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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.
The normal distribution shown in Figure 1 is
a specific examples of the general rule that 68% of the area
of any normal distribution is within one standard deviation of
the mean.
Figure 2 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.
Calculate
Area for a given X
Calculate
X for a given Area
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