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title: "Other moments - measures of shape"
canonical: "https://modelassist.epixanalytics.com/space/EA/26574928/Other%20moments%20-%20measures%20of%20shape"
format: markdown
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The mean and variance are called the first moment about zero and the second moment about the mean respectively. The third and fourth moments about the mean, called skewness and kurtosis, are also occasionally used in risk analysis as numerical descriptions of shape.

### ***Skewness S***

The skewness statistic is calculated from the following formulae:

Discrete variable:                      > Macro (inline-media-image)



> Macro (mathinline)



Continuous variable:                  > Macro (inline-media-image)



> Macro (mathinline)




This is often called the *standardized skewness*, since it is divided by *s*3 to give a unitless statistic. The skewness statistic refers to the *lopsidedness* of the distribution (see left panel below). If a distribution has a negative skewness (sometimes described as left skewed) it has a longer tail to the left than to the right. A positively skewed distribution (right skewed) has a longer tail to the right, and zero skewed distributions are usually symmetric.

###  

![image](media://b7e2557b-8723-4ab5-b755-de47dbf9e0e6)

### ***Kurtosis K***

The kurtosis statistic is calculated from the following formulae:


Discrete variable:                      > Macro (inline-media-image)



> Macro (mathinline)



Continuous variable:                  > Macro (inline-media-image)



> Macro (mathinline)




This is often called the *standardized kurtosis*, since it is divided by *s*4, again to give a unitless statistic. The kurtosis statistic refers to the *peakedness* of the distribution (see right panel above) - the higher the kurtosis, the more peaked the distribution. A Normal distribution has a kurtosis of 3, so kurtosis values for a distribution are often compared to 3. For example, if a distribution has a kurtosis below 3 it is flatter than a Normal distribution.


The following table gives some examples of skewness and kurtosis for common distributions.


|  |  |  |
| --- | --- | --- |
| ***Distribution*** | ***Skewness*** | ***Kurtosis*** |
| Binomial | -∞ to +∞ | 1 to +∞ |
| Chisq | 0 to 2.828 | 3 to 15 |
| Exponential | 2 | 9 |
| Lognormal | 0 to +∞ | 3 to +∞ |
| Normal | 0 | 3 |
| Poisson | 0 to +∞ | 3 to +∞ |
| Triangular | -0.562 to 0.562 | 2.388 |
| Uniform | 0 | 1.8 |





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