Download Lecture 3 - notes - for Dr. Jason P. Turner

The Normal Distribution
MARE 250
Dr. Jason Turner
Define Normal
A variable is normally distributed if it is in the
shape of a normal curve (Bell-Shaped Curve)
Define Normal
Normal Curve Associated with Normal Distribution
is:
Bell Shaped
Centered at μ
Range is ±3 SD from the mean
So, am I Normal?
Standardized Normal Distribution – Mean 0, Std
Dev 1
Associated curve – Standard Normal Curve
You can standardize a variable by subtracting its
Mean and then dividing by its SD
Properties of Normality
Total Area under Standard Normal Curve (SNC) is 1
SNC extends indefinitely in both directions,
approaching, but not touching the horizontal axis
SNC is symmetric about 0; mirror image right/left
Most area under SNC lies ±3 SD
Properties of Normality
Properties of Normality
1. 68.26% of all possible observation lie w/in 1 SD
of the mean μ – σ and μ + σ
2. 95.44% of all possible observation lie w/in 2 SD
of the mean μ – 2σ and μ + 2σ
3. 99.74% of all possible observation lie w/in 3 SD
of the mean μ – 3σ and μ + 3σ
Assessing Normality
Large samples: Histogram can give a rough
estimate of Normality
Small sample: difficult to tell with histogram
need a more sensitive graphical technique
Histogram of Weight
Normal
35
Mean
StDev
N
30
Frequency
25
20
15
10
5
0
0
80
160
240
Weight
320
400
480
192.2
110.5
143
Assessing Normality
Normal Probability Plot: plot of the observed
values of the variable versus the Normal Scores
(observations expected for a normally dist. variable)
Probability Plot of Weight
Normal
99.9
Mean
StDev
N
RJ
P-Value
99
Percent
95
90
80
70
60
50
40
30
20
10
5
1
0.1
-200
-100
0
100
200
Weight
300
400
500
600
192.2
110.5
143
0.955
<0.010
Assessing Normality
A normal distribution should have highly sample
data which is highly correlated (1:1 ratio, linear
relationship) with normally distributed values
Probability Plot of Weight
Normal
99.9
Mean
StDev
N
RJ
P-Value
99
Percent
95
90
80
70
60
50
40
30
20
10
5
1
0.1
-200
-100
0
100
200
Weight
300
400
500
600
192.2
110.5
143
0.955
<0.010
Guidelines for Probability Plots
Decision of whether PP plot is linear is
subjective
Using a of sample observations to assess all
Guidelines for Probability Plots
Plot is roughly linear – accept as reasonable
that variable is approximately normally
distributed
Plot shows deviations from linear – conclude
variable probably not normally distributed
Testing for Normality
How do we test for normality?
Use Linear Correlation Coefficient (we will use
this later to compare variables…ask “Are They Related?”)
Compute the linear correlation coefficient
between the sample data and normal scores
Testing for Normality
Essentially tests to determine if our data (red
dots) are significantly correlated with
Normally distributed data (blue line)
Probability Plot of Weight
Normal
99.9
Mean
StDev
N
RJ
P-Value
99
Percent
95
90
80
70
60
50
40
30
20
10
5
1
0.1
-200
-100
0
100
200
Weight
300
400
500
600
192.2
110.5
143
0.955
<0.010
Normality Tests
Many Statistical Tests require normal data
You must verify normality with a test
Three primarily utilized include:
Anderson-Darling
Ryan-Joiner (Shapiro-Wilk)
Kolmogorov-Smirnov
Probability Plots - PP
Probability Plot of Weight
H0 hypothesis: data
normally distributed
Normal
99.9
Mean
StDev
N
RJ
P-Value
99
80
70
60
50
40
30
20
10
5
Histogram of Weight
1
Normal
0.1
-200
-100
0
100
200
Weight
300
400
500
35
600
Mean
StDev
N
30
25
If p value is less than α,
then reject H0
Data does not follow a
normal distribution
Frequency
Percent
95
90
192.2
110.5
143
0.955
<0.010
20
15
10
5
0
0
80
160
240
Weight
320
400
480
192.2
110.5
143
This is not a Test…
Hypothesis testing – used for making
decisions or judgments
Hypothesis – a statement that something is
true
Hypothesis test typically involves two hypoth:
Null and Alternative Hypotheses
Next Time
Hypothesis Testing
Assumptions Tests
Means Tests
Hypothesis Testing 101