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In Depth Information
Deviating from the Middle
The highest point of the curve is the mean.
The width and height are determined by the standard deviation. The
larger the standard deviation, the wider and flatter the curve. You can
have two normal distributions with the same mean and different
standard deviations.
68.2 percent of the area under the curve is within one standard
deviation of the mean (both to the left and the right), 95.44 percent of the area
under the curve is within two standard deviations, and 99.72 percent of
the area under the curve is within three standard deviations.
The extreme left and right ends of the curve are called the tails. Extreme
values are found in the tails. For example, in a distribution of height,
very short heights are in the left tail, and very large heights are in the
right tail.
Figure 9-7:
Displaying a
in a graph.
Different sets of data almost always produce different means and standard
deviations and, therefore, a different-shaped bell curve. Figure 9-8 shows two
superimposed normal distributions. Each is a perfectly valid normal
distribution; however, each has its own mean and standard deviation, with the
narrower curve having a smaller standard deviation.
Analysis is often done with normal distributions to determine probabilities.
For example, what is the probability that a 10-year-old child’s height is 54
inches? Somewhere along the curve is a discrete point that represents this
height. Further computation (outside the scope of this discussion) returns
the probability. What about finding the probability that a 10-year-old is 54
inches high or greater ? Then the area under the curve is considered. These
are the type of questions and answers determined with normal distributions.
A good amount of analysis of normal distributions involves the values in the
tails — the areas to the extreme left and right of the normal distribution curve.
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