departures from normality

Departures from Normality


• Many statistical test depend on our population being normally distributed.



• How do we test if our population is normally distributed?




• compare mean and median





• graphically





• graphically

• goodness of fit (Shapiro-Wilk Hypothesis test)





• graphically


• using symmetry and kurtosis hypothesis testing





• graphically



• What do we do if our data are not normally distributed, but are Abby Normal?





• graphically




• Transformations





• graphically




• Transformations

• Non-parametric tests (coming later)

Non-Normal Data

0

50

100

Cou

nt

0 1 2 3 4 5 6 7 8Tail Length (cm)

Skewed Right (Positively)

0

20

40

60

80

100

Cou

nt

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7Toe Length (cm)

Skewed Left (Negatively)

Skewness

Non-Normal Data

0

50

100

Cou

nt

0 1 2 3 4 5 6 7 8Tail Length (cm)

Skewed Right (Positively)

0

20

40

60

80

100

Cou

nt

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7Toe Length (cm)

Skewed Left (Negatively)

Skewness

Platykurtic(flaty)

Leptokurtic

Nor

mal

Qua

ntile

Plo

t

-3 -2 -1 0 1 2 3

-2.33

-1.64-1.28

-0.67

0.0

0.67

1.281.64

2.33

0.5

0.8

0.9

0.2

0.10.050.02

0.950.98

Nor

mal

Qua

ntile

Plo

t

0 10 20 30 40 50 60 70 80 90Kurtosis

Graphical Assessments of Normality

Histograms

Normal Probability Plot or

Cumulative Density Function

Graphical Tests of NormalityNormal Quantile Plot/Normal Probability Plot

Normal- Black dots follow red line(straight)

Negatively skewedblack dots concave up compared

to red line

Graphical Tests of Normality

Normal- Black dots follow red line

Positively skewedblack dots concave down

compared to red line

-3.09

-2.33

-1.64-1.28-0.67

0.00.671.281.64

2.33

3.09

0.5

0.8

0.2

0.05

0.01

0.95

0.99

0.0011e-4

Nor

mal

Qua

ntile

Plo

t

0 1 2 3 4 5 6 7 8

Normal Quantile Plot/Normal Probability Plot

Graphical Tests of Normality

Platykurtic-black dots form backwards S

Leptokurticblack dots form an S

Normal Quantile Plot/Normal Probability Plot

-2.33

-1.64-1.28

-0.67

0.0

0.67

1.281.64

2.33

0.5

0.8

0.9

0.2

0.10.050.02

0.950.98

Nor

mal

Qua

ntile

Plo

t

0 10 20 30 40 50 60 70 80 90

-2.33

-1.64-1.28

-0.67

0.0

0.67

1.281.64

2.33

0.5

0.8

0.9

0.2

0.10.050.02

0.950.98

Nor

mal

Qua

ntile

Plo

t

-3 -2 -1 0 1 2 3

Graphical Tests of NormalityCumulative Density Function (CDF)

Normal- symmetric tails

Skewedone tail longer than the other

Statistical Tests of NormalityOverlay a normal

distribution with the same mean and

variance

Statistical Tests of NormalityOverlay a normal

distribution with the same mean and

variance

Perform Goodness-of-Fit Test

Skewness and Kurtosis

Choose “Customize Summary Statistics”



Many/most software will subtract 3 from the

kurtosis value.



But, is this -3 or not?

Many/most software will subtract 3 from the

kurtosis value.


OK, now that we know that, we need to do a hypothesis test.




Hypothesis Tests

Now What?

Transform the Data

Thanks to Andy Rhyne

departures from normality

Documents