Chapter 13
Normal Distributions
Chapter 13
4
Bell-Shaped Curve:
The Normal Distribution
of Population Values
Asymmetric Distributions
of the Population Values
1
Review
Parameters - fixed values, true for populations
population mean (µ) mu
population standard deviation (σ) sigma
Statistics - variables, calculated from samples
sample mean ( x )
x-bar
sample standard deviation (s)
!
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
Normal distn properties
•
•
•
•
•
Bell shaped
Symmetric about mu
Continuous
Area under curve = 1
Infinite number of normal curves, each
with its own mu and sigma
Standard normal distribution
has mean = 0 and sd=1
2
how to draw the normal curve
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”
–women
•mean: 65.0 inches
•standard deviation: 2.5 inches
• 68% of women are between 62.5” and 67.5”
• 95% of women are between 60” and 70”
• 99.7% of women are between 57.5” and 72.5”
3
Standardized Scores
• standardized score =
(observed value minus mean) / (std dev)
•
•
•
•
z is the standardized score
x is the observed value
µ is the population mean
σ is the population standard deviation
z=
x "µ
#
!
Table B: Percentiles of the
Standardized Normal Distribution
• See text for Table B (the “Standard Normal
Table”).
• 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).
Finding a Percentile from an observed value:
1. Find the standardized score
z=
x "µ
#
Don’t forget to keep the plus or minus sign.
2. Look up the percentile in Table B.
• Suppose your IQ score was 115.
!
• Standardized score = (115 – 100)/15 = +1
• Your IQ is 1 standard deviation above the mean.
• From Table B you would be at the 84th percentile.
• Your IQ would be higher than that of 84%
of the population.
4
Observed Value for a
Standardized Score
• observed value =
mean plus [(standardized score) × (std dev)]
•
•
•
•
!
x is the observed value
µ is the population mean
z is the standardized score
σ is the population standard deviation
x = µ + z"
Table B: Percentiles of the
Standardized Normal Distribution
• See Table B.
• 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.
Finding an Observed Value from a Percentile:
1. Look up the percentile in Table B and
find the corresponding standardized score.
2. Compute observed value
x = µ + z"
Tragically Low IQ
“Jury urges mercy for mother who killed baby. …
The mother had an IQ lower than 98 percent of the population.”
(Scotsman, March 8, 1994,p. 2)
!
• Mother was in the 2nd percentile.
• Table B gives her standardized score as approx –2.0
or 2 standard deviations below the mean of 100.
• Her IQ = 100 + (–2)(15) = 100 – 30 = 70
5
Calibrating Your GRE Score
GRE Exams have a mean
verbal score of 497 and a
standard deviation of 115.
(ETS, 1993)
Suppose your score was 650
and scores were bell-shaped.
• Standardized score z =
(650 – 497)/115 = +1.33.
• Table B, z = 1.33 is between
the 90th and 91st percentile.
• Your score was higher than
about 90% of the population.
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
6
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
Table B
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