Download Chapter 6 The Normal Distribution

Chapter 6
The Normal Distribution
Chapter 6 The Normal Distribution
Major Points
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Distributions and area
The normal distribution
The standard normal distribution
Setting probable limits on an observation
Measures related to z
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Chapter 6 The Normal Distribution
Distributions and Area
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We can apply the concepts of
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Area
Percentages
Probability
Addition of areas
Addition of probabilities
When using the normal distribution
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Chapter 6 The Normal Distribution
Convert Histogram to
Frequency Polygon of Distribution:
connect the dots……..
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N=50
9
8
7
6
5
4
3
2
1
0
1 to 10
11 to 20
21 to 30
31 to 40
41 to 50
51 to 60
61 to 70
71 to 80
81 to 90
91 to 100
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Chapter 6 The Normal Distribution
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The Normal Distribution
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“Normal” refers to the general shape of
the distribution
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See next slide
Remember our descriptors for distributions
The formula for the normal distribution
allows us to
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Predict a proportion of cases that fall under
the normal curve between 2 given x values
Cont.
Chapter 6 The Normal Distribution
Normal Distribution--cont.
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X is the value on the abscissa
Y is the resulting height of the curve at X
There are constants also in the formula
Do you need to memorize or use the
formula? No! It’s just there to show you how
they arrive at the percentages discussed in a
minute.
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Chapter 6 The Normal Distribution
The Standard Normal
Distribution
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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, can look at the proportion of cases
falling above mean and below mean
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Chapter 6 The Normal Distribution
Ex. of a transformed
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
z Scores
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Calculation of z
Xμ
z
σ
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where  is the mean of the population and  is its
standard deviation
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This is a simple linear transformation of X.
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Chapter 6 The Normal Distribution
Tables of z
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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
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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
Chapter 6 The Normal Distribution
Using the Tables
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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
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See next slide
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Chapter 6 The Normal Distribution
Calculating areas
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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
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Chapter 6 The Normal Distribution
Converting Back to X
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Assume  = 30 and  = 5
X 
z

Therefore X    z  
X  30  1.0  5  25
X  30  1.5  5  37.5
So 77% of the distribution is expected to
lie between 25 and 37.5
Chapter 6 The Normal Distribution
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Probable Limits
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X=+z
Our last example has  = 30 and  = 5
We want to cut off 2.5% in each tail, so
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z = + 1.96 (found from a z table, mean to z of
1.96 = .475, but we usually round to z = 2).
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.
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We have just shown that 95% of the normal
distribution lies between 20.2 and 39.8, or +/1.96 SD’s
Therefore the probability is .95 that an
observation drawn at random will lie between
those two values
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Chapter 6 The Normal Distribution
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Statistical Methods:
The Normal
Distributions
Walsh & Betz (1995), p29
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Chapter 6 The Normal Distribution
Measures Related to z
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Standard score
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Another name for a z score, there are other standard
scores, such as …..
T scores: Scores with a mean of 50 and and
standard deviation of 10, no negative values
Others include IQ, SAT’s, etc
Percentile score/rank
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Percent of cases which fall at or below a given score
in the norming sample- THIS is how we typically find
a percentile rank, not calculating using that somewhat
confusing formula
Cont.
Chapter 6 The Normal Distribution
Related Measures--cont.
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Stanines
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Scores which are integers between 1 and 9, with
mean = 5 and standard deviation = 2
Mainly used in education
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Chapter 6 The Normal Distribution
Review Questions
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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.
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How do we convert X to z?
How do we use the tables of z?
If we know your test score, how do we
calculate your percentile?
What is a T score and why do we care?
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