A correlation coefficient is a number between −1 and +1 that measures both the strength and direction of the linear relationship between two numeric variables. The closer the value of r is to +1 or −1, the stronger the relationship; r = 0 means no linear association at all. Knowing how to read and compute correlation coefficients is one of the most transferable skills in statistics — it appears in research papers, business dashboards, medical studies, and machine learning pipelines alike.
The most widely used measure is the Pearson correlation coefficient, written as r after Karl Pearson who formalised it in the 1890s. It is symmetric (the correlation of x with y equals the correlation of y with x), dimensionless (changing the units of measurement leaves r unchanged), and straightforward to compute with a standard formula. This article explains what a correlation coefficient means, walks through the coefficient of correlation formula step by step with a worked numeric example, and shows you exactly how to interpret any r value you encounter — including which values represent the strongest or weakest correlations.
What Is a Correlation Coefficient?
A correlation coefficient summarises how consistently two variables move together. When every increase in x accompanies an increase in y, the correlation is positive. When an increase in x accompanies a systematic decrease in y, the correlation is negative. When the two variables show no consistent linear pattern, r stays near zero.
The Pearson r is calculated for pairs of continuous measurements and assumes the relationship is approximately linear. Other correlation coefficients — Spearman’s ρ (rho) and Kendall’s τ (tau) — extend the concept to ranked data or non-linear monotone relationships, but Pearson’s r is the standard in most quantitative work and the one you will meet first in any statistics course.
Correlation coefficients appear in virtually every quantitative field. Economists look at whether income and educational attainment move together. Public-health researchers ask whether smoking rates correlate with lung-cancer incidence. Engineers test whether temperature correlates with product failure rates. The correlation coefficient puts a single, comparable number on what a scatter plot shows visually.
For a conceptual overview of correlation — what it means, what its different types are, and when to use it — see the article on what correlation means in statistics.
The Correlation Coefficient Formula
The Pearson correlation coefficient formula — sometimes written as the coefficient of correlation formula — expresses r as the ratio of the covariance of x and y to the product of their standard deviations:
r = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / √[ Σ(xᵢ − x̄)² · Σ(yᵢ − ȳ)² ]
where:
xᵢandyᵢare the individual paired data valuesx̄andȳare the sample means of x and yΣdenotes summation over all n pairs
An algebraically equivalent raw-score form avoids computing the means first and works well for mental arithmetic or a handheld calculator:
r = [ nΣxᵢyᵢ − (Σxᵢ)(Σyᵢ) ] / √{ [ nΣxᵢ² − (Σxᵢ)² ] · [ nΣyᵢ² − (Σyᵢ)² ] }
Both versions produce identical results. The deviation form (first formula) is conceptually transparent — the numerator is the sum of cross-products of deviations, and the denominator normalises it by the total spread of each variable. The raw-score form is faster when you are working from summary sums. The NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.11 — Bivariate Data and Scatter Plots presents these relationships and explains why dividing by the product of the two standard deviations forces r to stay inside the [−1, +1] interval.
The key insight in the coefficient and correlation formula is that the numerator captures the average co-movement of the two variables, while the denominator acts as a scale factor. When x and y both deviate from their means in the same direction (both above or both below), the cross-products are positive, pushing r toward +1. When they deviate in opposite directions, the cross-products are negative, pushing r toward −1. When the deviations are uncorrelated, the positive and negative terms cancel and r converges on 0.
How to Find the Correlation Coefficient: Step-by-Step Example
To see exactly how to find the correlation coefficient by hand, work through the following five-pair dataset:
| Pair i | xᵢ | yᵢ |
|---|---|---|
| 1 | 1 | 2 |
| 2 | 2 | 4 |
| 3 | 3 | 5 |
| 4 | 4 | 4 |
| 5 | 5 | 5 |
Step 1: Compute the sample means
Sum the x values: 1 + 2 + 3 + 4 + 5 = 15, so x̄ = 15 / 5 = 3.
