Linear Regression & Correlation Calculators

regression

Simple Linear Regression

Find the best-fit line ŷ = a + bx for paired data.

Example answerŷ = 0.6x + 2.2, R² = 0.6

This is the worked example that ships with the calculator. Replace the values above with your own and press Calculate.

X Values: 1, 2, 3, 4, 5
Y Values: 2, 4, 5, 4, 5

Simple linear regression: ŷ = a + bxFormulab = Σ(xᵢ-x̄)(yᵢ-ȳ) / Σ(xᵢ-x̄)², a = ȳ - b·x̄
n = 5
Means
x̄ = 3.0000, ȳ = 4.0000
Slope (b)Substituteb = 6.0000 / 10.0000
0.6000
Intercept (a)Substitutea = 4.0000 - 0.6000 · 3.0000
2.2000
Regression equation
ŷ = 2.2000 + 0.6000x
Correlation coefficient (r)
0.7746
Coefficient of determination (r²)Substituter² = 0.7746²
0.6000
Interpretation of r²
60.00% of the variation in y is explained by the linear relationship with x.
Residuals (yᵢ - ŷᵢ)
-0.8000, 0.6000, 1.0000, -0.6000, -0.2000

Free reference

Regression: the formulas

Simple linear regression fits the least-squares line through paired data so you can describe the relationship and predict y from x, while the correlation coefficient measures how tight that linear relationship is.

Least-Squares Regression Line: yhat = a + bx - Line that minimizes the sum of squared residuals.
Pearson Correlation: r = sum((x_i-xbar)(y_i-ybar)) / sqrt(sum((x_i-xbar)^2)sum((y_i-ybar)^2)) - Measures direction and strength of a linear association.

See the full formula reference

Worked example: Linear Regression

Five students studied 1, 2, 3, 4 and 5 hours and scored 2, 4, 5, 4 and 6 on a quiz. Fit the least-squares regression line predicting score from hours studied.

Inputs. x (hours): 1, 2, 3, 4, 5. y (score): 2, 4, 5, 4, 6.

Simple linear regression: ŷ = a + bx. b = Σ(xᵢ-x̄)(yᵢ-ȳ) / Σ(xᵢ-x̄)², a = ȳ - b·x̄ gives n = 5
Means. gives x̄ = 3.0000, ȳ = 4.2000
Slope (b). b = 8.0000 / 10.0000 gives 0.8000
Intercept (a). a = 4.2000 - 0.8000 · 3.0000 gives 1.8000
Regression equation. gives ŷ = 1.8000 + 0.8000x
Correlation coefficient (r). gives 0.8528
Coefficient of determination (r²). r² = 0.8528² gives 0.7273
Interpretation of r². gives 72.73% of the variation in y is explained by the linear relationship with x.
Residuals (yᵢ - ŷᵢ). gives -0.6000, 0.6000, 0.8000, -1.0000, 0.2000

Answer: ŷ = 0.8x + 1.8, R² = 0.7273

The fitted line is y-hat = 1.8 + 0.8x: each extra hour of study predicts 0.8 more points, and the linear relationship explains about 72.7% of the variation in scores.

Frequently asked questions

What does the slope of the regression line mean?

It is the predicted change in y for a one-unit increase in x. In the worked example the slope is 0.8, so each additional hour of study predicts 0.8 more quiz points on average.

What does R-squared tell you?

R-squared is the share of the variation in y that the linear relationship with x explains. The example gives R-squared = 0.7273, meaning about 72.7% of the differences in quiz scores are accounted for by hours studied.

What is the difference between correlation and regression?

Correlation (r) is a single unitless number between -1 and 1 that measures the strength and direction of a linear relationship. Regression goes further and produces an equation, y-hat = a + bx, that predicts y from x and quantifies the effect size in real units.