Chapter 7
The Normal
Distribution
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7-2 Properties of the Normal
Distribution
⚫ Many continuous variables have
distributions that are bell-shaped and are
called approximately normally distributed
variables.
⚫ The theoretical curve, called the normal
distribution curve, can be used to study
many variables that are not normally
distributed but are approximately normal.
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7-2 Properties of the
Theoretical Normal Distribution
⚫ The normal distribution curve is
bell-shaped.
⚫ The mean, median, and mode are
equal and located at the center of the
distribution.
⚫ The normal distribution curve is
unimodal (single mode).
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7-2 Properties of the
Theoretical Normal Distribution
⚫ The curve is symmetrical about the
mean.
⚫ The curve is continuous.
⚫ The curve never touches the x-axis.
⚫ The total area under the normal
distribution curve is equal to 1.
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7-2 Properties of the
Theoretical Normal Distribution
⚫ The area under the normal curve that
lies within
✓ one standard deviation of the mean is
approximately 0.68 (68%).
✓ two standard deviations of the mean is
approximately 0.95 (95%).
✓ three standard deviations of the mean is
approximately 0.997 (99.7%).
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7-2 Areas Under the Normal Curve
− − − + + +
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7-3 The Standard Normal
Distribution
⚫ The standard normal distribution is a
normal distribution with a mean of 0
and a standard deviation of 1.
⚫ All normally distributed variables can
be transformed into the standard
normally distributed variable by using
the formula for the standard score:
(see next slide)
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7-3 The Standard Normal
Distribution
value − mean
z=
standard deviation
or
X −
z=
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7-3 Area Under the Standard
Normal Curve - Example
⚫ Find the area under the standard
normal curve between z = 0 and
z = 2.34 P(0 z 2.34).
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7-3 Area Under the Standard
Normal Curve - Example
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7- 3 Area Under the Standard
Normal Curve - Example
⚫ Find the area under the standard
normal curve between z = 0 and
z = –1.75 P(–1.75 z 0).
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7-3 Area Under the Standard
Normal Curve - Example
−
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7-3 Area Under the Standard
Normal Curve - Example
−
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7-3 Area Under the Standard
Normal Curve - Example
⚫ Find the area to the right of z = 1.11
P(z 1.11).
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7-3 Area Under the Standard
Normal Curve - Example
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7-3 Area Under the Standard
Normal Curve - Example
⚫ Find the area to the left of z = –1.93
P(z –1.93).
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7-3 Area Under the Standard
Normal Curve - Example
−
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7-3 Area Under the Standard
Normal Curve - Example
⚫ Find the area between z = 2 and
z = 2.47 P(2 z 2.47).
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7-3 Area Under the Standard Normal Curve -
Example
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7-3 Area Under the Standard
Normal Curve - Example
⚫ Find the area between z = 1.68 and
z = –1.37 P(–1.37 z 1.68).
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7-3 Area Under the Standard Normal Curve -
Example
−
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7-3 Area Under the Standard
Normal Curve - Example
⚫ Find the area to the left of z = 1.99
P(z 1.99).
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7-3 Area Under the Standard Normal Curve -
Example
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7-3 Area Under the Standard
Normal Curve - Example
⚫ Find the area to the right of
z = –1.16 P(z –1.16).
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7-3 Area Under the Standard Normal Curve -
Example
−
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RECALL: The Standard Normal
Distribution
value − mean
z=
standard deviation
or
X −
z=
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7-4 Applications of the Normal
Distribution - Example
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7-4 Applications of the Normal
Distribution - Example
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7-4 Applications of the Normal
Distribution - Example
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7-4 Applications of the Normal
Distribution - Example
7-37
−
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7-4 Applications of the Normal
7-38 Distribution - Example
⚫
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GROUP ACTIVITY
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