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 |
Correlation analysis is a fundamental statistical technique used to measure the strength and direction of the linear relationship between two variables. The Pearson correlation coefficient (r) quantifies this relationship on a scale from -1 to +1, making it easy to interpret and compare across different datasets.
While correlation tells you how strongly two variables are related, linear regression takes it a step further by finding the equation of the best-fit line. The regression equation y = mx + b allows you to predict Y values for given X values, with R-squared telling you how reliable those predictions are.
Our correlation calculator computes the Pearson r, R-squared, regression equation (slope and intercept), and provides a complete step-by-step calculation table. Enter your X and Y data sets separated by commas to instantly analyze the relationship between your variables.
Last Updated: Mar 9, 2026
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