Statistical Characteristics of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 24

relative to the normal distribution. That is, datasets with high kurtosis tend to
have a distinct peak near the mean, decline rather rapidly, and have heavy
tails. For a time series data, x 1 , x 2 , …, x n , the formula for kurtosis can be
written as follows:

n

Ç

4

xx

i

i 1

J = (19)

4

ns 1

The kurtosis for a standard normal distribution is three, and hence some
authors provide the following definition of kurtosis, which is often referred to
as 'excess kurtosis' (Shahin et al., 1993):

n

Ç

4

xx

i

i 1

3

J = (20)

4

ns 1

A high kurtosis distribution has a sharper peak and longer, fatter tails ,
while a low kurtosis distribution has a more rounded peak and shorter, thinner
tails . Distributions with a zero excess kurtosis are called 'mesokurtic' or
'mesokurtotic'. On the other hand, distributions with a positive excess kurtosis
are called 'leptokurtic' or 'leptokurtotic'. In terms of shape, a leptokurtic
distribution has a more acute peak around the mean (i.e., a lower probability
than a normally distributed variable of values near the mean) and fatter tails
(i.e., a higher probability than a normally distributed variable of extreme
values). A distribution with a negative excess kurtosis is called 'platykurtic' or
'platykurtotic'. In terms of shape, a platykurtic distribution has a lower, wider
peak around the mean (i.e., a higher probability than a normally distributed
variable of values near the mean) and thinner tails (if viewed as the height of
the probability density—that is, a lower probability than a normally distributed
variable of extreme values).
Alternatively, peakedness of a data distribution may be described by
another measure known as 'percentile coefficient of kurtosis' (PCK). PCK is
mathematically expressed as follows (Shahin et al., 1993):

1
2

PP
PP

È Ø

75 25

PCK = É Ù (21)

Ê Ú

90 10

2.6 Statistical Measures for System Performance
Evaluation

Many theoretical and practical approaches have been proposed in the literature
for identifying and quantifying objectives and for considering multiple criteria/
objectives in water resources planning and management. Over the years, various