Probability &
Statistics
Sections 2.5 and 5.1
Introduction to Normal
Distributions and the
Standard
Distribution
Empirical Rule (68(68-9595-99.7%)
Empirical Rule
For data with a (symmetric) bell-shaped distribution, the
standard deviation has the following characteristics.
1. About 68% of the data lie within one standard
deviation of the mean.
2. About 95% of the data lie within two standard
deviations of the mean.
3. About 99.7% of the data lie within three standard
deviation of the mean.
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Empirical Rule (68(68-9595-99.7%)
99.7% within 3
standard deviations
95% within 2
standard deviations
68% within
1 standard
deviation
34%
34%
2.35%
2.35%
13.5%
–4
–3
–2
–1
13.5%
0
1
2
3
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4
3
Using the Empirical Rule
Example:
Example
The mean value of homes on a street is $125 thousand with a
standard deviation of $5 thousand. The data set has a bell
shaped distribution. Estimate the percent of homes between
$120 and $130 thousand.
68%
105
110
115
120
125
130
µ–σ
µ
µ+σ
135
140
145
68% of the houses have a value between $120 and $130 thousand.
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Using the Empirical Rule
Suppose the class average on a test is 87% and the
standard deviation of the class is 3. Find:
1. The percentage of students who scored > 90%.
2.
The percentage of students who scored between
84% and 90%.
3.
The percentage of students who scored < 84%.
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5
Properties of Normal Distributions
The most important probability distribution in
statistics is the normal distribution.
distribution
Normal curve
x
A normal distribution is a continuous probability
distribution for a random variable, x. The graph of a
normal distribution is called the normal curve.
curve
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Properties of Normal Distributions
Properties of a Normal Distribution
1. The mean, median, and mode are equal.
2. The normal curve is bell-shaped and symmetric about
the mean.
3. The total area under the curve is equal to one.
4. The normal curve approaches, but never touches the xaxis as it extends farther and farther away from the
mean.
5. Between µ − σ and µ + σ (in the center of the curve), the
graph curves downward. The graph curves upward to
the left of µ − σ and to the right of µ + σ. The points at
which the curve changes from curving upward to
curving downward are called the inflection points.
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Properties of Normal Distributions
Inflection points
Total area = 1
µ − 3σ
µ − 2σ
µ−σ
µ
µ+σ
µ + 2σ
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µ + 3σ
x
8
Means and Standard Deviations
A normal distribution can have any mean and
any positive standard deviation.
Inflection
points
The mean gives
the location of
the line of
symmetry.
Inflection
points
1 2 3 4 5 6
x
1 2
3 4 5
6 7
8
9 10 11
Mean: µ = 3.5
Mean: µ = 6
Standard
deviation: σ ≈ 1.3
Standard
deviation: σ ≈ 1.9
x
The standard deviation describes the spread of the data.
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Means and Standard Deviations
Example:
Example
1. Which curve has the greater mean?
2. Which curve has the greater standard deviation?
B
A
1
3
5
7
9
11
13
x
The line of symmetry of curve A occurs at x = 5. The line of symmetry
of curve B occurs at x = 9. Curve B has the greater mean.
Curve B is more spread out than curve A, so curve B has the greater
standard deviation.
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Interpreting Graphs
Example:
Example
The heights of fully grown magnolia bushes are normally
distributed. The curve represents the distribution. What
is the mean height of a fully grown magnolia bush?
Estimate the standard deviation.
µ=8
6
The inflection points are one
standard deviation away from the
mean.
σ ≈ 0.7
7
8
9
10
Height (in feet)
x
The heights of the magnolia bushes are normally
distributed with a mean height of about 8 feet and a
standard deviation of about 0.7 feet.
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