Statistical Characteristics of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 19

Fig. 2.3. Window diagram for the trimmed mean (Helsel and Hirsch, 2002).

2.2 Measures of Spread/Dispersion

It is very important to know the statistical dispersion or the variability of time
series data, which can be quantified by the measures of spread. Two widely
used measures of spread are described in subsequent sections.

2.2.1 Classical Measures

The 'sample variance' and 'sample standard deviation' (square root of sample
variance) are classical measures of spread (dispersion), which are the most
common measures of dispersion. Similar to the mean, the classical measures
of spread are strongly influenced by outlier values. The sample variance ( s 2 )
and the sample standard deviation ( s ) for a time series x 1 , x 2 , …, x n are
mathematically expressed as follows:

n

2

xx
n

Ç

i

s 2 =

(9)

1

i 1

n 2

xx
n

Ç

i

s = (10)

1

i 1

Both the classical measures ( s and s 2 ) of dispersion are computed using
the squares of deviations of data values from the mean of the time series, so
that magnitudes of the measures are even more influenced by outliers than
that for the mean. In presence of outliers in the time series, the classical
measures of dispersion become unstable and inflated. Under such condition,
the classical measures may indicate much greater spread than is indicated by
a majority of the hydrologic time series data.

2.2.2 Robust Measures

Robust measures of spreading about the mean include 'range', 'interquartile
range', 'coefficient of variation' and 'median absolute deviation'. As the value
of the range, standard deviation and coefficient of variation increases, the
population variability also increases (Helsel and Hirsch, 2002). The interquartile
range (IQR) is the most commonly used resistant measure of spread that
measures the range of the central 50% of the data in a time series, and is not