Sum the y values: 2 + 4 + 5 + 4 + 5 = 20, so ȳ = 20 / 5 = 4.
Step 2: Compute deviations, squared deviations, and cross-products
| xᵢ | yᵢ | xᵢ − x̄ | yᵢ − ȳ | (xᵢ − x̄)² | (yᵢ − ȳ)² | (xᵢ − x̄)(yᵢ − ȳ) |
|---|---|---|---|---|---|---|
| 1 | 2 | −2 | −2 | 4 | 4 | 4 |
| 2 | 4 | −1 | 0 | 1 | 0 | 0 |
| 3 | 5 | 0 | +1 | 0 | 1 | 0 |
| 4 | 4 | +1 | 0 | 1 | 0 | 0 |
| 5 | 5 | +2 | +1 | 4 | 1 | 2 |
| Σ | 10 | 6 | 6 |
Step 3: Apply the correlation coefficient formula
Plug the column sums into the deviation form of the formula:
r = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / √[ Σ(xᵢ − x̄)² · Σ(yᵢ − ȳ)² ]
= 6 / √(10 × 6)
= 6 / √60
= 6 / 7.746
≈ 0.775
What r ≈ 0.775 tells you
A correlation coefficient of approximately 0.775 indicates a strong positive linear relationship between x and y. As x increases, y tends to increase too, and the two variables track each other fairly closely. The coefficient of determination r² = 0.775² ≈ 0.600, meaning that x accounts for about 60 % of the variance in y.
Try the Calculator
Enter the same five pairs above — or substitute your own x and y values — to confirm the result instantly. The calculator computes r, r², and the number of paired values.
To work with larger datasets or explore the output labels in more detail, open the dedicated correlation coefficient calculator page. You can also browse all available tools on the calculators hub.
How to Interpret Correlation Coefficient r Values
Once you have r, interpreting it requires attending to two independent properties: direction and strength.
Direction
- Positive r (r > 0): Both variables tend to move in the same direction. Higher x values pair with higher y values more often than not. The scatter plot tilts upward from left to right.
- Negative r (r < 0): The variables move in opposite directions. Higher x values pair with lower y values. The scatter plot tilts downward from left to right.
- r = 0: No linear relationship. The scatter plot has no discernible slope, though a non-linear relationship (such as a U-curve) might still exist.
Strength
Statisticians use these conventional benchmarks, drawn from the treatment in Penn State STAT 501 — Regression Methods, Lesson 1:
| |r| value | Commonly described as | |------------|----------------------| | 0.9 – 1.0 | Very strong | | 0.7 – 0.9 | Strong | | 0.5 – 0.7 | Moderate | | 0.3 – 0.5 | Weak to moderate | | 0.0 – 0.3 | Weak or negligible |
These are conventions, not physical laws. In psychology, r = 0.40 can represent a meaningful effect; in precision engineering, r below 0.95 might be unacceptably noisy. Always interpret r relative to your field’s norms and your study’s sample size.
Which value of r indicates a stronger correlation?
When comparing two or more correlation coefficients, the one with the larger absolute value is always the stronger one. The sign indicates direction, not strength.
- r = −0.85 is stronger than r = +0.60, because |−0.85| = 0.85 > |+0.60| = 0.60.
- r = +0.92 is stronger than r = −0.45, because |+0.92| = 0.92 > |−0.45| = 0.45.
Determining which value of r indicates a stronger correlation always reduces to comparing absolute values.
Which r Value Represents the Strongest (or Weakest) Correlation?
This question appears frequently in statistics courses and standardised tests. The rule is the same every time:
- Strongest correlation = r value with the largest absolute value (closest to +1 or −1)
- Weakest correlation = r value with the smallest absolute value (closest to 0)
Example. Suppose you are given: r₁ = −0.90, r₂ = +0.75, r₃ = −0.30, r₄ = +0.10.
