Correlation appears in almost every branch of statistics — from medical research to finance to education — because it answers a straightforward question: do two measured quantities tend to move together? This article builds the correlation definition up from first principles, then covers positive, negative and zero correlation, what the number does and does not tell you, how to average several r values, and how to read a correlation table. The formula itself, a step-by-step calculation and the ways r can mislead are the job of the correlation coefficient guide; this page is about the concept.
The Correlation Definition
Correlation is a statistical measure of the strength and direction of the linear relationship between two quantitative variables. When one variable consistently rises as the other rises — or falls as the other falls — the variables are positively correlated. When one rises as the other falls, they are negatively correlated. When no systematic pattern links them, the correlation is near zero.
The standard measure is Pearson’s r (the Pearson product-moment correlation coefficient), named after the statistician Karl Pearson. It compresses both the strength and the direction of the relationship into a single number on a scale from −1 to +1.
| r value | Meaning |
|---|---|
| +1.0 | Perfect positive correlation |
| +0.7 to +0.9 | Strong positive correlation |
| +0.3 to +0.7 | Moderate positive correlation |
| 0.0 to +0.3 | Weak positive correlation |
| 0.0 | No linear correlation |
| −0.3 to 0.0 | Weak negative correlation |
| −0.7 to −0.3 | Moderate negative correlation |
| −1.0 to −0.7 | Strong negative correlation |
| −1.0 | Perfect negative correlation |
The boundaries above follow conventions from Cohen (1988) and are widely used in social-science research. What counts as “strong” depends on the field: an r of 0.50 may be unremarkable in psychology but exceptionally high in ecology, where dozens of variables compete to explain variance.
Where the Number Comes From
Pearson’s r is built from how each pair of observations deviates from its own mean: when x sits above its mean at the same time y sits above its mean (or both sit below), the pair pushes r toward +1; when the two deviate in opposite directions, the pair pushes r toward −1. Dividing by the spread of each variable keeps the result between −1 and +1 and strips out the units, so r is a pure number that can be compared across variables measured on completely different scales. The NIST/SEMATECH e-Handbook of Statistical Methods, section on the Linear Correlation Coefficient gives the derivation; the correlation coefficient guide works through the formula and a full hand calculation.
A Quick Example
A teacher records weekly study hours and end-of-term exam scores for five students:
| Student | Study hours (x) | Exam score (y) |
|---|---|---|
| A | 2 | 60 |
| B | 3 | 65 |
| C | 4 | 70 |
| D | 5 | 80 |
| E | 6 | 85 |
Every student who studied more scored higher, and the increases are nearly proportional — each extra hour adds roughly five to ten points. Running the five pairs through the formula gives r ≈ 0.991, very close to +1: as study hours increase, exam scores rise in an almost perfectly linear pattern. To compute r for your own data, the correlation coefficient calculator returns r, r², and the regression equation automatically.
Positive Correlation
Positive correlation occurs when both variables move in the same direction: as one increases, the other tends to increase as well. The r value falls between 0 and +1.
Common real-world examples of positive correlation:
- Height and weight. Taller people tend to weigh more (individual variation is large, but the trend is positive).
- Study time and exam score. As in the worked example above, more preparation typically produces a higher score.
- Temperature and ice-cream sales. Warmer days drive more purchases; as temperature rises, sales rise.
- Advertising spend and revenue. Higher marketing budgets often accompany higher sales figures, especially in early growth phases.
A scatter plot of two positively correlated variables shows a cloud of data points sloping upward from left to right. The tighter the cloud hugs that upward line, the closer r is to +1. When r = +1 exactly, every point lies on the same straight line — perfect positive correlation. In real data, r = +1 is essentially never observed because measurement error and unmeasured variables always introduce some scatter.
Negative Correlation
Negative correlation occurs when variables move in opposite directions: as one increases, the other tends to decrease. The r value falls between −1 and 0.
Examples of negative correlation:
- Exercise frequency and resting heart rate. People who exercise more regularly tend to have lower resting heart rates.
- Car age and resale value. Older vehicles generally command lower prices.
- Class absences and exam score. Students who miss more sessions tend to score lower.
A scatter plot of negatively correlated variables shows a cloud sloping downward from left to right. An r of −0.85 indicates a strong negative relationship — as meaningful as a positive r of +0.85, just in the opposite direction.
