Download The Normal Distribution

Chapter 6
The Normal Distribution
Fundamental Statistics for the
Behavioral Sciences, 5th edition
David C. Howell
©2003 Brooks/Cole Publishing Company/ITP
Chapter 6 The Normal Distribution
Major Points
• Distributions and area
• The normal distribution
• The standard normal distribution
• Setting probable limits on an observation
• Measures related to z
2
Chapter 6 The Normal Distribution
Distributions and Area
• The idea of a pie chart as representing
area
 See next slide
• Bar charts say the same thing
 See subsequent slide
3
Chapter 6 The Normal Distribution
Where are the Bad Guys?
4
5
Chapter 6 The Normal Distribution
Bar Chart of Bad Guys
Chart
75
P
e
r
c
e
n
t
a
g
e
50
25
0
Prison
Jail
Parole
Probation
6
Chapter 6 The Normal Distribution
Getting Closer
40
30
20
10
Std. Dev = 10.56
Mean = 49.1
N = 289.00
0
.5
81
.3
77
.1
73
.9
68
.7
64
.5
60
.3
56
.1
52
.9
47
.7
43
.5
39
.3
35
.1
31
.9
26
.7
22
.5
18
Behavior Problem Score
7
Chapter 6 The Normal Distribution
The Normal Distribution
• The general shape of the distribution
 See slide 9
• The formula for the normal distribution.
1
f (X ) 
e
 2
( X   ) 2
2 2
Cont.
Chapter 6 The Normal Distribution
Normal Distribution--cont.
• X is the value on the abscissa
• Y is the resulting height of the curve at X
•  and e are constants
8
9
Chapter 6 The Normal Distribution
The Distribution
1200
1000
800
600
400
200
Std. Dev = 1.00
Mean = -.01
N = 10000.00
0
50
3.
00
3.
50
2.
00
2.
50
1.
00
1.
0
.5
00
0.
0
-.5 0
.0
-1
0
.5
-1
0
.0
-2
0
.5
-2
0
.0
-3
0
.5
-3
0
.0
-4
X
Chapter 6 The Normal Distribution
The Standard Normal
Distribution
• We simply transform all X values to have
a mean = 0 and a standard deviation = 1
• Call these new values z
• Define the area under the curve to be 1.0
10
11
Chapter 6 The Normal Distribution
z Scores
• Calculation of z
Xμ
z
σ
 where  is the mean of the population and 
is its standard deviation
 This is a simple linear transformation of X.
12
Chapter 6 The Normal Distribution
Tables of z
• We use tables to find areas under the
distribution
• A sample table is on the next slide
• The following slide illustrates areas under
the distribution
13
Chapter 6 The Normal Distribution
z Table
Mean to
z
Larger
Portion
Smaller
Portion
0.000
.0000
.5000
.5000
0.100
.0398
.5398
.4602
0.200
.0793
.5793
.4207
1.000
.3413
.8413
.1587
1.500
.4332
.9332
.0668
1.645
.4500
.9500
.0500
1.960
.4750
.9750
.0250
z
14
Chapter 6 The Normal Distribution
Normal Distribution
Cutoff at +1.645
1200
1000
z=
1.64545
45
800
Area =
.05 .05
600
400
200
0
50
3.
00
3.
50
2.
00
2.
50
1.
00
1.
0
.5
00
0.
0
-.5
0
.0
-1
0
.5
-1
0
.0
-2
0
.5
-2
0
.0
-3
0
.5
-3
0
.0
-4
z
Chapter 6 The Normal Distribution
Using the Tables
• Define “larger” versus “smaller” portion
• Distribution is symmetrical, so we don’t
need negative values of z
• Areas between z = +1.5 and z = -1.0
 See next slide
15
16
Chapter 6 The Normal Distribution
Calculating areas
• Area between mean and +1.5 = 0.4332
• Area between mean and -1.0 = 0.3413
• Sum equals
0.7745
• Therefore about 77% of the observations
would be expected to fall between z =
-1.0 and z = +1.5
17
Chapter 6 The Normal Distribution
Converting Back to X
• Assume  = 30 and  = 5
• 77% of the distribution is expected to lie
between 25 and 37.5
z
X 

Therefore X    z  
X  30  1.0  5  25
X  30  1.5  5  37.5
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Chapter 6 The Normal Distribution
Probable Limits
• X=+z
• Our last example has  = 30 and  = 5
• We want to cut off 2.5% in each tail, so
 z = + 1.96
X    z 
X  30  1.96  5  39.8
X  30  1.96  5  20.2
Cont.
Chapter 6 The Normal Distribution
Probable Limits--cont.
• We have just shown that 95% of the
normal distribution lies between 20.2 and
39.8
• Therefore the probability is .95 that an
observation drawn at random will lie
between those two values
19
20
Chapter 6 The Normal Distribution
Measures Related to z
• Standard score
 Another name for a z score
• Percentile score
 The point below which a specified percentage of the
observations fall
• T scores
 Scores with a mean of 50 and a standard deviation of
10
Cont.
Chapter 6 The Normal Distribution
21
Review Questions
• Why do you suppose we call it the
“normal” distribution?
• What do we gain by knowing that
something is normally distributed?
• How is a “standard” normal distribution
different?
Cont.
Chapter 6 The Normal Distribution
Review Questions--cont.
• How do we convert X to z?
• How do we use the tables of z?
• Of what use are probable limits?
• If we know your test score, how do we
calculate your percentile?
• What is a T score and why do we care?
22