Statistical Analysis for Education and Psychology Researchers

Provided the pairs of sample observations are drawn at random from a population of
interest, Spearman’s rho, ρs (population rank order correlation), can be used to assess the
likelihood that two variables are associated in the population. The null hypothesis is, H 0 :
there is no association between the two variables and the alternative hypothesis, H 1 : is
that there is an association. This is a two-tailed alternative hypothesis. If we had specified
the nature of the relationship, i.e., positive or negative association, this would be a one-
tailed alternative hypothesis. In some introductory statistical texts, the null hypothesis is
specified as H 0 : ρs=0. Unlike the case with a parametric correlation, this does not
necessarily imply that the variables are independent. Only when values are normally
distributed does a correlation of 0 mean that variables are independent of one another.
The exact sampling distribution for Spearman’s rho for sample sizes1≤n≤ 10 has been
evaluated and is available in statistical tables, see for example, Kendall, (1948);
Documenta Geigy (1970); Zar (1972). There is no generally accepted procedure for
calculating confidence intervals for rs (sample rank order correlation) when sample sizes
are small, n<10.
When sample sizes are large, here, n≥ 10 rs approximates to that of Pearson’s product
moment correlation r (Kendall, 1948). Confidence intervals can therefore be constructed
by using a transformation of rs to z (Fisher’s z transform), which is approximately
normally distributed. Siegel and Castellan (1988) suggest sample size should be >20 for
the sampling distribution of rs to approximate to r. Use of confidence intervals for rs
when sample sizes are <20 are therefore of dubious value, recall as well that rs is most
likely to be used with small samples. If a 95 per cent confidence interval is calculated, we
would interpret it in the usual way, that is, we would be 95 per cent certain that the
obtained interval includes the true population value ρs. The confidence interval also
enables a test of the null hypothesis. If the obtained confidence interval excludes zero, we
can conclude that there is a significant correlation between the two variables. The
computational procedure for estimating confidence intervals for rs is identical to the
procedure for estimating the confidence interval for r and this is illustrated in Chapter 8.

Test Assumptions

Spearman’s rank order correlation is used when:

• observations do not come from a bivariate normal distribution;


• observations are ranked (given rank values);


• observations are ranked in two ordered series (one for each variable).

If observations represent a random sample from a specified population then rs can be
used to test whether there is a significant relationship between two variables in the
population.

Example from the Literature

In a study designed to investigate implementation of an integrated science curriculum,
Onocha and Okpala (1990) examined classroom interaction patterns of student and
practising teachers. They used a questionnaire to assess teachers’ reception of the science
curriculum and an observation schedule to identify teachers’ classroom interaction

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