Zero Correlation
When r is close to 0, the two variables have no linear correlation: knowing one variable’s value gives you no useful information about the other. A scatter plot looks like a roughly circular or randomly scattered cloud with no discernible slope.
Zero correlation does not mean the variables are completely unrelated. Two variables can follow a strong curved (non-linear) relationship — a U-shape or a step function, for instance — and still produce r ≈ 0, because Pearson’s formula measures only straight-line association. This is one reason why always plotting your data before computing r is good practice. Correlation is a summary, not a substitute for visualization.
Interpreting the Number
What a given r tells you depends on both its sign and its magnitude:
- Sign (+ or −): Positive means variables move together; negative means they move in opposite directions.
- Magnitude (distance from 0): Values near ±1 indicate a tight linear relationship; values near 0 indicate weak or no linear association.
- r² (the coefficient of determination): Squaring r gives the proportion of variance in one variable that is linearly explained by the other. An r of 0.70 yields r² = 0.49, meaning 49% of the variance in y is accounted for by the linear relationship with x. The remaining 51% comes from other variables or random variation.
Pearson’s r is dimensionless and bounded between −1 and +1, which makes it comparable across studies regardless of the units each variable was measured in. The Penn State STAT 501 Regression Analysis course notes, Introduction to Linear Correlation offer a detailed treatment of how r relates to the slope and intercept of the fitted regression line.
How to Average Several r Values
When you have Pearson correlations from several independent studies or subgroups and want to combine them into a single average, you cannot simply add the r values and divide. Pearson’s r is not measured on a linear scale, especially near ±1 — straightforward averaging of raw r values produces a biased (typically underestimated) result.
The correct approach uses Fisher’s z-transformation, introduced by Ronald Fisher in 1915.
Fisher’s z-Transformation Formula
Convert each r to a z’ value:
z' = 0.5 × ln[(1 + r) / (1 − r)]
This maps the bounded [−1, +1] scale onto an unbounded (−∞, +∞) scale where the values behave approximately like a normal distribution, making arithmetic averaging valid.
Step-by-Step Example
Suppose two studies report r₁ = 0.50 and r₂ = 0.80.
Step 1 — Convert each r to z’:
z'₁ = 0.5 × ln[(1 + 0.50) / (1 − 0.50)]
= 0.5 × ln(1.50 / 0.50)
= 0.5 × ln(3.00)
= 0.5 × 1.099
= 0.549
z'₂ = 0.5 × ln[(1 + 0.80) / (1 − 0.80)]
= 0.5 × ln(1.80 / 0.20)
= 0.5 × ln(9.00)
= 0.5 × 2.197
= 1.099
Step 2 — Average the z’ values:
z'_mean = (0.549 + 1.099) / 2 = 1.648 / 2 = 0.824
Step 3 — Back-transform to r:
r_mean = tanh(z'_mean) = tanh(0.824)
Using the definition of tanh:
r_mean = (e^(2 × 0.824) − 1) / (e^(2 × 0.824) + 1)
= (e^1.648 − 1) / (e^1.648 + 1)
≈ (5.197 − 1) / (5.197 + 1)
≈ 4.197 / 6.197
≈ 0.677
The average correlation is approximately r = 0.68. Averaging the raw values directly gives (0.50 + 0.80) / 2 = 0.65 — a slight underestimate, because the r scale compresses near ±1.
When studies differ in sample size, weight each z’ by (nᵢ − 3) before averaging:
z'_weighted = Σ[(nᵢ − 3) × z'ᵢ] / Σ(nᵢ − 3)
This gives larger samples proportionally more influence, producing a more precise estimate of the average correlation.
What Correlation Is and Is Not
Using r well means grasping both its power and its limits:
- Correlation measures only linear association. A strong non-linear pattern — a U-curve, an S-shape — can produce r ≈ 0 even when the variables are strongly related. Always check a scatter plot first.
- Correlation does not imply causation. A high r between two variables does not mean one causes the other. Both may be driven by an unmeasured third variable (a confounder), or the association may be entirely coincidental. See the fuller discussion in our article on correlation vs causation.
- Outliers can distort r substantially. A single extreme data point can pull r toward ±1 or toward 0. Report whether your analysis checked for outliers.
- Range restriction deflates r. If you sample only part of the natural spread of x — for instance, only participants above a certain score threshold — r will be artificially smaller than in a full-range sample.
- Sample size affects reliability. An r of 0.80 from n = 8 observations is far less trustworthy than the same r from n = 200. Statistical significance testing (a t-test for r, or a confidence interval) quantifies this uncertainty.
