10.1 P-Values and Statistical Significance

Bivariate Statistics

In the previous section, we explained the concept of correlation and how it measures the strength and direction of a linear relationship between two numeric variables. We also introduced the concept of R-squared (r²), which quantifies the proportion of variance in one variable that can be explained by another variable. When we calculate the correlation coefficient r, we often want to determine whether this observed correlation is statistically significant or if it could have occurred by random chance. This is where p-values come into play.

Understanding P-Values

A p-value is a statistical measure that helps us determine the significance of our results in hypothesis testing. Specifically, it quantifies the probability of observing a correlation as extreme as the one calculated (or more extreme) if the null hypothesis is true. The null hypothesis typically states that there is no relationship between the two variables being analyzed (i.e., the correlation coefficient r is equal to zero).

When we calculate a p-value for a correlation coefficient, we are essentially asking: “If there were truly no relationship between these two variables, what is the likelihood that we would observe a correlation as strong as the one we found in our sample?”

Interpreting P-Values

P-values are interpreted in the context of a predetermined significance level, often denoted as alpha (α). A common choice for α is 0.05, which means we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis (a Type I error).

• If the p-value is less than or equal to α (e.g., p ≤ 0.05), we reject the null hypothesis and conclude that there is a statistically significant correlation between the two variables.
• If the p-value is greater than α (e.g., p > 0.05), we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant correlation.