L-moments: definitions

L-moments for data samples

The first L-moment is the sample mean, a measure of location. The second L-moment is (a multiple of) Gini's mean difference statistic, a measure of the dispersion of the data values about their mean.

By dividing the higher-order L-moments by the dispersion measure, we obtain the L-moment ratios,

These are dimensionless quantities, independent of the units of measurement of the data. t₃ is a measure of skewness and t₄ is a measure of kurtosis -- these are respectively the L-skewness and L-kurtosis. They take values between -1 and +1 (exception: some even-order L-moment ratios computed from very small samples can be less than -1).

The L-moment analogue of the coefficient of variation (standard deviation divided by the mean), is the L-CV, defined by

It takes values between 0 and 1.

L-moments for probability distributions

L-moments are defined in terms of probability weighted moments, analogously to the sample L-moments:

L-moment ratios are defined by

Examples:

• Uniform (rectangular) distribution on (0,1):
  
  
• Normal distribution with mean 0 and variance 1:
  
  
• exponential distribution with mean 1:
  
  

[ IBM home page | Order | Search | Contact IBM | Help | (C) | (TM) ]