RESEARCH METHOD COHEN ok

526 QUANTITATIVE DATA ANALYSIS
Box 24.21
A2× 3contingencytableforchi-square
Science Arts Humanities
subjects subjects subjects
7.6 8 8.4
Males 14 4 6 24
11.4 12 12.6
Females 5 16 15 36
19 20 21 60
students opting for each of the particular subjects.
The figure is arrived at by statistical computation,
hence the decimal fractions for the figures. What
is of interest to the researcher is whether the actual
distribution of subject choice by males and females
differs significantly from that which could occur
by chance variation in the population of college
entrants (Box 24.21).
The researcher begins with the null hypothesis
that there is no statistically significant difference
between the actual results noted and what might
be expected to occur by chance in the wider population.
When the chi-square statistic is calculated,
if the observed, actual distribution differs from that
which might be expected to occur by chance alone,
then the researcher has to determine whether that
difference is statistically significant, i.e. not to
support the null hypothesis.
In our example of sixty students’ choices, the
chi-square formula yields a final chi-square value
of 14.64. This we refer to the tables of the
distribution of chi-square (see Table 4 in The
Appendices of Statistical Tables) to determine
whether the derived chi-square value indicates a
statistically significant difference from that occurring
by chance. Part of the chi-square distribution
table is shown here.
Degrees of Level of significance
freedom 0.05 0.01
3 7.81 11.34
4 9.49 13.28
5 11.07 15.09
6 12.59 16.81
The researcher will see that the ‘degrees of
freedom’ (a mathematical construct that is related
to the number of restrictions that have been
placed on the data) have to be identified. In
many cases, to establish the degrees of freedom,
one simply takes 1 away from the total number
of rows of the contingency table and 1 away from
the total number of columns and adds them; in
this case it is (2 − 1) + (3 − 1) = 3degreesof
freedom. Degrees of freedom are discussed in the
next section. (Other formulae for ascertaining
degrees of freedom hold that the number is
the total number of cells minus one – this is
the method set out later in this chapter.) The
researcher looks along the table from the entry
for the three degrees of freedom and notes that
the derived chi-square value calculated (14.64)
is statistically significant at the 0.01 level, i.e.
is higher than the required 11.34, indicating
that the results obtained – the distributions of
the actual data – could not have occurred simply
by chance. The null hypothesis is not supported
at the 0.01 level of significance. Interpreting
the specific numbers of the contingency table
(Box 24.21) in educational rather than statistical
terms, noting the low incidence of females in the
science subjects and the high incidence of females
in the arts and humanities subjects, and the high
incidence of males in the science subjects and the
low incidence of males in the arts and humanities,
the researcher would say that this distribution is
statistically significant – suggesting, perhaps, that
the college needs to consider action possibly to
encourage females into science subjects and males
into arts and humanities.
The chi-square test is one of the most widely
used tests, and is applicable to nominal data in
particular. More powerful tests are available for
ordinal, interval and ratio data, and we discuss
these separately. However, one has to be cautious
of the limitations of the chi-square test. Look at
the example in Box 24.22.
If one were to perform the chi-square test on
this table then one would have to be extremely
cautious. The chi-square statistic assumes that no
more than 20 per cent of the total number of
cells contain fewer than five cases. In the example
here we have one cell with four cases, another
with three, and another with only one case, i.e.