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