Pearson r
0.7746
R²
0.6000
n
5
5 values entered
5 values entered
0.7746
0.6000
There is a strong positive linear relationship (r = 0.7746). 60.0% of the variance in Y is explained by X.
y = 0.6000x + 2.2000
0.6000
2.2000
Same ranges apply for negative values
r = Σ((xᵢ - x̅)(yᵢ - y̅)) / √(Σ(xᵢ - x̅)² × Σ(yᵢ - y̅)²)Measures the strength and direction of the linear relationship between paired data. Values close to ±1 indicate a strong linear relationship.
Where:
xᵢ, yᵢ= Individual data points in the paired data setx̅, y̅= The means of the X and Y data setsΣ= Sum over all paired data pointsR² = r²The proportion of variance in Y explained by X. Calculated by squaring the Pearson correlation coefficient.
Where:
r= Pearson correlation coefficientR²= Proportion of explained variance (0 to 1)y = mx + b, where m = Σ((xᵢ-x̅)(yᵢ-y̅)) / Σ((xᵢ-x̅)²), b = y̅ - mx̅The least-squares regression line that minimizes the sum of squared residuals between predicted and actual Y values.
Where:
m= Slope (change in Y per unit change in X)b= Y-intercept (predicted Y when X = 0)x̅, y̅= Means of the X and Y data setsInputs
Result
With r = 0.77, there is a strong positive linear relationship. R² = 0.60 means 60% of the variance in Y is explained by X. The regression line predicts Y increases by 0.6 for each unit increase in X.
Inputs
Result
A perfect negative correlation (r = -1) means the data points fall exactly on a straight line with negative slope. Y decreases by exactly 2 for every unit increase in X.
Inputs
Result
With r = 0.40 and R² = 0.16, the linear relationship is weak. Only 15.8% of the variation in Y is explained by X, suggesting other factors dominate.
The Pearson correlation coefficient (r) measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and +1 indicates a perfect positive linear relationship.
| |r| Range | Strength | Example |
|---|---|---|
| 0.9 – 1.0 | Very strong | Height vs. arm span |
| 0.7 – 0.9 | Strong | Study hours vs. grades |
| 0.5 – 0.7 | Moderate | Income vs. happiness |
| 0.3 – 0.5 | Weak | Age vs. music preference |
| 0.0 – 0.3 | Negligible | Shoe size vs. IQ |
R-squared (the coefficient of determination) is the square of the correlation coefficient. It represents the proportion of variance in the dependent variable (Y) that is explained by the independent variable (X). An R² of 0.81 means 81% of the variation in Y can be explained by X.
| Pearson r | R² | % Variance Explained |
|---|---|---|
| ±0.3 | 0.09 | 9% |
| ±0.5 | 0.25 | 25% |
| ±0.7 | 0.49 | 49% |
| ±0.9 | 0.81 | 81% |
| ±1.0 | 1.00 | 100% |
The formula is r = Σ((xi - x̅)(yi - y̅)) / √(Σ(xi - x̅)² × Σ(yi - y̅)²). You subtract each value from its mean, multiply the paired deviations, sum them, and divide by the square root of the product of the squared deviation sums.
| Component | Formula | Purpose |
|---|---|---|
| Numerator | Σ(xi-x̅)(yi-y̅) | Covariance direction |
| Denominator | √(Σ(xi-x̅)²×Σ(yi-y̅)²) | Normalizes to [-1,1] |
| Result | Numerator / Denominator | Correlation strength |
Linear regression finds the best-fit line y = mx + b through the data points. The slope m indicates how much Y changes per unit change in X, and the intercept b is where the line crosses the Y-axis. The correlation coefficient r measures how closely the data follows this line.
| Concept | What It Measures | Range |
|---|---|---|
| r (correlation) | Strength of linear relationship | -1 to +1 |
| R² (determination) | Variance explained | 0 to 1 |
| m (slope) | Rate of change in Y per X | Any real number |
| b (intercept) | Y-value when X = 0 | Any real number |
No, correlation does not imply causation. A strong correlation between two variables means they move together, but it does not prove one causes the other. The relationship could be coincidental, reversed, or driven by a hidden third variable (confounding factor).
| Scenario | Correlation? | Causation? |
|---|---|---|
| Smoking & lung cancer | Yes (r ≈ 0.7) | Yes (proven via studies) |
| Ice cream & drownings | Yes (r ≈ 0.5) | No (confound: temperature) |
| Height & shoe size | Yes (r ≈ 0.8) | Common cause: genetics |
The Pearson correlation coefficient r ranges from –1.0 (perfect negative linear relationship) to +1.0 (perfect positive linear relationship), with 0 indicating no linear association. An |r| of 0.9–1.0 is very strong (example: height vs. arm span), 0.7–0.9 is strong (study hours vs. exam scores), 0.5–0.7 is moderate, and below 0.3 is negligible. The sign tells direction: positive means both variables increase together; negative means one increases as the other decreases.
