Areas of Normal Distributions2.42003/05/142003/07/11 15:06:17.313 GMT-5DavidLanelane@rice.eduDavidLanelane@rice.eduMatthewJeanesmjeanes@rice.eduareasundernormaldistributionstandarddeviationWays to compute the area under a normal distribution using a non-calculus approach.
Areas under portions of a normal distribution can be computed by
using calculus. Since this is a non-mathematical treatment of
statistics, 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 60
contains 68% of the distribution.
Normal distribution with a mean of 50 and standard deviation
of 10. 68% of the area is within one standard deviation (10)
of the mean (50).
shows a normal
distribution with a mean of 100 and a standard deviation of
20. As in Figure 1, 68% of the distribution is within one
standard deviation of the mean.
Normal distribution with a mean of 100 and standard
deviation of 20. 68% of the area is within one standard
deviation (20) of the mean (100).
The normal distributions shown in and
are specific examples of the general rule
that 68% of the area of any normal distribution is within one
standard deviation of the mean.
68% of the area of any normal distribution is within one
standard deviation of the mean
shows a normal
distribution with a mean of 75 and a standard deviation of
10. The shaded area contains 95% of the area and extends from
55.4 to 94.6. For all normal distributions, 95% of the area is
within 1.96 standard deviations of the mean. For quick
approximations, it is sometimes useful to round off and use 2
rather than 1.96 as the number of standard deviations you need
to extend from the mean so as to include 95% of the area.
A normal distribution with a mean of 75 and a standard
deviation of 10. 95% of the area is within 1.96 standard
deviations of the mean.
The Java applet "Calculate Area for a given X " can be used to
calculate areas under the normal distribution. Use it to find
the proportion of a normal distribution with a mean of 90 and
a standard deviation of 12 that is above 110. Set the mean to
90 and the standard deviation to 12. Then enter "110" in the
box to the right of the radio button "Above." At the bottom of
the display you will see that the shaded area is 0.04779. See
if you can use the applet to find that the area between 115
and 120 is 0.012401.
Display from applet showing the area above 110.
The applet "Calculate X for a given Area" works in
reverse. For example, say you wanted to find the score
corresponding to the 75th percentile of a normal distribution
with a mean of 90 and a standard deviation of 12. You enter 90
for the mean and 12 for the standard deviation. Then, enter
0.75 for the shaded area and click the "Below" button. The
area below 98.0939 is 0.75.
Display from applet showing that the 75th percentile is 98.093.