Departures from Normality Departures from Normality Many - PowerPoint PPT Presentation

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 100 100 80 Count Count 60 50 40 20 0 0 0 1 2 3 4 5 6 7 8 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Tail Length (cm) Skewness Toe Length (cm) Skewed Right Skewed Left (Positively) (Negatively)

Non-Normal Data 100 100 80 Count Count 60 50 40 20 0 0 0 1 2 3 4 5 6 7 8 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Tail Length (cm) Skewness Toe Length (cm) Skewed Right Skewed Left (Positively) (Negatively) -2.33 -3 -2 -1 0 1 2 3 Kurtosis 0 10 20 30 40 50 60 70 80 90 Platykurtic Leptokurtic (flaty)

Graphical Assessments of Normality Histograms Normal Probability Plot or Cumulative Density Function

Graphical Tests of Normality Normal Quantile Plot/Normal Probability Plot Normal- Black dots follow red line(straight) Negatively skewed black dots concave up compared to red line

Graphical Tests of Normality Normal Quantile Plot/Normal Probability Plot Normal- Black dots follow red line Normal Quantile Plot 3.09 2.33 0.99 1.64 0.95 1.28 0.8 0.67 0.0 0.5 -0.67 0.2 -1.28 -1.64 0.05 -2.33 0.01 -3.09 0.001 1e-4 Positively skewed black dots concave down compared to red line 0 1 2 3 4 5 6 7 8

Graphical Tests of Normality Normal Quantile Plot/Normal Probability Plot 2.33 Normal Quantile Plot 0.98 Leptokurtic 1.64 0.95 1.28 0.9 black dots form an S 0.8 0.67 2.33 Normal Quantile Plot 0.0 0.5 0.98 1.64 0.95 -0.67 0.2 1.28 0.9 -1.28 0.1 0.8 0.67 -1.64 0.05 0.02 0.0 0.5 -2.33 -0.67 0.2 -1.28 0.1 -1.64 0.05 0.02 -2.33 -3 -2 -1 0 1 2 3 Platykurtic-black dots form backwards S 0 10 20 30 40 50 60 70 80 90

Graphical Tests of Normality Cumulative Density Function (CDF) Normal- symmetric tails Skewed one tail longer than the other

Statistical Tests of Normality Overlay a normal distribution with the same mean and variance

Statistical Tests of Normality Overlay a normal distribution with the same mean and variance

Statistical Tests of Normality Overlay a normal distribution with the same mean and variance Perform Goodness-of-Fit Test

Statistical Tests of Normality Overlay 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” Many/most software will subtract 3 from the kurtosis value. But, is this -3 or not?

Skewness and Kurtosis Choose “Customize Summary Statistics” OK, now that we know that, we need to do a hypothesis test.

Skewness and Kurtosis Hypothesis Tests Choose “Customize Summary Statistics”

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