departures from normality

Departures from Normality

Departures from Normality

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

Departures from Normality

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

Departures from Normality

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

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

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

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

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

• goodness of fit (Shapiro-Wilk Hypothesis test)

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

• goodness of fit (Shapiro-Wilk Hypothesis test)

• using symmetry and kurtosis hypothesis testing

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

• goodness of fit (Shapiro-Wilk Hypothesis test)

• using symmetry and kurtosis hypothesis testing

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

• goodness of fit (Shapiro-Wilk Hypothesis test)

• using symmetry and kurtosis hypothesis testing

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

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

• goodness of fit (Shapiro-Wilk Hypothesis test)

• using symmetry and kurtosis hypothesis testing

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

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

• goodness of fit (Shapiro-Wilk Hypothesis test)

• using symmetry and kurtosis hypothesis testing

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

• Transformations

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

• goodness of fit (Shapiro-Wilk Hypothesis test)

• using symmetry and kurtosis hypothesis testing

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

• 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

Statistical Tests of NormalityOverlay a normal

distribution with the same mean and

variance

Perform Goodness-of-Fit Test

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”

Skewness and Kurtosis

Choose “Customize Summary Statistics”

Many/most software will subtract 3 from the

kurtosis value.

Skewness and Kurtosis

Choose “Customize Summary Statistics”

But, is this -3 or not?

Many/most software will subtract 3 from the

kurtosis value.

Skewness and Kurtosis

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

Choose “Customize Summary Statistics”

Skewness and Kurtosis

Choose “Customize Summary Statistics”

Hypothesis Tests

Skewness and Kurtosis

Choose “Customize Summary Statistics”

Skewness and Kurtosis

Choose “Customize Summary Statistics”

Now What?

Now What?

Now What?

Transform the Data

Thanks to Andy Rhyne