Skewness

A distribution is said to be left-skewed if the mode is less than the mean and right skewed if it is greater than the mean.  Whilst not a law of nature, it often seems that the distribution of desirable outcomes (e.g. the size of mineral deposits) are left skewed.

Two populations can have the same values for mean and standard deviation, but have very different characteristics as shown in the diagram below:

Both the normal and lognormal distributions in the above graphic have a mean of 1.0 and standard deviation of 1.0, yet the lognormal is highly left skewed.

Three measures of skewness are:

Moment Skewness

This is based on the ratio of the third moment about the mean to the standard deviation:

This statistic is also known as the Fishers skewness coefficient.

The weakness of the above definition is its sensitivity to anomalous observations at the extremes where the difference between the mean and the value is cubed.

Mode Skewness

A more intuitive definition of skewness is based on the normalised difference between the Mean and the Mode of a distribution:

This statistic is also known as Pearson's Mode Skewness.

This version is less sensitive to anomalous extreme values.

Quartile Skewness

A definition of skewness which uses only quartiles and does incorporate any moment based properties is known as quartile skewness:

This definition is least sensitive to anomalous observations and is also known as the Bowley coefficient of skewness.

A comparison of the different measures of skewness for example, are shown in the table below:

When skewness is a positive value, the distribution is said to be left-skewed with modal value less than the mean, whilst if it is right skewed, skewness is negative and the modal values is greater than the mean.

Page updated: 22-Nov-2004

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