 ## Principles for selecting a distribution

Usual practice in statistical analysis of hydrologic data is to fit several theoretical distributions to sample data and to make final choice based on several criteria.

• Visual inspection of probability plots

Probability plots are useful for several reasons. When sample data are plotted on probability paper for a particular distribution, then their approximate linear arrangement indicates that the distribution in question can be applicable for fitting the sample data.

A plot with both empirical and theoretical distribution can show whether the theoretical distribution is consistant with sample data. Some types of errors can also be easily spotted on probability plots. For example, erroneously calculated standard deviation will produce a line on normal probability paper with slope which differs from the slope of sample data.

• Goodness-of-fit tests

The purpose of these tests is to examine the differences between a proposed theoretical distribution and the empirical distribution of sample data, and to determine whether they are statistically significant or not.

When several distributions are fitted to sample data, it is usual that more than one is acceptable according to the goodness-of-fit tests, showing similar differences between theoretical and empirical distributions. In these cases other criteria for "best" distribution have to be used. Goodness-of-fit tests can sometimes be more valuable if they can prove that a proposed distribution should be rejected.

• Moment ratios and diagrams

Values of moment ratios (Cv and Cs) can be useful to rule out some theoretical distribution. For example, if the sample coefficient of skewness Cs differs significantly from zero, then there is no point in fitting data with normal distribution for which Cs = 0. Similarly, a sample Cs close to 1.14 is an argument in favour of the Gumbel distribution.

For many theoretical distributions there is a specific relationship between moment ratios Cv and Cs. Such relationships are usually plotted on a diagram with Cv on one axis and Cs / Cv on another. Plotting sample statistics on such a diagram allows making choice between alternative distributions.   