Methods for Testing Normality of Hydrologic Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 32

Thus, the best way to decide whether time series data are normal or not is to
apply both graphical and statistical methods for examining normality.
This chapter presents an overview of commonly used graphical and
statistical methods for checking the normality of time series data. The step-
by-step procedures for applying these methods are also included in this chapter.
This chapter can serve as a guideline for the researchers and scientists as well
as for practising engineers in selecting an appropriate normality test for their
time series data.

3.1 Graphical Methods

Graphical methods provide detailed information about a hydrologic time series
that may not be apparent from statistical methods. Histograms, stem-and-leaf
plots, and normal probability plots are some of the graphical methods which
are useful for determining whether or not a given set of time series data follow
a normal distribution curve (USEPA, 1996). Both the histogram and stem-
and-leaf plot of a normal distribution are bell-shaped. The normal probability
plot of a time series having normal distribution follows a straight line. However,
for the non-normally distributed data, there are large deviations from the
straight line in the tail or middle of a normal probability plot. The subsequent
section deals with six graphical methods used for checking normality of time
series data.

3.1.1 Frequency Plots/Histogram
Two classical methods for summarizing hydrologic time series data are
'frequency plot' (Fig. 3.1) and 'histogram' (Fig. 3.2). The basic principles
used by both the histogram and the frequency plot to display the data are
almost same: dividing the data range into units/bins, counting the number of
data points within the units/bins, and displaying the data as the height or area
within a bar graph (Walpole and Myers, 1985). Besides similarity, both the
histogram and the frequency plot slightly differ from each other. The frequency
plot represents the relative density of the data points by the relative height of
the bars, while in a histogram, the area within the bar represents the relative
density of the data points. A more distinct difference between the two plots
can be seen by using unequal box sizes (USEPA, 1996). Structure/patterns of
the histogram and frequency plot reveal about the symmetry and variability of
the data. If the data are symmetric, then the structure of these plots will be
symmetric around a central point such as mean or median. The histogram and
frequency plots will generally indicate if the data are skewed and the direction
of the skewness (Dixon and Massey Jr., 1983). Step-by-step procedures for
generating a frequency plot and histogram are given below:
Step 1:

Select suitable data intervals that cover the range of entire observations
of time series. The data intervals should be of equal widths, if possible.
A rule of thumb is to have between 7 to 11 intervals (USEPA, 1996).