MARE 250 Dr. Jason Turner The Normal Distribution
Dec 19, 2015
MARE 250Dr. Jason Turner
The Normal Distribution
Define NormalA 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 Normality1. 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 histogramneed a more sensitive graphical technique
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
Assessing Normality
Weight
Perc
ent
6005004003002001000-100-200
99.9
99
95
90
80706050403020
10
5
1
0.1
Mean
<0.010
192.2StDev 110.5N 143RJ 0.955P-Value
Probability Plot of WeightNormal
Probability Plots - PP
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 TestsMany Statistical Tests require normal data
You must verify normality with a test
Three primarily utilized include:Anderson-Darling
Ryan-Joiner (Shapiro-Wilk)
Kolmogorov-Smirnov
More powerful
Probability Plots - PP
Weight
Perc
ent
6005004003002001000-100-200
99.9
99
95
90
80706050403020
10
5
1
0.1
Mean
<0.010
192.2StDev 110.5N 143RJ 0.955P-Value
Probability Plot of WeightNormal
Weight
Frequency
480400320240160800
35
30
25
20
15
10
5
0
Mean 192.2StDev 110.5N 143
Histogram of WeightNormal
H0 hypothesis: data normally distributed
If p value is less than α, then reject H0
Data does not follow a normal distribution
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