Correlation Coefficients Explained: What r Actually Measures

A correlation coefficient is a single number between -1 and +1 that summarizes how two variables move together. Most researchers can interpret one long before they can say what it's actually calculating. This guide opens the box: where r comes from, why it's bounded at -1 and +1, how to compute it by hand, what r squared adds, and how correlation connects to regression.

This is the mechanics article. If what you need is which coefficient to use for your data, or how to report one in APA format, that's covered in the companion guide to using correlation coefficients in research papers, which walks through Pearson, Spearman, Kendall's tau, and the rest, along with reporting conventions and worked disciplinary examples. Come back here when you want to know what the number is doing.

Start with Covariance

You can't understand r without covariance, because r is covariance with the units removed. Covariance answers a simple question: when one variable is above its mean, does the other tend to be above its mean too?

For each observation, you take how far x sits from its mean, multiply it by how far y sits from its mean, and average those products across the sample. When both variables tend to be above (or both below) their means at the same time, the products are positive and the covariance is positive. When one tends to be high while the other is low, the products are negative and the covariance is negative. When there's no consistent pattern, positives and negatives cancel and the covariance sits near zero.

cov(x, y) = Σ (x − M _x)(y − M _y) / (n − 1)

That n − 1 is the same sample correction used in the variance and standard deviation formulas, and for the same reason: you're estimating a population quantity from a sample, and you used the sample means to do it.

Covariance has one fatal flaw for reporting. Its size depends entirely on the units of measurement. The covariance between height and weight is a different number in centimeters and kilograms than in inches and pounds, even though the relationship is identical. A covariance of 340 tells you the direction is positive, and nothing else. You can't tell whether that's a strong relationship or a weak one.

Why r Is Bounded Between -1 and +1

The fix is standardization. Divide the covariance by the product of the two standard deviations, and the units cancel out. What's left is a pure number that can be compared across any two variables, in any units, in any study.

r = cov(x, y) / (SD _x × SD _y)

This division is also what creates the boundaries. Covariance is at its largest, in absolute terms, when the two variables move in perfect lockstep. In that case the covariance exactly equals the product of the two standard deviations, so the ratio is exactly 1 (or -1, if they move in perfectly opposite directions). Any less-than-perfect relationship produces a covariance smaller than that product, so the ratio falls somewhere in between. The scale isn't an arbitrary convention. It falls out of the arithmetic.

This is why r is comparable in a way covariance never is. An r of 0.60 between study hours and exam scores means the same strength of linear relationship as an r of 0.60 between plasma concentration and blood pressure reduction, despite the variables having nothing in common and no shared units.

The Pearson Formula, Worked by Hand

Seeing r calculated once makes the formula stop being abstract. Here's a small dataset: study hours per week and exam score, for eight students.

Step 1: Find both means

Study hours sum to 56, so M _x = 56 / 8 = 7.0. Exam scores sum to 634, so M _y = 634 / 8 = 79.25.

Step 2: Get the deviations and their products

For each student, subtract the mean from each value, then multiply the two deviations together. The last column is what feeds the covariance.

Notice that every product here is positive. Students below the mean on hours are also below the mean on scores, and vice versa. That's what a positive relationship looks like at the level of individual observations.

Step 3: Sum the columns

The products sum to 247.00. The squared x deviations sum to 92.00. The squared y deviations sum to 677.50.

Step 4: Compute r

The computational form of Pearson's r divides the sum of products by the square root of the product of the two sums of squares. This is algebraically identical to dividing covariance by the two standard deviations, with the n − 1 terms canceling out.

r = Σ(x − M _x)(y − M _y) / √[Σ(x − M _x)² × Σ(y − M _y)²]

r = 247.00 / √(92.00 × 677.50) = 247.00 / √62,330.00 = 247.00 / 249.66 = 0.989

An r of 0.99 is a very strong positive linear relationship, which matches what the raw data shows: exam scores climb steadily with study hours, with almost no exceptions. Real data is rarely this tidy, but the tidiness makes the arithmetic legible.

What r Squared Adds

Square the correlation coefficient and you get r squared, the coefficient of determination. It answers a different question than r does. Where r describes the strength and direction of the relationship, r squared describes how much of the variance in one variable is accounted for by the other.

The shift from r to r squared is not intuitive, because squaring a number below 1 makes it smaller, sometimes dramatically. This table shows how quickly the explained variance drops off.

