Inferential Testing (A Level Only) (AQA A-Level Psychology): Revision Notes

Tests of Correlation: Spearman's & Pearson's

Overview of correlation tests

Correlation tests help us determine whether there is a relationship between two variables and measure the strength of that relationship. Two key tests are used in psychology research: Spearman's rho and Pearson's r. The choice between these tests depends on the type of data you have and whether certain statistical assumptions are met.

Spearman's rho

When to use Spearman's rho

Spearman's rho is a non-parametric correlation test designed for use with ordinal data, though it can also be used with interval data when the assumptions for parametric tests are not met.

The test works by converting raw scores into ranks, then calculating the correlation between these ranks. This makes it less sensitive to extreme values compared to Pearson's r.

Calculating Spearman's rho

The calculation involves several systematic steps:

Step 1: Rank each set of scores separately for each variable, from lowest to highest. If scores are tied, calculate the mean of their combined ranks.

Step 2: Calculate the difference between each pair of ranks (d), then square these differences (d²).

Step 3: Apply the Spearman's formula:

$\rho = 1 - \frac{6\sum d^2}{N(N^2-1)}$

Where $\rho$ (rho) is the correlation coefficient, $\sum d^2$ is the sum of squared differences, and N is the number of participants.

Step 4: Compare the calculated value with the critical value from statistical tables to determine significance.

Pearson's r

When to use Pearson's r

Pearson's r is a parametric correlation test used when both variables are measured at the interval level. This test is more powerful than Spearman's rho when its assumptions are met, but these assumptions are more stringent.

Pearson's r measures the strength of linear relationships between variables, making it ideal for investigations where you expect variables to change proportionally.

Calculating Pearson's r

The calculation requires several preparatory steps before applying the main formula:

Step 1: Create a table with columns for each variable (x and y), their squares (x², y²), and their products (xy).

Step 2: Calculate the sums: $\sum x$, $\sum y$, $\sum x^2$, $\sum y^2$, and $\sum xy$.

Step 3: Apply Pearson's formula:

$r = \frac{N\sum xy - (\sum x)(\sum y)}{\sqrt{[N\sum x^2 - (\sum x)^2][N\sum y^2 - (\sum y)^2]}}$

Where r is the correlation coefficient and N is the number of participants.

Step 4: Compare the calculated value with the critical value to determine significance.

Key differences between the tests

The fundamental distinction lies in the type of data each test can handle and their underlying assumptions.

Spearman's rho converts all data to ranks, making it robust against outliers and suitable for non-normal distributions. Pearson's r works with raw data and is more sensitive to the actual values, making it more powerful when its assumptions are satisfied.

Understanding critical values and significance