Statistical Characteristics of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 15

aquifer or stream reach, or all streamflows over some time at a particular site.
All such data are seldom available to us. It may be impossible both physically
and economically to collect all data of interest (all the groundwater in an
aquifer over the study period). Alternatively, a subset of the entire data called
'sample' is selected and measured in such a way that conclusions about the
sample may be extended to the entire population. Statistical characteristics
computed from the sample are only inferences or estimates about characteristics
of the entire population, such as location or central tendency, spread or
dispersion, skewness and kurtosis. Measures of location are usually the sample
mean and sample median. Measures of spread include the sample standard
deviation and sample interquartile range. Use of the term 'sample' before each
statistic explicitly demonstrates that they only estimate the population value,
the population mean or median, etc. As sample estimates are far more common
than the measures based on the entire population, the term 'mean' used in this
topic should be interpreted as the 'sample mean', and similarly other statistics
should be interpreted.

2.1 Measures of Location

Out of six measures of location (mean, median, mode, geometric mean,
harmonic mean, and trimmed mean), the 'mean' and 'median' are two
commonly used measures of location.

2.1.1 Classical Measure: Arithmetic Mean

The arithmetic mean ( ¯ ) is calculated by summing up of all data values, x i
and dividing the sum by the sample size n :

n

x
n

Ç

i

¯ = (1)

i 1

For data which are in one of n groups, Eqn. (1) can be rewritten to show
that the overall mean ( ¯ ) depends on the mean for each group, weighted by
the number of observations n i in each group (Shahin et al., 1993; Helsel and
Hirsch, 2002):

n

n

Ç

i

¯ = x (2)

i

n

i 1

where x is the mean for i th group. The influence of any single observation x (j)
on the mean can be seen by placing all but that single observation in one
'group', or

( ) n 1

¯ = x x (3)

j j

n n