# 72 Correlations

Correlation coefficients are used to measure the strength of the relationship between numeric variables. The most common correlation coefficients are Pearson’s r and spearman’s rho (rho (ρ), or rs.) which both can range from -1 to +1. If the coefficient is between 0 and 1, then as one variable increases, the other also increases (positive correlation). If the correlation coefficient is between -1 and 0, as one variable increases the other decreases (negative correlation). Note that unlike the Pearson correlation coefficient, the Spearman correlation does not require continuous-level data (interval or ratio), because it uses ranks instead of assumptions about the distributions of the two variables.  This allows us to analyze the association between variables of ordinal measurement levels. A Spearman correlation analysis can therefore be used in many cases in which the assumptions of the Pearson correlation (continuous-level variables, linearity, heteroscedasticity, and normality) are not met. In your papers, correlations can be presented in two ways:

• The descriptive statistics are presented for all the variables (refer to the Table 10.4.2.6)
• A correlation matrix is produced. Typically, a correlation matrix is “square”, with the same variables shown in the rows and columns (see Table 10.6.1)

Table 10.6 - Sample Correlation Matrix
Always studies with partners Always participates in class Always completes assignments Always visits office hours
Always study with partners 1 0.87 -0.65 0.48
Always participates in class 0.87 1 0.77 0.66
Always completes assignments -0.65 0.77 1 0.89
Always visits office hours 0.48 0.66 0.89 1

## Reporting Correlations

When reporting correlations, you need to record the correlation value (r), sample size (N) and significance (P) and whether the test is two-tailed or one tailed. Below is an example:

A correlation analysis was done to determine the relationship between studying with partners and participation in classes This relationship was not statistically significant (r = -.65, N =1525, p <.460 two-tailed.

Note:

• If the relationship was statistically significant (p <.05), then we would calculate   r2 (.054 x .054) to determine the strength of the relationship.  r2 tells us the explained variance.
• A  r2 value of 0 to.3 can be interpreted as weak; 0.31 to .59 can be seen as moderate and .6 or higher is good (note that social scientists do not have consensus on this. This is just a rule of thumb)