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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