Statistical Analysis for Education and Psychology Researchers

the distributions continuous, for example, normal, bivariate normal with inferences

based on the normal distribution?

These three issues will now be appraised in more detail.

Research Questions

Correlation, relationship and association

In many research studies more than one research question is addressed, consequently
more than one type of statistical test may be used. If the purpose of the study is to test for
a relationship between observations or scores then correlation type statistics are used.
Statistics about relationships are often called correlations. They are frequently used but
generally not well understood. The concept of significance in correlation is not very
helpful. Correlation represents a measure of the degree of closeness (co-relation) between
two variables. The correlation coefficient provides an indication of the strength of the
relationship. Even very weak correlations (small correlation coefficients) can be
statistically significant with large sample sizes,
When data is category ranked, Spearman’s Rho, rs is an appropriate correlation
statistic to use. When data is continuous and has an underlying normal distribution, the
Pearson correlation r should be used. In both cases, the null hypotheses are:

H 0 :ρ=0, that is the population correlation (ρ, rho) is zero.

The one sample χ^2 (Chi square) test of independence (Goodness-of-fit test) is often
considered as a correlation type statistic for nominal data. Goodness-of-fit refers to the
extent to which observed frequencies correspond to expected frequencies. This test
provides a measure of the degree of statistical independence of two variables, or put
simply a measure of the relationship between two variables when data is categorical (two
or more categories). This procedure is appropriate when one sample from a population
can be cross classified into two or more categories on two variables. The null hypothesis
is that the two variables are independent that is no relationship exists between them.
The r×2 sample χ^2 test of homogeneity is actually a test of the equality of the
distributions of two sets of proportions (the distribution of proportions in each population
for an example see Chapter 4 Table 4.1). It is appropriate when data is categorical. The
null hypothesis is that the distribution of proportions is the same in each population. A
two sample χ^2 refers to only two populations, the r refers to the number of categories or
rows in an r×2 contingency table, and the 2 refers to the two populations.

Dependence and prediction

When the research question of interest focuses on prediction then a regression type
analysis should be considered. The simplest form of statistical linear regression is when a
response variable, Y, is dependent on a predictor variable X and both Y and X are
continuous variables with a linear relationship. A regression equation can be used to
predict the dependence of Y on X. The line which best represents the linear relationship

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