By Rajiv S. Jhangiani, I-Chant A. Chiang, Carrie Cuttler and Dana C. Leighton, adapted by Marc Chao and Muhamad Alif Bin Ibrahim

When studying relationships between quantitative variables, Pearson’s r (the correlation coefficient) is a widely used measure that quantifies the strength and direction of the relationship between two variables. However, to determine whether this observed relationship is statistically significant, that is, unlikely to have occurred by chance. A formal test of the correlation coefficient is required. This test follows the same fundamental principles as other null hypothesis tests, aiming to make inferences about the population based on the sample data.

The null and alternative hypotheses for testing the significance of Pearson’s r are as follows:

• Null hypothesis (ρ = 0): There is no relationship between the variables in the population. The population correlation, represented by the Greek letter rho (ρ), is equal to zero.
• Alternative hypothesis (ρ ≠ 0): There is a relationship between the variables in the population. The population correlation is not equal to zero.

The null hypothesis assumes that any observed correlation in the sample is due to random variation, while the alternative hypothesis asserts that the correlation reflects a true relationship in the population.

Statistical Approach to Testing Pearson’s r

There are two primary methods for conducting the test:

1. Using a t score: The sample correlation coefficient (r) can be converted into a t score using the formula for a t statistic. This t score is evaluated against a t distribution with N − 2 degrees of freedom, where N is the sample size. From this point, the procedure aligns with the standard approach for a t-test. The t score provides a basis for calculating the p value, which helps determine whether the null hypothesis should be rejected.
2. Direct Use of Pearson’s r: Because of the way it is computed, Pearson’s r itself can function as the test statistic. Modern statistical tools like Excel, SPSS, or online calculators typically compute Pearson’s r and provide the corresponding p value automatically. These tools simplify the process, allowing researchers to quickly assess the significance of the correlation without manual calculations.

Interpreting the Results

If the p value is ≤ 0.05, we reject the null hypothesis and conclude there is a relationship between the variables in the population. However, if the p value is > 0.05, we retain the null hypothesis, meaning there is not enough evidence to conclude a relationship exists.

If calculating by hand, you can compare the sample’s correlation coefficient to critical values in a table like Table 11.4.1. The critical value depends on the sample size (N) and whether the test is one-tailed or two-tailed. If the sample’s correlation coefficient is more extreme than the critical value, it is statistically significant.

Example: Testing a Correlation Coefficient

A health psychologist is exploring whether there is a correlation between individuals’ calorie estimates for food and their body weight. She does not have a specific expectation about whether the relationship will be positive or negative. Therefore, she opts for a two-tailed test, which allows her to detect a correlation in either direction.

The psychologist collects data from a sample of 22 university students, recording each student’s calorie estimates for a particular food item along with their body weight. Using this data, she calculates the correlation coefficient, Pearson’s r, which measures the strength and direction of the relationship. The resulting value is −0.21, indicating a weak negative correlation. This suggests that, in this sample, higher calorie estimates tend to be associated with lower body weights, though the relationship is not strong.

To assess whether this observed correlation is statistically significant, meaning that it likely reflects a true relationship in the population rather than random chance, the psychologist uses statistical software. The software computes the p value associated with her correlation coefficient and reports it as 0.348.

• Interpreting the p value: The p value represents the probability of obtaining a sample correlation as extreme as −0.21, or more extreme, if the null hypothesis is true (i.e., if there is no correlation in the population). In this case, the p value is 0.348, which is substantially greater than the conventional significance threshold of 0.05.

Because the p value exceeds 0.05, the psychologist retains the null hypothesis, concluding that the data does not provide sufficient evidence to suggest a significant relationship between calorie estimates and weight in the population.

If the psychologist were calculating the test manually, she would refer to a table of critical values for Pearson’s r to confirm whether the correlation is statistically significant. For a sample size of 22 participants, the degrees of freedom are calculated as N − 2, or 20.

• Critical value for two-tailed test: From the critical value table (e.g., Table 11.4.1), she finds that for a two-tailed test with α = 0.05 and 20 degrees of freedom, the critical value of r is ±0.444.

To determine significance, the absolute value of the sample correlation coefficient (|−0.21| = 0.21) is compared to the critical value:

• If ∣r∣ > 0.444, the correlation is statistically significant.
• If ∣r∣ ≤ 0.444, the correlation is not statistically significant.

Since 0.21 is less extreme than the critical value of ±0.444, the psychologist confirms that the p value is greater than 0.05. This further supports her decision to retain the null hypothesis.

Based on the analysis, the psychologist concludes that there is no statistically significant relationship between calorie estimates and weight in the population. The weak negative correlation observed in the sample (−0.21) could be due to random sampling variability rather than a true association.