if m even 
if m odd 
if m even 
if m odd 
criterion compares a quadratic function of 
the sums of ranks R[ of the samples with 
chi-square distribution critical values, 
where the degrees of freedom are the 
number of samples minus one: 
C-l.###Kruskall-Wallis' test on the 
homogeneity of variance components for 
several variances of independent samples. 
This test must be used instead of classical 
Bartlett's 
test, when population distributions aren't 
normal, although samples must be 
independent. It is a generalisation of 
Siegel-Tukey's rank test for two variances 
under the same hypotheses. 
Therefore the ranks are assigned from the 
smallest absolute residual to the largest 
one, after having put all samples together 
and sorting their residuals, so that the 
dispersion of each appears immediately. 
The test criterion compares a quadratic 
function of the sums of ranks R t of the 
samples with chi-square distribution 
critical values, where the degrees of 
freedom are the number of samples minus 
one: 
H 0 : 
P(X 2 u=/n-i(a / 2) < H < x 2 u=m-i(1 - a / 2)) = 1 - a 
H 0 : 
P(X 2 u= m -i(a / 2) < H < x 2 v=m-1 (1 - a / 2)) = 1 - a 
where: 
D. Kruskall-Wallis' test or Friedman's test on 
the variance analysis of several means of 
independent or correlated samples, 
respectively. 
### These tests must be used instead of 
classical Fisher's test, when population 
distributions aren't normal or there are 
inequalities of dispersion of the samples, 
respectively, in case of independent 
samples or correlated ones. The first test 
generalises Mann-Withney's rank test for 
two means under the same hypotheses, 
whilst the second one generalises 
Thompson's sign test. 
The test procedure is the same explained 
for the homogeneity of variance 
components; the ranks are assigned to the 
arguments of the elements, obviously. 
where: 
12 
R? 
N(N+1) /=1 n, 
3(/V+1) 
N= In, 
/'=1 
C-2. Friedman's test on the homogeneity of 
variance components for several variances 
of correlated samples.### 
This test is less powerful than the above 
mentioned Kruskall-Wallis' test, but it 
permits to analyse correlated samples. This 
is a generalisation of Thompson's sign test 
for two variances under the same 
hypotheses. 
Therefore the ranks are assigned to the 
absolute residuals, element by element 
across the samples in increasing order, 
after having paired all samples. The test 
Aknoledgment 
The authors thank, Mr. Pasquale Pellicano 
(civil engineer student) and Mr Consolato 
Dattola (UTE Reggio Calabria), for their 
support during the measurements. 
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GPS Associates, 1986. 
T. Bellone, L. Mussio: Trattamento delle 
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G. Inghilleri: Topografia generale. UTET, 
Torino, 1974. 
F. Sansò: Il trattamento statistico dei dati. 
CLUP, Milano, 1990.