Other moments (measures of shape) | Vose Software

# Other moments (measures of shape)

The mean and variance are called the first raw 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.

They can also be applied when fitting a distribution to data through Method of Moments, if there are three or more parameters to estimate.

### Skewness S

The skewness statistic is calculated from the following formulae:

Discrete variable:

Continuous variable:

This is often called the standardised skewness, since it is divided by s3 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.

### Kurtosis K

The kurtosis statistic is calculated from the following formulae:

Discrete variable:

Continuous variable:

This is often called the standardised kurtosis, since it is divided by s4, 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 +∞ Triangle -0.562 to 0.562 2.388 Uniform 0 1.8