Download Chapter 5 normal distribution

CHAPTER 5: THE
NORMAL
DISTRIBUTION
Leon-Guerrero and Frankfort-Nachmias,
Essentials of Statistics for a Diverse Society
Chapter 5: The Normal Distribution
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Properties of the Normal Distribution
Shapes of Normal Distributions
Standard (Z) Scores
The Standard Normal Distribution
Transforming Z Scores into Proportions
Transforming Proportions into Z Scores
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Normal Distributions

Normal Distribution – A bell-shaped and
symmetrical theoretical distribution, with the
mean, the median, and the mode all
coinciding at its peak and with frequencies
gradually decreasing at both ends of the
curve.
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The normal distribution is a theoretical ideal distribution. Real-life empirical
distributions never match this model perfectly. However, many things in life
do approximate the normal distribution, and are said to be “normally
distributed.”
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Shape of a Normal
Distribution: The Normal Curve
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Shape of a Normal
Distribution
Notice the shape of the
normal curve in this graph.
Some normal distributions
are tall and thin, while
others are short and wide.
All normal distributions,
though, are wider in the
middle and symmetrical.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Scores “Normally Distributed?”
Table 5.1 Final Grades in Social Statistics of 1,200 Students (1983-1993)
Midpoint
Cum. Freq.
Cum %
Score Frequency Bar Chart Freq.
(below)
%
(below)
40 *
4
4
0.33
0.33
50 *******
78
82
6.5
6.83
60 ***************
275
357
22.92
29.75
70 ***********************
483
840
40.25
70
80 ***************
274
1114
22.83
92.83
90 *******
81
1195
6.75
99.58
100 *
5
1200
0.42
100
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Is this distribution normal?
There are two things to initially examine: (1)
look at the shape illustrated by the bar chart,
and (2) calculate the mean, median, and mode.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Scores Normally Distributed!
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The Mean = 70.07
The Median = 70
The Mode = 70
Since all three are essentially equal, and this
is reflected in the bar graph, we can assume
that these data are normally distributed.
Also, since the median is approximately
equal to the mean, we know that the
distribution is symmetrical.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Different Shapes of the Normal
Distribution
Notice that the standard deviation changes the relative width
of the distribution; the larger the standard deviation, the wider
the curve.
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Areas Under the Normal Curve by
Measuring Standard Deviations
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Standard (Z) Scores
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A standard score (also called Z score) is the
number of standard deviations that a given
raw score is above or below the mean.
Y Y
Z
Sy
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Where Sy = standard deviation
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
The Standard Normal Table
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A table showing the area (as a proportion, which
can be translated into a percentage) under the
standard normal curve corresponding to any Z
score or its fraction
Area up to
a given score
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding the Area Between the
Mean and a Positive Z Score
Using the data presented in Table 5.1, find
the percentage of students whose scores
range from the mean (70.07) to 85.
(Standard deviation = 10.27)
(1) Convert 85 to a Z score:
 Z = (85-70.07)/10.27 = 1.45
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(2) Look up the Z score (1.45) in Column A (in
appendix B in the textbook), finding the
proportion (.4265)
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding the Area Between the Mean
and a Positive Z Score
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding the Area Between the
Mean and a Negative Z Score
Using the data presented in Table 5.1, find
the percentage of students scoring between
65 and the mean (70.07)
(1) Convert 65 to a Z score:
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Z = (65-70.07)/10.27 =
-.49
(2) Since the curve is symmetrical and
negative area does not exist, use .49 to find
the area in the standard normal table: .1879
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding the Area Between the Mean
and a Negative Z Score
(3) Convert the proportion (.1879) to a percentage (18.79%); this is the
percentage of students scoring between 65 and the mean (70.07)
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding Area Above a Positive Z
Score or Below a Negative Z Score
Find the percentage of students who did (a) very
well, scoring above 85, and (b) those students
who did poorly, scoring below 50.
(a) Convert 85 to a Z score, then look up the value
in Column C of the Standard Normal Table:
Z = (85-70.07)/10.27 = 1.45  7.35%
(b) Convert 50 to a Z score, then look up the
value (look for a positive Z score!) in Column C:
Z = (50-70.07)/10.27 = -1.95  2.56%
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding Area Above a Positive Z Score or
Below a Negative Z Score
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding a Z Score Bounding an Area
Above It
Find the raw score that bounds the top 10
percent of the distribution (Table 5.1)
(1) 10% = a proportion of .10
(2) Using the Standard Normal Table, look in
Column C for .1000, then take the value in
Column A; this is the Z score (1.28)
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83.22
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding a Z Score Bounding an Area
Above It
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding a Z Score Bounding an Area
Below It
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Find the raw score that bounds the lowest 5
percent of the distribution (Table 5.1)
(1) 5% = a proportion of .05
(2) Using the Standard Normal Table, look in
Column C for .05, then take the value in Column
A; this is the Z score (-1.65); negative, since it is
on the left side of the distribution
(3) Finally convert the Z score to a raw score:
Y=70.07 + -1.65 (10.27) = 53.12
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications
Finding a Z Score Bounding an
Area Below It
Leon-Guerrero/Frankfort-Nachmias: Essentials of Social Statistics for a Diverse Society
© 2012 SAGE Publications