166
m the number of rows;
n the number of columns;
di :the difference between the sum of column i and the mean of
these sums;
Tf S (t3 tj), where tj is the extent of a tie on row j.
The actual distribution of W is only known for small values of
m and n, but a good approximation is given by Fisher's z-distri-
bution. The following formules give the relation between W and z:
(m i) W
z i 1° i W
With Vj and v2 as degrees of freedom, z may be tested in a table
of the z-distribution, which can be found in [i] and [2]. As a level
of significance is chosen 0.05so from all cases where the hypothesis
is correct, we will reject it nevertheless 5 of the time.
The result of the test is that the hypothesis H0: no difference
between the colours, is rejected. The z-values found lie far in the
critical region, in fact is the probability of finding z 1465 with
H0 correct much smaller than 1 Thus the conclusion is that there
is difference between the colours; the next question is of course
which is the best. It can be shown that the colour belonging to the
column with the smallest sum of ranks, is the best, in a certain
sense connected with least squares. The colour giving the best
results is also the normal white one. A disadvantage of testing
with ranks as has been done here, is that it is not possible to say
how much better one colour is than another but that is scarcely
necessary for our purposes.
To confirm the conclusions, the same test may be performed on
the number of signals in classes a and b or a, b, c and d taken
together; as the classes a and b are rather empty in some places,
here the second possibility {a b c -f d) will be chosen. In this
case the highest number will have rank 1. This is done in tables 4a
and 4b separately for the two parts of the area as well as for the
different scales. In this way a possible although not probable dif
ference between scales may also be detected. In two parts of table
4b, viz. 1 and 2, a certain refinement has been applied, in order to
allow for the large number and extent of the ties.
i i 4 21 |a2? |i.2a
Let r,,„dMir)= („'I';, Sw),-
i
2
v2 (m i) Vj.
