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Chapter 13
Normal Distributions
Chapter 13
1
Thought Question 1
Birth weights of babies born in the United
States follow, at least approximately, a bellshaped curve. What does that mean?
Thought Question 2
What does it mean if a person’s SAT score
falls at the 20th percentile for all people
who took the test?
Chapter 13
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Thought Question 3
A study in 1976-80 found that males (ages 18-24)
have a mean height of 70 inches and a standard
deviation of 2.8 in., while females (ages 18-24)
have a mean height of 65 in. and a standard
deviation of 2.5 in. A “standardized score” is the
number of standard deviations an individual falls
above or below the mean for the whole group (is
positive for values above the mean, and negative
for those below the mean). Thus, a man who is
72.8 inches tall has a standardized score of 1.
What is the standardized score for your height?
Chapter 13
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Thought Question 4
Many measurements in nature tend to follow a
similar pattern. The pattern is that most of the
individual measurements take on values that
are near the average, with fewer and fewer
measurements taking on values that are
farther from the average in either direction.
Describe what shape the distribution of such
measurements would have.
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Bell-Shaped Curve:
The Normal Distribution
of Population Values
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Asymmetric Distributions
of the Population Values
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The Normal Distribution
standard deviation
mean
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With the Mean and Standard
Deviation of the Normal Distribution
We Can Determine:
 What
proportion of individuals fall into
any range of values
 At what percentile a given individual
falls, if you know their value
 What value corresponds to a given
percentile
Chapter 13
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Empirical Rule for
Any Normal Curve
 68%
of the values fall within one
standard deviation of the mean
 95% of the values fall within two
standard deviations of the mean
 99.7% of the values fall within three
standard deviations of the mean
“68-95-99.7 Rule”
Chapter 13
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Empirical Rule for
Any Normal Curve
68%
-1sd
+1sd
95%
-2 sd
+2 sd
99.7%
-3 sd
+3 sd
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 Heights
of adults, ages 18-24
– women
 mean:
65.0 inches
 standard deviation: 2.5 inches
– men
 mean:
70.0 inches
 standard deviation: 2.8 inches
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)

Empirical Rule
– women
68% are between 62.5 and 67.5 inches
[mean  1 std dev = 65.0  2.5]
 95% are between 60.0 and 70.0 inches
 99.7% are between 57.5 and 72.5 inches

– men
68% are between 67.2 and 72.8 inches
 95% are between 64.4 and 75.6 inches
 99.7% are between 61.6 and 78.4 inches

Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
proportion of men are less than
72.8 inches tall? 68%
(by Empirical Rule)
32% / 2 = 16%
?
-1
+1
standard
deviations: -3
-2?
= 84%
-1
height values: 61.6
64.4
67.2
70
72.8
(height values)
0
1
2
3
70
72.8
75.6
78.4
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
proportion of men are less than
68 inches tall?
?
68 70
(height values)
standard
deviations: -3
-2
-1
0
1
2
3
height values: 61.6
64.4
67.2
70
72.8
75.6
78.4
Chapter 13
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Standardized Scores
 How
many standard deviations is 68
from 70?
 standardized score =
(observed value minus mean) / (std dev)
[ = (68 - 70) / 2.8 = -0.71 ]
 The
value 68 is 0.71 standard
deviations below the mean 70.
Chapter 13
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Standardized Scores
 standardized
score =
(observed value minus mean) / (std dev)




