Linear Regression Calculator
Find the linear relationship between two variables by computing the slope, correlation coefficient, and R² from paired data points. Use it when analyzing trends in sales, scientific experiments, or any dataset where you want to predict one variable from another.
Last updated: May 2026
About this calculator
Simple linear regression models the relationship between an independent variable X and a dependent variable Y using the equation Ŷ = a + bX, where b is the slope and a is the intercept. The slope is calculated as b = (n·ΣXY − ΣX·ΣY) / (n·ΣX² − (ΣX)²). The Pearson correlation coefficient r = (n·ΣXY − ΣX·ΣY) / √[(n·ΣX² − (ΣX)²)(n·ΣY² − (ΣY)²)] measures the strength and direction of the linear association, ranging from −1 to +1. The coefficient of determination R² = r² tells you what proportion of variance in Y is explained by X. For example, R² = 0.81 means 81% of the variation in Y is accounted for by the linear model. These metrics together describe both the direction and the predictive power of the relationship.
How to use
Suppose you have three data points: X = 1, 2, 3 and Y = 2, 4, 5. Step 1: n = 3, ΣX = 6, ΣY = 11, ΣXY = (1×2)+(2×4)+(3×5) = 25, ΣX² = 14, ΣY² = 45. Step 2: Slope b = (3×25 − 6×11) / (3×14 − 6²) = (75 − 66) / (42 − 36) = 9 / 6 = 1.5. Step 3: r = (3×25 − 6×11) / √[(3×14 − 36)(3×45 − 121)] = 9 / √[6 × 14] = 9 / √84 ≈ 0.982. Step 4: R² ≈ 0.964, meaning 96.4% of the variance in Y is explained by X.
Frequently asked questions
What does the correlation coefficient r tell me about my data?
The correlation coefficient r measures both the strength and direction of the linear relationship between two variables, ranging from −1 to +1. A value near +1 indicates a strong positive relationship (as X increases, Y increases), while a value near −1 indicates a strong negative relationship. Values close to 0 suggest little to no linear association. Note that r only captures linear relationships — two variables can have a strong curved relationship yet show r ≈ 0. Always visualize your data with a scatter plot alongside calculating r.
What is R-squared and what is a good R-squared value?
R² (the coefficient of determination) equals r² and represents the proportion of variability in Y that is explained by the linear regression model. An R² of 0.75 means 75% of the variation in Y is accounted for by X. What constitutes a 'good' R² depends heavily on the field — physical sciences often expect R² above 0.99, while social sciences may consider 0.3 meaningful. A high R² does not guarantee the model is appropriate; always check residual plots to verify linear assumptions are met.
How is the slope of a regression line used for prediction?
The slope b tells you how much Y is expected to change for each one-unit increase in X. Once you have the slope and intercept from the regression, you can substitute any new X value into Ŷ = a + bX to generate a predicted Y. For instance, a slope of 2.5 in a height-weight regression means each additional centimeter of height is associated with 2.5 kg more weight on average. Predictions are most reliable when the new X falls within the range of your original data; extrapolating far beyond that range increases the risk of inaccurate predictions.