RESEARCH METHOD COHEN ok

528 QUANTITATIVE DATA ANALYSIS
has lost its freedom. In our example then df = 4, that
is N − 1 = 5 − 1 = 4.
Suppose now that we are told to select any five
numbers, the first two of which have to total 9, and
the total value of all five has to be 25. One restriction
is apparent when we wish the total of the first two
numbers to be 9. Another restriction is apparent in
the requirement that all five numbers must total 25.
In other words we have lost two degrees of freedom
in our example. It leaves us with df = 3, that is,
N − 2 = 5 − 2 = 3.
(Cohen and Holliday 1996: 113)
For a cross-tabulation (a contingency table),
degrees of freedom refer to the freedom with
which the researcher is able to assign values to
the cells, given fixed marginal totals, usually given
as (number of rows − 1) + (number of columns −
1). There are many variants of this, and readers will
need to consult more detailed texts to explore this
issue. We do not dwell on degrees of freedom here,
as it is automatically calculated and addressed in
subsequent calculations by most statistical software
packages such as SPSS.
Measuring association
Much educational research is concerned with
establishing interrelationships among variables.
We may wish to know, for example, how
delinquency is related to social class background;
whether an association exists between the
number of years spent in full-time education
and subsequent annual income; whether there
is a link between personality and achievement.
What, for example, is the relationship, if any,
between membership of a public library and social
class status Is there a relationship between social
class background and placement in different strata
of the secondary school curriculum Is there a
relationship between gender and success or failure
in ‘first time’ driving test results
There are several simple measures of association
readily available to the researcher to help her test
these sorts of relationships. We have selected the
most widely used ones here and set them out in
Box 24.23.
Of these, the two most commonly used
correlations are the Spearman rank order
correlation for ordinal data and the Pearson
product-moment correlation for interval and ratio
data. At this point it is pertinent to say a few
words about some of the terms used in Box 24.23
to describe the nature of variables. Cohen and
Holliday (1982; 1996) provide worked examples
of the appropriate use and limitations of the
correlational techniques outlined in Box 24.23,
together with other measures of association such
as Kruskal’s gamma, Somer’s d, and Guttman’s
lambda (see http://www.routledge.com/textbooks/
9780415368780 – Chapter 24, file 24.13.ppt and
SPSS Manual 24.6).
Look at the words used at the top of Box 24.23 to
explain the nature of variables in connection with
the measure called the Pearson product moment,
r. Thevariables,welearn,are‘continuous’andat
the ‘interval’ or the ‘ratio’ scale of measurement.
Acontinuousvariableisonethat,theoretically
at least, can take any value between two points
on a scale. Weight, for example, is a continuous
variable; so too is time, so also is height. Weight,
time and height can take on any number of
possible values between nought and infinity, the
feasibility of measuring them across such a range
being limited only by the variability of suitable
measuring instruments.
Turning again to Box 24.23, we read in
connection with the second measure shown there
(rank order or Kendall’s tau) that the two
continuous variables are at the ordinal scale of
measurement.
The variables involved in connection with
the phi coefficient measure of association
(halfway down Box 24.23) are described as
‘true dichotomies’ and at the nominal scale of
measurement. Truly dichotomous variables (such
as sex or driving test result) can take only two
values (male or female; pass or fail).
To conclude our explanation of terminology,
readers should note the use of the term ‘discrete
variable’ in the description of the third correlation
ratio (eta) in Box 24.23. We said earlier that a
continuous variable can take on any value between
two points on a scale. A discrete variable, however,