Chapter 13
Normal Distributions
Chapter 13
1
The Normal Distribution
Figure 13.7 Two Normal
curves. The standard deviation
fixes the spread of a Normal
curve.
Chapter 13
2
Normal Density Curve
The normal curves are symmetric, bell-shaped
curve has these properties:
•A specific normal curve is completely
determined by its mean and its standard
deviation.
•The mean determines the center of the
distribution. It is located at the center of
symmetry of the curve.
•The standard deviation determines the shape of
the curve.
Chapter 13
3
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
4
“68-95-99.7 Rule”
• 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
Chapter 13
5
Chapter 13
6
Health and Nutrition Examination Study of 1976-1980
• 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
7
Health and Nutrition Examination Study of 1976-1980
• 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
8
Health and Nutrition Examination Study of 1976-1980
• 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
9
Health and Nutrition Examination Study of 1976-1980
• 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
10
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
11
Standard Scores




z is the standard score
x is the observed value
m is the population mean
s is the population standard deviation
x-m
z
s
Chapter 13
12
Health and Nutrition Examination Study of 1976-1980
• What proportion of men are less than 68
inches tall?
?
68 70
-0.71
0
Chapter 13
(height values)
(standard values)
13
Table B: Percentiles of the standard
Normal Distribution
• See pg. 599 in text for Table B (the “Standard
Normal Table”).
• Look up the closest standard score in the
table.
• Find the percentile corresponding to the
standard score (this is the percent of values
below the corresponding standard score or zvalue).
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14
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
15
Table B: Percentiles of the standard
Normal Distribution
Standard Score
(Z)
Percentile
-0.8
21.19
-0.7
24.20
-0.6
27.42
Chapter 13
16
Health and Nutrition Examination Study of 1976-1980
• What proportion of men are less than 68
inches tall?
24.2%
68 70
-0.71
0
Chapter 13
(height values)
(standard values)
17
Health and Nutrition Examination Study of 1976-1980
• What height value is the 10th percentile for
men aged 18 to 24?
10%
? 70
Chapter 13
(height values)
18
Table B: Percentiles of the standard
Normal Distribution
• See pg. 599 in text for Table B.
• Look up the closest percentile in the table.
• Find the corresponding standard score.
• The value you seek is that many standard
deviations from the mean.
Chapter 13
19
Table B: Percentiles of the standard
Normal Distribution
Standard Score
(Z)
Percentile
-1.4
8.08
-1.3
9.68
-1.2
11.51
Chapter 13
20
Health and Nutrition Examination Study of 1976-1980
• What height value is the 10th percentile for
men aged 18 to 24?
10%
? 70
-1.3
0
Chapter 13
(height values)
(standard values)
21
Observed Value for a standard Score
• What height value is the 10th percentile for
men aged 18 to 24?
• observed value = mean + [(standard 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 standard Score
• observed value = mean + [(standard score)  (std dev)]

x is the observed value

m is the population mean
z is the standard score
s is the population standard deviation


x  m  zs
Chapter 13
23
Key Concepts
• Population values are distributed with
differing shapes, some normal, some nonnormal.
• Empirical Rule (“68-95-99.7 Rule”)
• standard Score
• Percentile
• Standard Normal Table
Chapter 13
24
Exercise 13.7 – 13.9
IQ Test Scores. Figure 13.13 is a stemplot of the IQ
test of 74 seventh-grade students. This distribution
is very close to Normal with mean 111 and
standard deviation 11. It includes all the seventhgraders in a rural Midwest school except for 4 low
outliers who were dropped because they may have
been ill or otherwise not pay attention to the test.
Take the Normal distribution with mean 111 and
standard deviation as a description of the IQ test
score of all rural Midwest seventh-grade students.
Chapter 13
25
Figure 13.13 Stemplot of the IQ scores of 74 seventh-grade students.
• 13.7 Between what values do the IQ scores of
95% of all rural Midwest seventh-graders lie?
• 13.8 what percentage of all students have IQ
scores are less than 100?
• 13.9 what percentage of all students have IQ
scores 144 or higher? None of the 74 students
in our sample school had scores this high. Are
you surprised at this? Why?
Chapter 13
27
Exercise 13.14
Comparing IQ Scores. The Wechsler Adult
Intelligence Scale ( WAIS) is an IQ test. Scores on
the WAIS for the 20 to 34 age group are
approximately Normally distributed with mean
110 and standard deviation 15. Scores for the 60
to 64 age group are approximately Normally
distributed with mean 90 and standard deviation
15. Sarah, who is 30, scores 130 on the WAIS. Her
mother, who is 60, takes the test and scores 115.
Chapter 13
28
• (a) Express both scores as standard scores that
show where each woman stands within her own
age group.
• (b) Who scored higher relative to her age group,
Sarah or her mother? Who has the higher
absolute level of the variable measured by the
test?
Chapter 13
29
Exercise 13.21
NCAA rules for athletes. The NCAA requires
Division II athletes to get a combined score of at
least 820 on the Mathematics and Critical Reading
sections of the SAT exam in order to compete in
their first college year. In 2007, the combined
scores of the millions of college-bound seniors
taking the SATs were approximately Normal with
mean 1017 and standard deviation approximately
210. What percentage of all college-bound seniors
had scores less than 820?
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30
Exercise 13.22
More NCAA rules. For Division I athletes the NCAA
uses a sliding scale, based on both core GPA and
the combined Mathematics and Critical Reading
SAT score, to determine eligibility to compete in
the first year of college. For athletes with a core
GPA of 3.0, a score of at least 620 on the combined
Mathematics and Critical Reading sections of the
SAT exam is required. Use the information in the
previous exercise to find the percentage of all SAT
scores of college-bound seniors that are less than
620.
Chapter 13
31
Exercise 13.23
800 on the SAT. It is possible to score higher than
800 on the SAT, but scores above 800 are reported
as 800. ( That is, a student can get a reported score
of 800 without a perfect paper.) In 2007, the
scores of college-bound senior men on the SAT
Math test followed a Normal distribution with
mean 533 and standard deviation 116. What
percentage of scores were above 800 ( and so
reported as 800 ) ?
Chapter 13
32