RESEARCH METHOD COHEN ok

STATISTICAL SIGNIFICANCE 517
hands and the size of feet. The task is not to support
the hypothesis, i.e. the burden of responsibility is
not to support the null hypothesis. If we can
show that the hypothesis is not supported for 95
per cent or 99 per cent or 99.9 per cent of the
population, then we have demonstrated that there
is a statistically significant relationship between
the size of hands and the size of feet at the 0.05,
0.01 and 0.001 levels of significance respectively.
These three levels of significance – the 0.05,
0.01 and 0.001 levels – are the levels at which
statistical significance is frequently taken to have
been demonstrated, usually the first two of these
three levels. The researcher would say that the
null hypothesis (that there is no statistically
significant relationship between the two variables)
has not been supported and that the level of
significance observed (ρ) is at the 0.05, 0.01 or
0.001 level. Note here that we have used the
terms ‘statistically significant’, and not simply
‘significant’; this is important, for we are using
the term in a specialized way.
Let us take a second example. Let us say
that we have devised a scale of 1–8 which
can be used to measure the sizes of hands and
feet. Using the scale we make the following
calculations for eight people, and set out the results
thus (see http://www.routledge.com/textbooks/
9780415368780 – Chapter 24, file 24.9.ppt):
Hand size
Foot size
Subject A 1 1
Subject B 2 2
Subject C 3 3
Subject D 4 4
Subject E 5 5
Subject F 6 6
Subject G 7 7
Subject H 8 8
We can observe a perfect correlation between the
size of the hands and the size of feet, from the
person who has a size 1 hand and a size 1 foot
to the person who has a size 8 hand and also a
size 8 foot. There is a perfect positive correlation
(as one variable increases, e.g. hand size, so the
other variable – foot size – increases, and as one
variable decreases so does the other). We can
use the mathematical formula for calculating
the Spearman correlation (this is calculated
automatically in SPSS):
r = 1 − 6 ∑ d 2
N(N 2 − 1)
where d = the difference between each pair of
scores, = the sum of, and N = the size of
the population. We calculate that this perfect
correlation yields an index of association – a
coefficient of correlation – which is +1.00.
Suppose that this time we carry out the
investigation on a second group of eight people
and report the following results:
Hand size
Foot size
Subject A 1 8
Subject B 2 7
Subject C 3 6
Subject D 4 5
Subject E 5 4
Subject F 6 3
Subject G 7 2
Subject H 8 1
This time the person with a size 1 hand has a size 8
foot and the person with the size 8 hand has a size
1 foot. There is a perfect negative correlation
(as one variable increases, e.g. hand size, the
other variable – foot size – decreases, and as one
variable decreases, the other increases). Using the
same mathematical formula we calculate that this
perfect negative correlation yields an index of
association – a coefficient of correlation – which
is −1.00.
Now, clearly it is very rare to find a perfect positive
or a perfect negative correlation; the truth
of the matter is that looking for correlations will
yield coefficients of correlation which lie somewhere
between −1.00 and +1.00. How do we
know whether the coefficients of correlation are
statistically significant or not (See http://www.
routledge.com/textbooks/9780415368780 –
Chapter 24, file 24.10.ppt.)
Let us say that we take a third sample of eight
people and undertake an investigation into their
hand and foot size. We enter the data case by case
(Subject A to Subject H), indicating their rank
Chapter 24