Methods for Testing Normality of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 46

3.2.8 Coefficient of Variation
The coefficient of variation (CV), a well-known term in statistics, can be used
to quickly decide whether or not the time series data follow a normal distribution
curve by comparing the value of sample CV with 1. However, checking
normality based on the CV is somewhat a weak approach. This method is
valid only for some hydrologic and environmental applications if the data
represent a non-negative characteristic such as rainfall amount or pollutant
concentration (USEPA, 1992). If the value of CV is greater than 1, the data
should not be modelled with a normal distribution curve. However, the opposite
statement is not correct, i.e., we cannot conclude that the data can be modelled
with a normal distribution curve if the CV is less than 1 (USEPA, 1992). The
CV test is generally recommended to be used along with other statistical tests
or when the graphical representation of data indicates extreme departures
from normality.
3.2.9 Range Tests
The range tests are based on the fact that almost entire area of a normal
distribution curve lies within ±5 standard deviations from the mean. There are
two types of range tests: Studentized range test, and Geary's test. Both of
these tests use a ratio of an estimate of the sample range to the sample
standard deviation (Madansky, 1988; USEPA, 1996). A brief description about
these range tests is provided below.
3.2.9.1 Studentized Range Test
The Studentized range test uses the ratio of range of a sample to the sample
standard deviation. Tables of critical values for sample sizes up to 1000 are
available for checking whether the absolute value of this ratio is significantly
large (Madansky, 1988). The Studentized range test does not perform well if
the data points are not symmetric or if the tails of the data points are heavier
than that for the normal distribution. Also, this test may be sensitive to outlier
or extreme data points. Unfortunately, lognormally distributed data, which are
quite common in hydrological and environmental applications, have these
characteristics. If the data appear to be lognormally distributed, this test should
not be used (USEPA, 2006). In most cases, the Studentized range test performs
the same as the Shapiro-Wilk test but is much easier to apply.
3.2.9.2 Geary's Test
Test-statistic of the Geary's test is defined as the ratio of mean deviation of a
sample to the sample standard deviation. This ratio indicates whether time
series data follows a standard normal distribution or deviates from the normal
distribution (Madansky, 1988). This test is not as strong as the Shapiro-Wilk
test or the Studentized range test. However, since the Geary's test-statistic is
based on the normal probability distribution, critical values for all possible