RESEARCH METHOD COHEN ok

518 QUANTITATIVE DATA ANALYSIS
order for hand size and then for foot size. This
time the relationship is less clear because the rank
ordering is more mixed, for example, Subject A
has a hand size of 2 and 1 for foot size, Subject B
has a hand size of 1 and a foot size of 2 etc.:
Hand size
Foot size
Subject A 2 1
Subject B 1 2
Subject C 3 3
Subject D 5 4
Subject E 4 5
Subject F 7 6
Subject G 6 7
Subject H 8 8
Using the mathematical formula for calculating
the correlation statistic, we find that the coefficient
of correlation for the eight people is 0.7857. Is it
statistically significant From a table of significance
(Tables 2 and 3 in the Appendices of Statistical
Tables), we read off whether the coefficient is
statistically significant or not for a specific number
of cases, for example:
Number of cases
Level of significance
0.05 0.01
6 0.93 0.96
7 0.825 0.92
8 0.78 0.875
9 0.71 0.83
10 0.65 0.795
20 0.455 0.595
30 0.36 0.47
We see that for eight cases in an investigation the
correlation coefficient has to be 0.78 or higher, if
it is to be significant at the 0.05 level, and 0.875
or higher, if it is to be significant at the 0.01 level
of significance. As the correlation coefficient in
the example of the third experiment with eight
subjects is 0.7857 we can see that it is higher
than that required for significance at the 0.05
level (0.78) but not as high as that required for
significance at the 0.01 level (0.875). We are safe,
then, in stating that the degree of association
between the hand and foot sizes does not support
the null hypothesis and demonstrates statistical
significance at the 0.05 level.
The first example above of hands and feet is
very neat because it has 100 people in the sample.
If we have more or less than 100 people how do
we know if a relationship between two factors is
statistically significant Let us say that we have
data on 30 people; in this case, because sample size
is so small, we might hesitate to say that there is
astrongassociationbetweenthesizeofhandsand
size of feet if we observe it occurring in 27 people
(i.e. 90 per cent of the population). On the other
hand, let us say that we have a sample of 1,000
people and we observe the association in 700 of
them. In this case, even though only 70 per cent
of the sample demonstrate the association of hand
and foot size, we might say that because the sample
size is so large we can have greater confidence in
the data than in the case of the small sample.
Statistical significance varies according to the
size of the number in the sample (as can be seen
also in the section of the table of significance
reproduced above) (see http://www.routledge.
com/textbooks/9780415368780 – Chapter 24, file
24.11.ppt). In order to be able to determine significance
we need to have two facts in our possession:
the size of the sample and, in correlational research,
the coefficient of correlation or, in other
kinds of research, the appropriate coefficients or
data (there are many kinds, depending on the
test being used). Here, as the selection from the
table of significance reproduced above shows, the
coefficient of correlation can decrease and still
be statistically significant as long as the sample
size increases. (This resonates with Krejcie’s and
Morgan’s (1970) principles for sampling, observed
in Chapter 4, namely as the population increases
the sample size increases at a diminishing rate in
addressing randomness.) This is a major source of
debate for critics of statistical significance, who
argue that it is almost impossible not to find statistical
significance when dealing with large samples,
as the coefficients can be very low and still attain
statistical significance.
To ascertain statistical significance from a table,
then, is a matter of reading off the significance
level from a table of significance according to
the sample size, or processing data on a computer
program to yield the appropriate statistic. In the