Statistical Characteristics of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 23

A positively skewed distribution of hydrologic time series with right
extended tail has a positive coefficient of skewness, whereas a time series
with negative-skewed distribution with left extended tail has a negative
coefficient of skewness. The presence of outliers has a significant influence
on the coefficient of skewness ( g ). For instance, an otherwise symmetric
distribution having one outlier will produce a large (and possibly misleading)
measure of skewness.

2.3.2 Robust Measure of Skewness
A robust measure of skewness is the 'quartile skew coefficient ( qs )', which is
defined as the difference in distances of the upper and lower quartiles from the
median, divided by the IQR (Kenney and Keeping, 1954). Mathematically, it
is expressed as:

PP PP
PP

75 50 50 25

qs = (17)

75 25

Similar to the coefficient of skewness, a right-skewed distribution has a
positive quartile skew coefficient and a left-skewed distribution has a negative
quartile skew coefficient. Also, similar to the trimmed mean and IQR, the
quartile skew coefficient uses the central 50% of the data.

2.4 Additional Robust Measures

As an additional robust measure, percentile values other than three quartiles
may be used to produce a series of robust measures of location, spread and
skewness. For example, the 15% trimmed mean can be coupled with the range
between the 15 th and 85 th percentiles as a measure of spread, and a
corresponding measure of skewness to produce a consistent series of robust
statistics. The robust measure of skewness for 15% trimmed mean ( qs 15 ) is
mathematically expressed as:

PP PP
PP

85 50 50 15

qs 15 = (18)

85 15

Geologists have used the 16 th and 84 th percentiles for many years to
compute a similar series of robust measures of the distributions of sediment
particles (Inman, 1952). However, the measures based on quartiles have become
generally standard, and additional measures should be clearly defined prior to
their use (Helsel and Hirsch, 2002). The median, IQR and quartile skew can
be easily summarized graphically using box and whisker plots (see Section
3.1.3 of Chapter 3), which are widely used by scientists and researchers.

2.5 Measures of Peakedness or Flatness

'Kurtosis' is a measure of peakedness or flatness of a data series distribution