Simple Linear Regression & OLS Line of Best Fit Calculator

Mathematical Foundations of Ordinary Least Squares (OLS) Regression

Linear regression models the relationship between an independent explanatory variable (X) and a dependent response variable (Y) by finding the straight line that minimizes the sum of squared vertical residuals:

• The Least Squares Criterion: The line minimizes \^2, ensuring the residuals sum to zero) and have zero covariance with X.

• Coefficient of Determination (R^2): Represents the proportion of variance in the dependent variable explained by the regression model (ranging from 0.0 to 1.0). An R^2 of 0.95 indicates that 95% of variability in Y is explained by X.

• Pearson Correlation Coefficient (r): Measures linear strength and direction). The slope m and correlation r always share the same sign).

• Standard Error of Estimate (S_e): Measures the standard deviation of data points around the fitted regression line, defining prediction interval dispersion.

How to Use This Calculator

Enter your X Values (Independent Variable, comma or space separated) (e.g. 1, 2, 3, 4, 5, 6, 7, 8), Y Values (Dependent Variable, comma or space separated) (e.g. 2.5, 3.8, 5.2, 7.1, 8.4, 10.1, 11.8, 13.5), and Forecast / Predict Y for X Value (e.g. 9). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Ordinary Least Squares (OLS) Simple Linear Regression equations:

x = (Σ x_i) / (n), y = (Σ y_i) / (n)

SS_xx = Σ (x_i - x)^2, SS_yy = Σ (y_i - y)^2, SS_xy = Σ (x_i - x)(y_i - y)

Slope m = β_1 = SS_xySS_xx

Y-Intercept b = β_0 = y - mx

Pearson Correlation r = SS_xy√(SS_xx) · SS_yy

Coefficient of Determination R^2 = r^2 = 1 - SS_resSS_total

Standard Error of Estimate S_e = √((Σ (y_i - y)_i)^2) / (n - 2)

Worked Example

For example, analyzing 8 paired observations \ and With \ and \ and . The calculated slope is \ and Y-intercept is \ = 0.7071, giving fitted equation \ with . For predicted .

Understanding Your Result

Statistical results require context: a mean alone is misleading without knowing the standard deviation (spread). Normal distribution reference: 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, 99.7% within ±3 SD. For small sample sizes (n < 30), use t-distributions rather than z-scores. Statistical significance (p < 0.05) does not imply practical significance — effect size (Cohen's d, R², odds ratio) measures real-world importance. Correlation coefficient r: |r| < 0.3 weak, 0.3–0.7 moderate, > 0.7 strong.