Download Lecture 6 - 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)
Normal Curve Associated with a Normal Distribution is:
Bell Shaped
Centered at μ
Range is between +3 and -3 std dev 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 Std Dev
Properties of Normality
1. Total Area under Standard Normal Curve (SNC) is 1
2. SNC extends indefinitely in both directions,
approaching, but not touching the horizontal axis
3. SNC is symmetric about 0; mirror image right/left
4. Most area under SNC lies between -3 and 3 (std dev)
Properties of Normality
1. 68.26% of all possible observation lie w/in 1 std. dev. of the
mean μ – σ and μ + σ
2. 95.44% of all possible observation lie w/in 2 std. dev. of the
mean μ – 2σ and μ + 2σ
3. 99.74% of all possible observation lie w/in 3 std. dev. 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
Assessing Normality
Normal Probability Plot: plot of the observed
values of the variable versus the Normal Scores
(observations expected for a normally dist.
variable)
A normal distribution should have highly sample data
which is highly correlated (1:1 ratio, linear
relationship) with normally distributed values
Probability Plots - PP
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
When Using 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:
Compute the linear correlation coefficient
between the sample data and normal scores
Normality Tests
Many Statistical Tests require normal data
You must verify normality with a test
Three primarily utilized include:
Anderson-Darling
More powerful
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 hypothesis:
Null and Alternative Hypotheses
Hypothesis Testing 101