Statistical Analysis for Education and Psychology Researchers

and the joint distribution of these variables should be normal (for all values of xi, the

conditional distribution of yi is normal and vice versa). This particular assumption is

required for hypothesis tests of statistical relationships to be valid.

Cautionary comments about use of r

These are not strictly assumptions but are typical research situations when Pearson’s r
either should be interpreted with great caution, or should not be used at all.

• When the variances of the two measures are very different, often associated with
  different ranges or possibly a restricted range for one variable, then the sample
  correlation is affected. For example, if one variable was to suffer from range
  restriction, (part of the score range not used or not appropriate) then this would tend to
  attenuate (lower) the correlation between the two variables. The reader is referred to
  an informative chapter on reliability and validity (Bartram, 1990) for discussion of this
  problem.


• When outliers are present, r should be interpreted with caution.


• When the relative precision of measurement scales/instruments differ. Measurement
  error will reduce the size of a correlation and r should therefore be interpreted with
  caution.


• When observations are taken from a heterogeneous group, for example, different
  subgroups in the sample such as age groups, then r should be interpreted with caution.


• When data is sparse (too few measures available), r should not be used. With too few
  values it is not possible to tell whether the bivariate relationship is linear. The Pearson
  correlation r is most appropriate for larger samples, (n>30).


• The correlation r should not be used when the values on one of the variables are fixed in
  advance.

Example from the Literature

In an empirical study to examine the relationships between adolescent attitudes towards
Christianity, interest in science and other socio-demographic factors, Fulljames, Gibson
and Francis (1991) used an attitude questionnaire to survey 729 secondary school pupils.
Attitude towards Christianity was measured by a 24-item scale with known reliability and
validity indices, index of interest in science was measured by a four-item scale, and
pupils’ mothers’ and fathers’ church attendance were each measured on a five-point scale
from ‘never’ to ‘weekly’.
The authors reported that the first stage of the analysis involved assessing the
psychometric [measurement] properties of the indices, (reliability coefficients and
Cronbach alphas were reported but data distributions were not), the second stage of
analysis involved inspection of bivariate relationships (correlations). Part of the
correlation matrix of the variables reported by the authors is shown in Table 8.3:

Table 8.3: Correlation matrix of the variables in

Fulljames’ et al., study of adolescent attitudes

(^) Attitude towards Fathers’ church Interest in
Statistical analysis for education and psychology researchers 282