A critical limitation: r only measures linear relationships. Two variables can have a perfect nonlinear relationship (like a parabola) and still show r ≈ 0. Always visualize data with a scatter plot before interpreting correlation. A dataset where Y = X² for X from –10 to +10 has essentially zero Pearson correlation despite a perfect functional relationship. Use the Statistics Calculator for descriptive statistics on individual variables before running correlation analysis.
| |r| Range | Strength | Typical Example | R² (Variance Explained) |
|---|---|---|---|
| 0.9–1.0 | Very strong | Height vs. arm span | 81–100% |
| 0.7–0.9 | Strong | Study hours vs. grades | 49–81% |
| 0.5–0.7 | Moderate | Income vs. life satisfaction | 25–49% |
| 0.3–0.5 | Weak | Age vs. music preference | 9–25% |
| 0.0–0.3 | Negligible | Shoe size vs. IQ | 0–9% |
R² (the coefficient of determination) is simply the square of the Pearson r. It tells you the proportion of variance in Y that is explained by X. An r of 0.9 gives R² = 0.81, meaning 81% of Y’s variability is accounted for by the linear relationship with X. The remaining 19% is due to other factors, measurement error, or nonlinear effects.
The jump from r to R² is often surprising. A "moderate" correlation of r = 0.5 yields R² = 0.25 — only 25% of variance explained, meaning 75% of Y’s variation has nothing to do with X. This is why correlations below 0.7 (R² < 0.49) are generally too weak for reliable prediction. In academic research, R² values of 0.30–0.50 are common in social sciences, while physical sciences often achieve 0.90+ because relationships are governed by deterministic laws.
| Pearson r | R² | Variance Explained | Prediction Quality |
|---|---|---|---|
| ±0.3 | 0.09 | 9% | Very poor |
| ±0.5 | 0.25 | 25% | Weak |
| ±0.7 | 0.49 | 49% | Moderate |
| ±0.9 | 0.81 | 81% | Strong |
| ±1.0 | 1.00 | 100% | Perfect |
Linear regression finds the line y = mx + b that minimizes the sum of squared vertical distances (residuals) between the data points and the line. The slope m = Σ((xᵢ–x̅)(yᵢ–y̅)) / Σ((xᵢ–x̅)²) tells you how much Y changes per unit change in X. The intercept b = y̅ – m×x̅ is where the line crosses the Y-axis.
The regression line always passes through the point (x̅, y̅) — the means of X and Y. The sign of the slope always matches the sign of the correlation coefficient. A positive r means an upward-sloping line; a negative r means a downward-sloping line. For the dataset X={1,2,3,4,5}, Y={2,4,5,4,5}: r = 0.7746, R² = 0.6, and the regression equation is y = 0.6x + 2.2. For each unit increase in X, Y increases by 0.6 on average.
Never extrapolate far beyond your data range. A regression equation computed from X = 1 to 5 may not hold at X = 50. The linear relationship is only validated within the observed range of the data.
The most frequently cited statistical principle is often ignored in practice. A strong correlation between two variables does not prove that one causes the other. Three alternative explanations exist for any observed correlation: reverse causation (Y causes X instead of X causing Y), confounding (a third variable drives both X and Y), and coincidence (the correlation is spurious, especially in large datasets with many tested relationships).
The classic example: ice cream sales and drowning deaths correlate strongly (r ≈ 0.5). Ice cream does not cause drowning — summer heat (the confounding variable) increases both swimming and ice cream consumption. Establishing causation requires controlled experiments, natural experiments, or strong theoretical frameworks combined with techniques like instrumental variables. The Confidence Interval Calculator helps assess whether an observed correlation is statistically significant before investigating causality.
| Observed Correlation | Causal? | Explanation |
|---|---|---|
| Smoking ↔ lung cancer | Yes (proven) | Decades of controlled studies confirmed mechanism |
| Ice cream ↔ drownings | No | Confounding variable: temperature (summer) |
| Height ↔ shoe size | Common cause | Genetics and growth drive both |
| Pirates ↔ global warming | No (spurious) | Coincidence in time-series trends |
Last Updated: Mar 26, 2026
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