The Normal Distribution

  The normal distributions are a very important class of statistical distributions. All normal distributions are symmetric and have bell-shaped density curves with a single peak.

To speak specifically of any normal distribution, two quantities have to be specified: the mean , where the peak of the density occurs, and the standard deviation , which indicates the spread or girth of the bell curve. (The greek symbol is pronounced mu and the greek symbol is pronounced sig-ma.)

Different values of and yield different normal density curves and hence different normal distributions. Try the applet below for example. You should be able to change the mean and the standard deviation using the sliders and see the density change.

The normal density can be actually specified by means of an equation. The height of the density at any value x is given by

Although there are many normal curves, they all share an important property that allows us to treat them in a uniform fashion.

The 68-95-99.7% Rule

  All normal density curves satisfy the following property which is often referred to as the Empirical Rule.

68%

of the observations fall within 1 standard deviation of the mean, that is, between and .

95%

of the observations fall within 2 standard deviations of the mean, that is, between and .

99.7%

of the observations fall within 3 standard deviations of the mean, that is, between and .

The check buttons below will help you realize the appropriate percentages of the area under the curve.

Remember that the rule applies to all normal distributions. Also remember that it applies only to normal distributions.

An Example

Let us apply the Empirical Rule to Example 1.17 from Moore and McCabe.

The distribution of heights of American women aged 18 to 24 is approximately normally distributed with mean 65.5 inches and standard deviation 2.5 inches. From the above rule, it follows that

68%

of these American women have heights between 65.5 - 2.5 and 65.5 + 2.5 inches, or between 63 and 68 inches,

95%

of these American women have heights between 65.5 - 2(2.5) and 65.5 + 2(2.5) inches, or between 63 and 68 inches.

Again, you can try this out with the example below.

Therefore, the tallest 2.5% of these women are taller than 70.5 inches. (The extreme 5% fall more than two standard deviations, or 5 inches from the mean. And since all normal distributions are symmetric about their mean, half of these women are the tall side.)

Almost all young American women are between 58 and 73 inches in height if you use the 99.7% calculations.

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