5.6 Areas of normal distributions

Areas under portions of a normal distribution can be computed by using calculus. Since this is a non-mathematical treatment ofstatistics, we will rely on computer programs and tables to determine these areas. shows a normal distribution with a mean of 50 and a standard deviation of 10. The shaded area between 40 and 60contains 68% of the distribution.

shows a normal distribution with a mean of 100 and a standard deviation of20. As in Figure 1, 68% of the distribution is within one standard deviation of the mean.

The normal distributions shown in and are specific examples of the general rule that 68% of the area of any normal distribution is within onestandard deviation of the mean.

shows a normal distribution with a mean of 75 and a standard deviation of10. The shaded area contains 95% of the area and extends from 55.4 to 94.6. For all normal distributions, 95% of the area iswithin 1.96 standard deviations of the mean. For quick approximations, it is sometimes useful to round off and use 2rather than 1.96 as the number of standard deviations you need to extend from the mean so as to include 95% of the area.

The Java applet "Calculate Area for a given X " can be used to calculate areas under the normal distribution. Use it to findthe proportion of a normal distribution with a mean of 90 and a standard deviation of 12 that is above 110. Set the mean to90 and the standard deviation to 12. Then enter "110" in the box to the right of the radio button "Above." At the bottom ofthe display you will see that the shaded area is 0.04779. See if you can use the applet to find that the area between 115and 120 is 0.012401.

The applet "Calculate X for a given Area" works in reverse. For example, say you wanted to find the scorecorresponding to the 75th percentile of a normal distribution with a mean of 90 and a standard deviation of 12. You enter 90for the mean and 12 for the standard deviation. Then, enter 0.75 for the shaded area and click the "Below" button. Thearea below 98.0939 is 0.75.