Shapes of Distributions

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Shapes of distributions can differ in skew and/or kurtosis.

Skew

ClipCapIt-140527-183917.PNG

Distributions with positive skew have

  • tails that extend to the right.
  • (normally) larger means than medians.

Example

Central-tendency-compare2.jpg

  • Histogram above shows the salaries of major league baseball players (in tens of thousands of dollars).
  • It shows a distribution with a very large positive skew.
  • The mean and median of the baseball salaries are $1,183,417 and $500,000 respectively.
  • Thus, for this highly-skewed distribution, the mean is more than twice as high as the median.

Measures of skew

Pearson's measure of skew

The relationship between skew and the relative size of the mean and median led the statistician Pearson to propose the following simple and convenient numerical index of skew:

Pearson skew.gif

  • The standard deviation of the baseball salaries is 1,390,922.
  • Therefore, Pearson's measure of skew for this distribution is 3(1,183,417 - 500,000)/1,390,922 = 1.47.


Third moment about the mean

  • Although Pearson's measure is a good one, the following measure is more commonly used.
  • It is sometimes referred to as the third moment about the mean.

Skew-third-moment-about-the-mean.gif

Kurtosis

The following measure of kurtosis is similar to the definition of skew. The value "3" is subtracted to define "no kurtosis" as the kurtosis of a normal distribution. Otherwise, a normal distribution would have a kurtosis of 3.

Kurtosis formula.gif


Quiz

1 What would be Pearson's measure of skew for the following distribution: Mean = 50, Median = 60, and Variance = 100?

Answer >>

-3

3(50-60)/sqrt(100) is -3


2 Which of the following distributions would have a positive skew?

Mean = 80, Median = 70, SD = 20
Mean = 40, Median = 60, SD = 30
Mean = 80, Median = 80, SD = 40

Answer >>

Mean is 80, Median is 70, SD is 20

When the mean is larger than the median, the distribution has a positive skew.


3 This distribution has

Pos skew.gif

Negative skew
No skew
Positive skew

Answer >>

Positive skew

The tail is on the right so the skew is positive. Also called skewed to the right.


4 This distribution has

Pos kurt.gif

Negative kurtosis
No kurtosis
Positive kurtosis

Answer >>

Positive skew

This distributon has relatively long tails.