Absolute values: |r₁| = 0.90, |r₂| = 0.75, |r₃| = 0.30, |r₄| = 0.10.
- Which of the following r values represents the strongest correlation? r₁ = −0.90, because its absolute value (0.90) is the largest.
- Which of the following r-values represents the strongest correlation? Same answer: r₁ = −0.90.
- Which of these r values represents the weakest correlation? r₄ = +0.10, because its absolute value (0.10) is the smallest.
- Which of these r-values represents the weakest correlation? Same answer: r₄ = +0.10.
The most common mistake on exam questions is selecting the largest positive r as the strongest. A large negative r can be just as strong — or stronger. r = −0.95 is a near-perfect correlation; it simply runs in the opposite direction from r = +0.95.
Positive vs Negative Correlation Coefficients
Positive correlation coefficients
A positive r means that higher values of x tend to accompany higher values of y. Positive correlation coefficients arise when two variables are driven by a common underlying factor or when one genuinely influences the other in the same direction.
Common real-world examples:
- Study hours and exam scores (more hours → higher scores)
- Height and arm span (taller people tend to have longer arm spans)
- Monthly advertising spend and revenue (within a range)
When r = +1, every point in the scatter plot falls exactly on an upward-sloping straight line — a perfect positive relationship that rarely occurs in messy, real data.
Negative correlation coefficients
A negative r means that higher values of x tend to accompany lower values of y. Negative correlation coefficients are just as informative as positive ones; they simply describe an inverse relationship.
Common real-world examples:
- Altitude and air temperature (higher elevation → lower temperature)
- Exercise frequency and resting heart rate (more exercise → lower resting heart rate)
- Number of absences and final grade (more absences → lower grade)
When r = −1, every point falls exactly on a downward-sloping straight line — a perfect negative relationship. Again, real data rarely achieves this.
The key takeaway: both strong positive and strong negative correlation coefficients indicate that the two variables have a tight, predictable linear relationship. Only the direction differs.
Correlation Does Not Imply Causation
A high r value frequently tempts readers to conclude that one variable causes the other. It does not. Two variables can be strongly correlated for several reasons:
- Direct causation: A causes B (or B causes A).
- A common cause: A third variable C drives both A and B independently, creating the appearance of a link between A and B even though neither affects the other.
- Coincidence: Spurious correlations arising from chance, especially in small samples or when many pairs are tested simultaneously.
A frequently cited example: ice-cream sales and drowning incidents correlate positively across months of the year. Neither causes the other — hot weather drives both, creating a spurious association.
Establishing causation requires controlled experiments, careful study design, or formal causal inference methods. A correlation coefficient on its own, however large, cannot distinguish between these scenarios. Read the dedicated article on correlation vs causation for a fuller discussion.
Limitations of the Pearson Correlation Coefficient
Understanding what r does not measure is as important as knowing what it does.
Pearson r only captures linear relationships. If x and y have a strong U-shaped or any other non-linear relationship, r can sit near zero even though the variables are clearly associated. Always inspect a scatter plot before reporting or interpreting r in isolation.
Outliers can distort r substantially. A single extreme data point can inflate or deflate the correlation coefficient. If your dataset contains outliers, consider computing Spearman’s rank correlation (ρ) as a more robust alternative — it is based on the ranks of the data rather than raw values.
r is not a proportion of relationship. r = 0.60 does not mean the relationship is “60 % complete.” What is meaningful as a proportion is r² = 0.36, which tells you that 36 % of the variance in y is statistically accounted for by x. Squaring r to get the coefficient of determination is the correct step when you want a proportion.
Range restriction reduces r. If you measure correlation within a narrow slice of one variable (students at a single high school, products in a single price range), the observed r will underestimate the true relationship in the full population. This is a common source of bias in applied research.
Sample size matters. A correlation of r = 0.50 in a sample of n = 10 is not statistically significant at conventional levels, while the same r in a sample of n = 50 would be. The interpretation of r should always be accompanied by a significance test or confidence interval, especially in small samples.