z is the standardized score
x is the observed value
m is the population mean
s is the population standard deviation
z
x-m
s
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
proportion of men are less than
68 inches tall?
?
68 70
-0.71
0
(height values)
(standardized values)
Chapter 13
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Table B: Percentiles of the
Standardized Normal Distribution
Table B (the “Standard Normal Table”) in
back of the text (or back of the supplement).
 Look up the closest standardized score in
the table.
 Find the percentile corresponding to the
standardized score (this is the percent of
values below the corresponding
standardized score or z-value).
 See
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Table B
Standard
Score
–3.4
–3.3
–3.2
–3.1
–3.0
–2.9
–2.8
–2.7
–2.6
–2.5
–2.4
–2.3
–2.2
–2.1
–2.0
–1.9
–1.8
–1.7
–1.6
–1.5
–1.4
–1.3
–1.2
Percentile
0.03
0.05
0.07
0.10
0.13
0.19
0.26
0.35
0.47
0.62
0.82
1.07
1.39
1.79
2.27
2.87
3.59
4.46
5.48
6.68
8.08
9.68
11.51
Standard
Score
–1.1
–1.0
–0.9
–0.8
–0.7
–0.6
–0.5
–0.4
–0.3
–0.2
–0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Percentile
13.57
15.87
18.41
21.19
24.20
27.42
30.85
34.46
38.21
42.07
46.02
50.00
53.98
57.93
61.79
65.54
69.15
72.58
75.80
78.81
81.59
84.13
86.43
Chapter 13
Standard
Score
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
Percentile
88.49
90.32
91.92
93.32
94.52
95.54
96.41
97.13
97.73
98.21
98.61
98.93
99.18
99.38
99.53
99.65
99.74
99.81
99.87
99.90
99.93
99.95
99.97
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Table B: Percentiles of the
Standardized Normal Distribution
Standard Score
(Z)
Percentile
-0.8
21.19
-0.7
24.20
-0.6
27.42
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
proportion of men are less than
68 inches tall?
24.20%
68 70
-0.71
0
(height values)
(standardized values)
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
height value is the 10th percentile
for men ages 18 to 24?
10%
? 70
(height values)
Chapter 13
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Table B: Percentiles of the
Standardized Normal Distribution
Table B (the “Standard Normal Table”) in
back of the text (or back of the supplement).
 See
 Look
up the closest percentile in the table.
 Find
the corresponding standardized score.
 The
value you seek is that many standard
deviations from the mean.
Chapter 13
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Table B: Percentiles of the
Standardized Normal Distribution
Standard Score
(Z)
Percentile
-1.4
8.08
-1.3
9.68
-1.2
11.51
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
height value is the 10th percentile
for men ages 18 to 24?
10%
? 70
-1.3
0
(height values)
(standardized values)
Chapter 13
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Observed Value for a
Standardized Score
 What
height value is the 10th percentile
for men ages 18 to 24?
 observed value =
mean plus [(standardized score)  (std dev)]
= 70 + [(-1.3 )  (2.8)]
= 70 + (-3.64) = 66.36
 The
value 66.36 is approximately the
10th percentile of the population.
Chapter 13
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Observed Value for a
Standardized Score
 observed
value =
mean plus [(standardized score)  (std dev)]

x is the observed value

m is the population mean
z is the standardized score
s is the population standard deviation


x  m  zs
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 RECALL:
What proportion of men are
less than 68 inches tall?
NOW: what proportion
of men are greater
than 68 inches tall?
?
100%-24.2%
24.20% = 75.8%
68 70
-0.71
0
(height values)
(standardized values)
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 The
average height of males ages 18–
24 years old was 70.0 inches with a
standard deviation of 2.8 inches.
 It
is also known that this distribution of
heights follows a normal or bell-shaped
curve.
 What
proportion of men are between
68 inches tall and 74 inches tall?
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 First,
draw and label a normal curve.
m = 70 in.
s = 2.8 in.
standard
deviations: -3
-2
-1
0
1
2
3
height values: 61.6
64.4
67.2
70
72.8
75.6
78.4
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 Shade
on the graph the range of
heights between 68 and 74 inches.
?
(height values)
(standardized values)
68 70
?
0
Chapter 13
74
?
31
Standardized Scores
 How
many standard deviations is 68
from 70?
 standardized score =
(observed value minus mean) / (std dev)
[ = (68 - 70) / 2.8 = –0.71 ]
 The
value 68 is 0.71 standard
deviations below the mean 70.
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
proportion of men are less than 68
inches tall?
?%
(height values)
68 70
(standardized values) -0.71
0
Chapter 13
74
?
33
Table B: Percentiles of the
Standardized Normal Distribution
Standard Score
(Z)
Percentile
-0.8
21.19
-0.7
24.20
-0.6
27.42
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
proportion of men are less than 68
inches tall?
24.20%
(height values)
68 70
(standardized values) -0.71
0
Chapter 13
74
?
35
Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
proportion of men are less than 74
inches tall?
?%
24.20%
(height values)
68 70
(standardized values) -0.71
0
Chapter 13
74
?
36
Standardized Scores
 How
many standard deviations is 74
from 70?
 standardized score =
(observed value minus mean) / (std dev)
[ = (74 - 70) / 2.8 = 1.43 ]
 The
value 74 is 1.43 standard
deviations above the mean 70.
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
proportion of men are less than 74
inches tall?
?%
24.20%
(height values)
68 70
(standardized values) -0.71
0
Chapter 13
74
1.43
38
Table B: Percentiles of the
Standardized Normal Distribution
Standard Score
(Z)
Percentile
1.3
90.32
1.4
91.92
1.5
93.32
Chapter 13
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Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
proportion of men are less than 74
inches tall?
91.92%
24.20%
(height values)
68 70
(standardized values) -0.71
0
Chapter 13
74
1.43
40
Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What
proportion of men are between
68 inches tall and 74 inches tall? 67.72%
91.92%
24.20%
(height values)
91.92% – 24.20%
= 67.72%
68 70
(standardized values) -0.71
0
Chapter 13
74
1.43
41
Key Concepts
 Population
values are distributed with
differing shapes, some normal, some
non-normal.
 Empirical Rule (“68-95-99.7 Rule”)
 Standardized Score
 Percentile
 Standard Normal Table
Chapter 13
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