Methods for Testing Normality of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 45

distribution and the ordered data points of a hydrologic time series. If the
normal probability plot is approximately linear (i.e., the data follow a normal
probability distribution), the test-statistic value will be relatively high. In
contrast, if the normal probability plot is nonlinear, the test-statistic value will
be relatively low. The S-W test has been recommended by the US
Environmental Protection Agency (USEPA, 2006) as well as in many statistical
texts (Gilbert, 1987) for testing normality in the environmental time series. In
recent years, the S-W test has become the preferred test of normality due to its
good power properties as compared to a wide range of alternative tests (Mendes
and Pala, 2003).
The test-statistic ( W ) of the S-W test examines whether a random sample,
x 1 , x 2 , ..., x n of a hydrologic variable comes from (specifically) a normal
distribution. This test-statistic is given as follows (Shapiro and Wilk, 1965):

2

n

È Ø

Ç

ax

É Ù

i )

Ê Ú i

n

1

W = (7)

Ç

2

xx

i

i 1

where x (i) = ordered (increasing ordered) sample values and a = constants
generated from the means, variances and covariances of the order statistics of
a sample of size n from a normal distribution (Pearson and Hartley, 1972).
Small values of the test-statistic, W indicate departure from normality.
Percentage points for the W , obtained by Monte Carlo simulations, are given
in Pearson and Hartley (1972). Since the computation of W is not much easier,
available statistical packages such as GraphPad Prism, Dataplot, DataQUEST
or STATISTICA can be used for analyzing the time series data by using
Shapiro-Wilk test.

3.2.7 Probability Plot Correlation Coefficient
Probability plot correlation coefficient (PPCC) test is considered as an extension
of the Shapiro-Wilk test. It is also known as 'Filliben's test-statistic' (Filliben,
1975). It measures the linearity of the data on a normal probability paper. Like
the S-W test, if the normal probability plot is approximately linear (i.e., the
hydrologic data follow a normal distribution curve), the correlation coefficient
value will be relatively high (USEPA, 1992). On the other hand, if the normal
probability plot contains several data points deviating from linearity (i.e., the
data do not follow a normal distribution curve), the correlation coefficient will
be relatively low. Although the Filliben's test-statistic is easier to compute
than the test-statistic of the S-W test, it is still difficult to compute by hand.
Therefore, statistical software like DataQUEST can be used to calculate the
Filliben's test-statistic.