Frequently Asked Questions
What is a correlation coefficient?
A correlation coefficient is a standardised measure — always between −1 and +1 — of the strength and direction of the linear relationship between two numeric variables. The most common form is Pearson’s r. A value near +1 indicates a strong positive linear relationship, near −1 indicates a strong negative linear relationship, and near 0 indicates no linear association.
What is the difference between a correlation coefficient and covariance?
Covariance is the unstandardised version of r. It measures the same co-movement of two variables but is expressed in the original units (e.g., kg · cm), which makes it hard to compare across datasets. Dividing the covariance by the product of the two standard deviations produces r, which is dimensionless and always within [−1, +1]. In other words, the correlation coefficient formula is a scaled version of the covariance formula.
How do I find the correlation coefficient by hand?
To find the correlation coefficient for n paired values: (1) compute the sample means x̄ and ȳ; (2) subtract each mean from its column of values to get deviations; (3) multiply each x-deviation by the corresponding y-deviation (cross-product); (4) sum all cross-products — this is the numerator; (5) sum the squared x-deviations and squared y-deviations separately; (6) multiply those two sums and take the square root — this is the denominator; (7) divide numerator by denominator to obtain r. The five-pair example earlier in this article walks through all seven steps with a full table.
Which of the following r values represents the strongest correlation?
Whichever r value has the largest absolute value represents the strongest correlation. For candidates such as −0.90, +0.75, −0.30, and +0.10, the answer is −0.90 because |−0.90| = 0.90 is the largest. This rule answers “which of the following r-values represents the strongest correlation” equally — the sign is irrelevant to strength; only the absolute value matters.
Which of these r values represents the weakest correlation?
Whichever r value has the smallest absolute value (closest to 0) represents the weakest correlation. In the example above, +0.10 is the weakest because |+0.10| = 0.10 is the smallest. The same rule applies when the question is phrased as “which of these r-values represents the weakest correlation.”
What r value indicates a stronger correlation: −0.80 or +0.65?
−0.80 indicates a stronger correlation, because |−0.80| = 0.80 is greater than |+0.65| = 0.65. When comparing which value of r indicates a stronger correlation, always compare the absolute values; the direction (positive or negative) is not part of the strength comparison.
Can two variables have r = 0 yet still be related?
Yes. Pearson r measures only linear association. If two variables have a perfect U-shaped or quadratic relationship — say, y = x² — their Pearson r can be exactly 0 because the upward and downward halves cancel out symmetrically. This is one reason why inspecting a scatter plot before computing and interpreting r is always recommended.
What counts as a good correlation coefficient?
There is no universal threshold. In psychology and social science, r > 0.50 is typically described as a strong effect. In epidemiology, r = 0.30 can be highly meaningful at a population level. In physical chemistry or precision engineering, r below 0.95 may be unacceptably imprecise. The appropriate standard depends on your discipline, the purpose of the analysis, and the practical consequences of errors. Always interpret r in context.
Summary
The correlation coefficient r is a compact, interpretable measure of linear association. The correlation coefficient formula standardises the covariance of two variables by their standard deviations, producing a dimensionless value between −1 and +1. A positive r indicates that the variables rise together; a negative r indicates that one rises as the other falls; the absolute value of r indicates the strength of the relationship.
Key interpretation rules:
- Strongest correlation among a set of r values = the one with the largest |r| (closest to 1 in absolute value)
- Weakest correlation = the one with the smallest |r| (closest to 0)
- r² (the coefficient of determination) = the proportion of variance in y explained by x
- Sign tells you direction only, never strength
Always view a scatter plot alongside r, check for outliers that could distort the value, and never mistake correlation for causation. The step-by-step example above (r ≈ 0.775 for five paired values) illustrates the full computation; the calculator widget lets you apply the same logic instantly to any dataset you have.