Ahnboyoung: /* Skew */

2014-05-27T17:39:25Z

Skew

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{{Cat|Summarizing Distributions| 04}}

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

= Skew =
[[File:ClipCapIt-140527-183917.PNG]]

Distributions with positive skew have
*tails that extend to the right.
*(normally) larger means than medians.

==Example==
[[File:Central-tendency-compare2.jpg|400px]]

* 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:

[[File: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.

[[File: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.

[[File:Kurtosis formula.gif]]

=Quiz=
<quiz display=simple>

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

|type="{}"}
{ -3 }

{
{{Show Answer|
-3

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

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

|type="()"}
+Mean = 80, Median = 70, SD = 20
-Mean = 40, Median = 60, SD = 30
-Mean = 80, Median = 80, SD = 40

{
{{Show Answer|
Mean is 80, Median is 70, SD is 20

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

{ This distribution has

[[File:Pos skew.gif|200px]]

|type="()"}
-Negative skew
-No skew
+Positive skew

{
{{Show Answer|
Positive skew

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

{ This distribution has

[[File:Pos_kurt.gif|200px]]

|type="()"}
-Negative kurtosis
-No kurtosis
+Positive kurtosis

{
{{Show Answer|
Positive skew

This distributon has relatively long tails.
}}
}

</quiz>

Shapes of Distributions - Revision history

Ahnboyoung: /* Skew */