# Chapter 13

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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 bell-
shaped 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                 2
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                  3
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.

Chapter 13                   4
Bell-Shaped Curve:
The Normal Distribution
of Population Values

Chapter 13        5
Asymmetric Distributions
of the Population Values

Chapter 13         6
The Normal Distribution

standard deviation

mean

Chapter 13              7
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                    8
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               9
Empirical Rule for
Any Normal Curve

68%                         95%
-1sd   +1sd           -2 sd          +2 sd

99.7%
-3 sd                +3 sd

Chapter 13              10
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          11
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                12
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                             70       72.8         (height values)
deviations: -3        ?
-2     = 84%
-1      0        1           2      3
height values: 61.6   64.4    67.2     70      72.8        75.6   78.4

Chapter 13                             13
Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What    proportion of men are less than
68 inches tall?

?
standard                        68 70   (height values)
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                        14
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                  15
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
x-m
z
s
Chapter 13                 16
Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What proportion of men are less than
68 inches tall?

?
68 70    (height values)
-0.71   0   (standardized values)

Chapter 13                   17
Table B: Percentiles of the
Standardized Normal Distribution
 See  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).

Chapter 13               18
Table B
Standard                 Standard                Standard
Score     Percentile     Score     Percentile    Score     Percentile
–3.4       0.03          –1.1      13.57          1.2      88.49
–3.3       0.05          –1.0      15.87          1.3      90.32
–3.2       0.07          –0.9      18.41          1.4      91.92
–3.1       0.10          –0.8      21.19          1.5      93.32
–3.0       0.13          –0.7      24.20          1.6      94.52
–2.9       0.19          –0.6      27.42          1.7      95.54
–2.8       0.26          –0.5      30.85          1.8      96.41
–2.7       0.35          –0.4      34.46          1.9      97.13
–2.6       0.47          –0.3      38.21          2.0      97.73
–2.5       0.62          –0.2      42.07          2.1      98.21
–2.4       0.82          –0.1      46.02          2.2      98.61
–2.3       1.07           0.0      50.00          2.3      98.93
–2.2       1.39           0.1      53.98          2.4      99.18
–2.1       1.79           0.2      57.93          2.5      99.38
–2.0       2.27           0.3      61.79          2.6      99.53
–1.9       2.87           0.4      65.54          2.7      99.65
–1.8       3.59           0.5      69.15          2.8      99.74
–1.7       4.46           0.6      72.58          2.9      99.81
–1.6       5.48           0.7      75.80          3.0      99.87
–1.5       6.68           0.8      78.81          3.1      99.90
–1.4       8.08           0.9      81.59          3.2      99.93
–1.3       9.68           1.0      84.13          3.3      99.95
–1.2      11.51           1.1      86.43          3.4      99.97

Chapter 13                                       19
Table B: Percentiles of the
Standardized Normal Distribution
Standard Score      Percentile
(Z)

-0.8                21.19

-0.7                24.20

-0.6                27.42

Chapter 13           20
Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What proportion of men are less than
68 inches tall?

24.20%
68 70      (height values)
-0.71   0   (standardized values)

Chapter 13                    21
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                22
Table B: Percentiles of the
Standardized Normal Distribution
 SeeTable B (the “Standard Normal Table”) in
back of the text (or back of the supplement).
 Look   up the closest percentile in the table.
 Find   the corresponding standardized score.
 Thevalue you seek is that many standard
deviations from the mean.

Chapter 13               23
Table B: Percentiles of the
Standardized Normal Distribution
Standard Score      Percentile
(Z)

-1.4                8.08

-1.3                9.68

-1.2                11.51

Chapter 13           24
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)
-1.3   0   (standardized values)

Chapter 13                    25
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
 Thevalue 66.36 is approximately the
10th percentile of the population.

Chapter 13                   26
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                   27
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      (height values)
-0.71   0   (standardized values)

Chapter 13                    28
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.
 Itis 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               29
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                         30
Health and Nutrition Examination
Study of 1976-1980
(HANES)
 Shade   on the graph the range of
heights between 68 and 74 inches.

?

(height values)         68 70        74
(standardized values)   ?   0        ?

Chapter 13        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                  32
Health and Nutrition Examination
Study of 1976-1980
(HANES)
 What  proportion of men are less than 68
inches tall?

?%
(height values)        68 70          74
(standardized values) -0.71   0       ?

Chapter 13        33
Table B: Percentiles of the
Standardized Normal Distribution
Standard Score      Percentile
(Z)

-0.8                21.19

-0.7                24.20

-0.6                27.42

Chapter 13           34
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        74
(standardized values) -0.71   0     ?

Chapter 13         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        74
(standardized values) -0.71   0     ?

Chapter 13         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                  37
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        74
(standardized values) -0.71   0     1.43

Chapter 13          38
Table B: Percentiles of the
Standardized Normal Distribution
Standard Score     Percentile
(Z)

1.3                90.32

1.4                91.92

1.5                93.32

Chapter 13            39
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        74
(standardized values) -0.71     0       1.43

Chapter 13          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%

91.92% – 24.20%
24.20%             = 67.72%

(height values)            68 70            74
(standardized values) -0.71     0           1.43

Chapter 13              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                 42

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