For related computations, the linear regression calculator fits the least-squares line through your paired data and reports r, r², slope, and intercept together.
How to Read a Correlation Table
A correlation table — also called a correlation matrix — displays pairwise Pearson r values for three or more variables at once. Each cell shows the correlation between the variable named in that row and the variable named in that column.
Here is an example matrix for three variables: study time (X), exam score (Y), and sleep hours (Z):
| X | Y | Z | |
|---|---|---|---|
| X | 1.00 | 0.85 | −0.12 |
| Y | 0.85 | 1.00 | 0.30 |
| Z | −0.12 | 0.30 | 1.00 |
How to read it:
- The main diagonal is always 1.00 — every variable is perfectly correlated with itself.
- The matrix is symmetric: the r for X and Y is the same as r for Y and X.
- X and Y share a strong positive correlation (0.85): more study time strongly accompanies higher exam scores.
- X and Z share a weak negative correlation (−0.12): study time has almost no linear relationship with sleep hours.
- Y and Z share a weak positive correlation (0.30): higher exam scores are weakly associated with more sleep.
Which Table Shows a Positive Correlation?
A correlation table shows a positive correlation when its off-diagonal cells contain positive values noticeably above zero — roughly 0.3 or higher for a meaningful relationship. In the matrix above, both the X–Y cell (0.85) and the Y–Z cell (0.30) indicate positive correlation.
In contrast, a table showing a negative correlation has negative off-diagonal values (e.g., −0.65), and a table showing no correlation has off-diagonal values hovering near 0. When answering a multiple-choice question about which table shows a positive correlation, look for the table where the off-diagonal cells contain positive numbers between 0 and +1.
Larger correlation tables (five or more variables) are read the same way — scan the off-diagonal half for strong positive values, note their variable-pair labels, and check both the sign and magnitude.
Other Correlation Coefficients
Pearson’s r covers linear relationships between two continuous variables. Other coefficients handle different data types:
- Spearman’s rₛ (rank correlation): Ranks all x values and all y values first, then applies the Pearson formula to those ranks. Robust to outliers and appropriate for monotone (consistently increasing or decreasing but not necessarily linear) relationships, or when data contain ordinal ranks.
- Kendall’s τ (tau): Counts the proportion of concordant minus discordant pairs across all possible pairings. More conservative than Spearman for small samples and more interpretable as a probability statement.
- Point-biserial r: Measures the correlation between one continuous variable and one binary (0/1) variable. It is mathematically equivalent to the Pearson formula applied to the continuous-versus-0/1 pair.
All three share the same −1 to +1 interpretation as Pearson’s r; the choice depends on data type, sample size, and whether a strictly linear relationship is a reasonable assumption.
Frequently Asked Questions
What is the difference between positive and negative correlation?
Only the direction. With r between 0 and +1 both variables increase together or both decrease together; with r between −1 and 0 one increases as the other decreases. A correlation of +0.80 is exactly as strong as −0.80. Zero correlation means no systematic linear relationship exists.
Which table shows a positive correlation?
Look at the off-diagonal cells: a matrix whose off-diagonal entries are positive numbers clearly above zero (say, 0.72) is describing variables that rise and fall together. Cells near 0 show little or no linear relationship, and cells near −1 show a strong negative one. In a multiple-choice problem, the table with clearly positive off-diagonal entries is the one to pick.
Can you average several correlation coefficients?
Not directly. Because the r scale is compressed near ±1, raw r values from different studies or subgroups cannot simply be added and divided. Convert each r to Fisher’s z’ = 0.5 × ln[(1 + r)/(1 − r)], average the z’ values (weighting by nᵢ − 3 if sample sizes differ), then back-transform with r = tanh(z’_mean), as in the worked example above.
Summary
Correlation is a foundational concept in statistics. The correlation definition — a measure of the strength and direction of the linear relationship between two variables — underlies regression analysis, factor analysis, and countless research designs. A positive correlation (r > 0) means variables rise and fall together; a negative correlation (r < 0) means one rises as the other falls. For combining r values across studies, Fisher’s z-transformation is the statistically correct approach. And when reading a correlation matrix, a positive relationship shows up as off-diagonal values that are clearly positive and above the noise threshold of roughly 0.3.
To compute correlations and the associated regression line for your own data, visit the correlation coefficient calculator. For the full suite of statistical tools, the calculators hub has everything in one place.