Source: Mathematical Statistics, University of Central Florida
Tags: correlation, Pearson correlation, correlation coefficient, r value, bivariate data, scatter plot, linear relationship, covariance, statistical association
Difficulty: Intermediate | Prerequisites: Descriptive statistics (mean, standard deviation), summation notation, basic algebra
Correlation analysis is one of the foundational tools in statistics for measuring how two variables move together. It sits at the heart of the broader regression and inference unit in most mathematical statistics courses, and nearly every applied field (psychology, economics, biology, engineering) uses it routinely. Before you can build predictive models or run hypothesis tests on relationships, you need to quantify the strength and direction of association between variables. If you are comfortable with means, standard deviations, and summation notation, you have what you need to follow this material.
Correlation measures the strength and direction of a linear relationship between two variables. The Pearson correlation coefficient (r) ranges from -1 to +1, where values near the extremes indicate strong linear association and values near 0 indicate weak or no linear relationship. Correlation does not imply causation, and it only captures linear patterns.
Bivariate data
A dataset where each observation consists of two paired measurements taken on the same subject or unit. In the source data, each person's Form 1 score and Form 2 score on the same item number constitute a bivariate pair.
In simple terms, this means you are tracking two things at once for each case, like height and weight for the same group of people.
Correlation
A statistical measure that describes the extent to which two variables change together. Positive correlation means both variables tend to increase together; negative correlation means one tends to decrease as the other increases.
Think of it as a single number that tells you how tightly two variables follow a straight-line pattern.
Pearson correlation coefficient (r)
The most commonly used measure of linear association between two continuous variables. It is defined as the ratio of the covariance of the two variables to the product of their standard deviations. Values range from -1 (perfect negative linear relationship) to +1 (perfect positive linear relationship), with 0 indicating no linear relationship.
In simple terms, r tells you how close the data points fall to a straight line when you plot them.
Covariance
A measure of how much two variables vary together. Positive covariance means the variables tend to move in the same direction; negative covariance means they move in opposite directions. Unlike correlation, covariance is not standardised, so its magnitude depends on the units of the variables.
Think of it as the raw, unscaled version of correlation. It tells you the direction of association but not how strong it is in any absolute sense.
Coefficient of determination (r²)
The square of the Pearson correlation coefficient. It represents the proportion of variance in one variable that is explained by the linear relationship with the other variable. An r of 0.80 gives an r² of 0.64, meaning 64% of the variance is accounted for.
In simple terms, r² tells you what fraction of the movement in one variable you can "explain" by knowing the other.
Scatter plot
A graph that displays bivariate data as points on a two-dimensional plane, with one variable on each axis. The visual pattern of the points reveals the direction, strength, and form of the relationship.
Think of it as the picture version of correlation: you can often eyeball whether the relationship is strong, weak, positive, or negative just from the shape of the cloud.
Population correlation coefficient (ρ, rho)
The true correlation between two variables across the entire population. In practice, we almost never know ρ directly; we estimate it using r from a sample.
In simple terms, ρ is the "real answer" and r is our best guess from the data we have.
Spearman rank correlation (rₛ)
A non-parametric version of correlation that works on the ranks of the data rather than the raw values. Useful when data are ordinal (like Likert scale responses) or when the relationship is monotonic but not strictly linear.
Think of it as Pearson correlation applied after converting all your data to rankings. It is more robust to outliers and does not require the data to be normally distributed.
A high correlation between two variables means they move together, but it says nothing about whether one causes the other.
A third, unmeasured variable (a confounding variable) may drive both.
Classic example: ice cream sales and drowning rates are positively correlated, but the confounding variable is warm weather.
Establishing causation requires controlled experiments or carefully designed observational studies with appropriate controls.
Positive correlation (r > 0): as one variable increases, the other tends to increase.
Example: hours studied and exam score.
Negative correlation (r < 0): as one variable increases, the other tends to decrease.
Example: hours spent on social media and GPA.
Zero or near-zero correlation (r ≈ 0): no consistent linear pattern.
Important caveat: r ≈ 0 does not mean the variables are unrelated. They may have a strong non-linear relationship (e.g., a U-shape) that r cannot detect.
|r| = 0.00 to 0.19: very weak or negligible
|r| = 0.20 to 0.39: weak
|r| = 0.40 to 0.59: moderate
|r| = 0.60 to 0.79: strong
|r| = 0.80 to 1.00: very strong
These are rough guidelines, not rigid thresholds. In some fields (e.g., physics), r = 0.70 would be considered low; in social sciences, r = 0.40 might be noteworthy.
Both variables continuous and roughly normal: Pearson r is the standard choice.
Ordinal data or non-normal distributions: Spearman rₛ is more appropriate. The source data (Likert-type scales from 0 to 5) falls into this category, so Spearman would often be the better fit for this kind of exercise.
One dichotomous variable and one continuous: point-biserial correlation (a special case of Pearson r).
Correlation analysis is the starting point for nearly all predictive modelling. In medicine, researchers correlate biomarkers with disease outcomes to identify diagnostic indicators. In finance, portfolio managers track correlations between asset returns to diversify risk: if two stocks are strongly positively correlated, holding both does not reduce risk as much as holding two uncorrelated stocks.
Students often assume that a correlation of 0 means the variables have nothing to do with each other. It only means there is no linear relationship; a strong curved pattern would still produce r ≈ 0.
Students frequently interpret r as a percentage. An r of 0.50 does not mean the variables are "50% related." It is r² (0.25, or 25%) that gives the proportion of shared variance.
Students sometimes think a strong correlation automatically means the relationship is important or meaningful. With very large sample sizes, even trivially small correlations can be statistically significant.
Students often forget that Pearson r is sensitive to outliers. A single extreme data point can dramatically inflate or deflate the correlation.
Continued in Part 2: Formulas, Computation, and Worked Examples