This is why a "moderate" correlation of 0.30 can feel underwhelming once you see it explains 9 percent of the variance. Both numbers are honest; they just answer different questions. Reporting r squared alongside r keeps readers from reading more into a correlation than it supports.

One caution: r squared is symmetric and non-directional. Saying that x explains 25 percent of the variance in y is a statement about shared variance, not about x causing anything. The explanatory language is a mathematical convention, not a causal claim.

r as an Effect Size

Correlation coefficients do double duty. Beyond describing a relationship, r is itself a standardized effect size, which is why it appears in power analyses, meta-analyses, and effect-size tables alongside measures like Cohen's d.

This works because r is already unitless. Effect sizes exist to make findings comparable across studies that measured things differently, and r arrives pre-standardized by construction. A meta-analyst can average correlations across a dozen studies using different instruments in a way that would be meaningless with raw covariances or unstandardized slopes.

It also means a correlation carries information that a p-value does not. Statistical significance tells you an effect is unlikely to be chance; r tells you how large it is. Because significance depends heavily on sample size, a trivially small correlation can be highly significant in a large sample. Reporting r (and r squared) alongside the p-value is what keeps the distinction visible. For the underlying logic of significance testing, see the guide to hypothesis testing.

Partial Correlation: Holding a Third Variable Constant

A plain correlation between two variables can be inflated or masked by a third variable related to both. Partial correlation estimates the relationship between x and y with the influence of a third variable, z, statistically removed from both.

The classic illustration: ice cream sales and drowning deaths correlate strongly. Both are driven by temperature. Compute the partial correlation between sales and drownings while holding temperature constant, and the relationship largely disappears. The original correlation was real as an association but misleading as an explanation.

Mechanically, partial correlation works by removing the part of each variable that z can predict, then correlating what's left over. Those leftovers are the residuals, the variation in x and y that z doesn't account for. The residual logic is exactly what regression formalizes, which makes partial correlation the natural bridge to the next topic.

From Correlation to Regression

Correlation and simple linear regression describe the same relationship from two angles. Correlation gives you a symmetric summary: r between x and y is identical to r between y and x, with no direction implied. Regression is asymmetric: it fits a line that predicts y from x, producing a slope and an intercept, and reversing the roles gives a different equation.

They're connected by more than intuition. In simple linear regression with one predictor, the r squared of the model is exactly the square of the correlation between the two variables. And the standardized regression slope is exactly r. Regression, at this level, is correlation with a direction and a prediction equation attached.

That's the reason this article sits where it does in the cluster. Understanding r as standardized covariance, and r squared as explained variance, means the regression output that follows, the slope, the intercept, the model r squared, is already half familiar when you meet it.

Common Misunderstandings About r

• Reading r as a percentage. An r of 0.50 does not mean 50 percent of anything. The percentage interpretation belongs to r squared, which for r = 0.50 is 25 percent.
• Assuming r near zero means no relationship. It means no linear relationship. A perfect U-shaped relationship can produce an r near zero. Plot the data before trusting the coefficient.
• Treating r as directional. r is symmetric. It carries no information about which variable influences which, or whether either does.
• Comparing r values as if the scale were even. The step from 0.80 to 0.90 adds far more explained variance than the step from 0.10 to 0.20. Because r squared grows non-linearly, equal gaps in r are not equal gaps in explanatory power.
• Forgetting that r is sensitive to restricted range. If your sample covers only part of a variable's natural range, r shrinks toward zero even when the full-range relationship is strong. This is common in studies of selected populations, such as correlating test scores with performance among people who all passed the same admissions cutoff.
• Letting a single outlier drive the result. Because deviations are multiplied together, one extreme point far from both means can dominate the sum of products, especially in a small sample.

Where to Go Next

This article covered what the coefficient is doing. Two neighboring questions have their own guides. For choosing between Pearson, Spearman, Kendall's tau, and the other coefficients based on your data type, and for the conventions journals expect when you write the result up, see using correlation coefficients in research papers. For the descriptive statistics that feed the formula, see the guides to mean, median, and mode and standard deviation and variance.

When your analysis is written up, the statistics section is where a subject-matter editor earns their place, because a reviewer who spots a correlation described in causal language or an r reported without context will question everything around it. Editor World's academic editing, dissertation editing, and journal article editing services include review of statistical reporting, and you choose your own editor by field so the person reading your results knows your discipline's conventions.

Frequently Asked Questions

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