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Areas Under Normal Distributions
Prerequisites
Distributions,
Central Tendency, Variability,
Introduction to Normal Distributions
Learning Objectives
- State the proportion of a normal distribution within 1 and within 2
standard deviations of the mean
- Use the calculator "Calculate Area for a given X"
- Use the calculator "Calculate X for a given Area."
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. Figure 1 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.
Figure 2 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.
The normal distributions shown in Figures 1 and
2 are specific examples of the general rule that 68% of the area
of any normal distribution is within one standard deviation of
the mean.
Figure 3 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.
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.
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.
Calculate
Area for a given X
Calculate
X for a given Area
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