Skew and Kurtosis Skew and Kurtosis
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Skew and Kurtosis
Skew and Kurtosis
We have illustrated the central tendency of a distribution and the dispersion of scores within a distribution.
There are two other categories of statistics that help tell the story of a distribution:
• Skew
• Kurtosis
Skew and Kurtosis
Skew
Distributions (aggregations of observations) can be spread evenly around both sides of the central tendency, like so:
Skew and Kurtosis
Skew
Distributions (aggregations of observations) can be spread evenly around both sides of the central tendency, like so:
1 2 3 4 5 6 7 8 9
1
2
3
4
5
Skew and Kurtosis
Skew
Such distributions are considered symmetrical with no skew.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
Skew and Kurtosis
Skew
In symmetrical distributions the mean and the median are identical.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
5
5
Mean =
Median =
Skew and Kurtosis
Skew
As scores are weighted and distribute unevenly around the median, the mean is “pulled” toward the extreme outlier and it diverges away from the median.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
5
5
Mean =
Median =
Skew and Kurtosis
Skew
As scores are weighted and distribute unevenly around the median, the mean is “pulled” toward the extreme outlier and it diverges away from the median.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
75
Mean =
Median =
2120
Skew and Kurtosis
Skew
When the outlying scores are on the higher end of the scale the distribution becomes positively skewed.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
2120
+
Skew and Kurtosis
Skew
When the outlying scores are on the lower end of the scale the distribution becomes negatively skewed.
1 2 3 4 5 6 7 8 9
1
2
3
4
5
2120
_
Skew and Kurtosis
Skew Review
Skew and Kurtosis
Skew Review
Mean = 3
Median = 5
Negatively Skewed
Distribution
Mostly higher scores
Skew and Kurtosis
Skew Review
Mean = 5
Median = 5
Mean = 3
Median = 5
Normal Distribution
Negatively Skewed
Distribution
Mostly higher scores
Scores Evenly Distributed
Skew and Kurtosis
Skew Review
Mean = 5
Median = 5
Mean = 7
Median = 5
Mean = 3
Median = 5
Normal Distribution
Positively Skewed
Distribution
Negatively Skewed
Distribution
Mostly higher scores
Scores Evenly Distributed
Mostly lower scores
Skew and Kurtosis
Skew Review
A simple way to screen for the direction of the skew is to subtract the median from the mean.
Mean = 5
Median = 5
Mean = 7
Median = 5
Mean = 3
Median = 5
Normal Distribution
Positively Skewed
Distribution
Negatively Skewed
Distribution
Mostly higher scores
Scores Evenly Distributed
Mostly lower scores
Skew and Kurtosis
Skew Review
If mean – median results in a positive number then the skew is positive.
Mean = 5
Median = 5
Mean = 7
Median = 5
Mean = 3
Median = 5
Positively Skewed
Distribution
Negatively Skewed
Distribution
Mostly higher scores
Scores Evenly Distributed
Mostly lower scores
= +2
Normal Distribution
Skew and Kurtosis
Skew Review
If mean – median results in a negative number then the skew is negative.
Mean = 5
Median = 5
Mean = 7
Median = 5
Mean = 3
Median = 5
Positively Skewed
Distribution
Negatively Skewed
Distribution
Mostly higher scores
Scores Evenly Distributed
Mostly lower scores
= -2
Normal Distribution
Skew and Kurtosis
Skew Review
If mean – median results in zero then the skew is zero.
Mean = 5
Median = 5
Mean = 7
Median = 5
Mean = 3
Median = 5
Positively Skewed
Distribution
Negatively Skewed
Distribution
Mostly higher scores
Mostly lower scores
= 0
Normal Distribution
Scores Evenly Distributed
Skew and Kurtosis
Skew and the Mean
In practice almost no distributions are perfectly symmetrical.
Most are skewed to some degree.
At some point they become so extremely skewed that we have to make adjustments to the scales or use alternative inferential techniques in our analyses.
Skew and Kurtosis
Kurtosis
Kurtosis represents the attribute of flatness
Scores are fairly evenly spread out.
Skew and Kurtosis
Kurtosis
Kurtosis represents the attribute of flatness
Or peakedness of a distribution
Scores are fairly evenly spread out.
Scores cluster around a particular point
Skew and Kurtosis
Kurtosis
Distributions with the same range of scores and central tendency may nonetheless be shaped differently.
Skew and Kurtosis
Kurtosis
Distributions with the same range of scores and central tendency may nonetheless be shaped differently.
Skew and Kurtosis
Kurtosis
Distributions with the same range of scores and central tendency may nonetheless be shaped differently.
Skew and Kurtosis
Kurtosis
Distributions with the same range of scores and central tendency may nonetheless be shaped differently.
1 9
1 9
Mean = 5Median = 5 Mode = 5
Mean = 5Median = 5 Mode = 5
Skew and Kurtosis
Kurtosis
Distributions with the same range of scores and central tendency may nonetheless be shaped differently.
1 9
1 9
Same Range
Mean = 5Median = 5 Mode = 5
Mean = 5Median = 5 Mode = 5
Skew and Kurtosis
Kurtosis
Distributions with the same range of scores and central tendency may nonetheless be shaped differently.
1 9
1 9
Same Mean and Median and Mode
Mean = 5Median = 5 Mode = 5
Mean = 5Median = 5 Mode = 5
Skew and Kurtosis
Is your question dealing with SKEW?
or KURTOSIS?
Skew and Kurtosis
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