Therefore the classical significance tests 
should be used, when the, a priori, hypotheses 
on the behaviour of statistical populations are 
reasonably satisfied. On the contrary when 
these, a priori, hypotheses cannot be met, 
hence distribution-free significance tests must 
be used. 
There are several situations that require for 
their use: 
♦ ###departure from the normality of 
population distributions; 
♦ ###interdependence among the samples 
and/or within one or more samples; 
♦ ###inequality of dispersion (e.g., 
variances) of the samples; 
and, last but not least, samples too small, for 
the hypotheses on the behaviour of statistical 
populations could be verified. 
Note that robust statistics, useful when data are 
affected by outliers, belongs to non-parametric 
statistics, at least in terms of statistical 
inference, because the behaviour of population 
distributions could be modelled by 
approximate models only. 
Non-parramelric Robust Parametric 
statistics statistics statistics 
The role of non-parametric, robust and 
parametric statistics. 
B) DISTRIBUTION-FREE 
SIGNIFICANCE TESTS 
There exist many different significance 
distribution-free tests, as above mentioned. 
Nevertheless for the sake of brevity, only few 
multiple significance distribution-free tests are 
presented in the following, concerning: 
♦ ###goodness of fit of population 
distributions; 
♦ ###independence against interdependence 
or dependence (e.g., collinearity); 
♦ ###homogeneity of variance components 
for several variances; 
♦ ###variance analysis of several means; 
where the last two tests are repeated for 
independent and correlated samples, 
respectively. 
The following figures show the partition of a 
population distribution (the normal function, in 
the example) into three regions: the null 
hypothesis central zone and two critical sided 
zones, and the sketch of a power curve 
(according to the normal function, in the 
example). Note that these schemes are 
basically always the same, although tests and 
population distributions are different in many 
ways. In the following m samples whose sizes 
are ni or if all samples have the same size, 
are considered. 
Power curve according to the normal function.