}$

The standard normal curve (or Z-curve) has the following properties:

$\fbox{ \parbox{5.5in}{ \vspace*{1ex} {\bf Properties of the $Z$ -curve:}\\ 1. T... ....0) = .95$\\ \hspace*{3em}iii. $P(-3.0 \leq Z \leq 3.0) = .997$\vspace*{1ex} }}$

The empirical rule for the SD is really a special case of the normal curve applied to data. We restate the empirical rule here

$\fbox{ \parbox{5.5in}{ \vspace*{1ex} {\bf Empirical Rule:} \\ If the data histog... ...99.7\% of the observations will fall within 3 SD's of the mean \vspace*{1ex} }}$

Exercise: Revisit Figure 4.2, and examine how closely the data satisfies the empirical rule.

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2003-09-08