Linear Regression Visual Tool Math

y = ...
r = ...
r² = ...

Click on the canvas to add a point. Drag existing points to move them. The regression line updates automatically.

What is the Linear Regression Visual Tool?

Quick Answer: An interactive scatter plot that fits a least-squares regression line (y = mx + b) to points you place by clicking. Drag points to see how the slope, intercept, correlation coefficient (r), and R-squared change instantly. Residuals are shown as dashed lines.

Theory of Linear Regression

Simple linear regression finds the line that minimizes the sum of squared vertical distances (residuals) between observed y values and predicted y values. Given n points (xi, yi), the slope m = Σ(xi - x¯)(yi - y¯) / Σ(xi - x¯)2, and intercept b = y¯ - m x¯. The Pearson correlation coefficient r measures linear association strength (-1 to 1). R-squared = r2 is the proportion of variance in y explained by x. This tool demonstrates how adding or moving points changes the fit.

Step-by-Step Examples

Example 1: Perfect Positive Correlation

  1. Click a few points that lie almost on a straight rising line. The r value should approach 1.
  2. Observe the line passing through the centroid (mean x, mean y).

Example 2: Outlier Effect

  1. Create a tight cluster of points with positive trend, then add a single point far away in the opposite direction.
  2. Watch how the regression line tilts and r decreases. This shows the high leverage of outliers.

Example 3: No Correlation

  1. Add points randomly scattered across the canvas. The line will be nearly horizontal and r close to 0.

Frequently Asked Questions

How do I add data points?

Simply click anywhere on the canvas grid. A new point appears, and the regression statistics update immediately.

Can I move existing points?

Yes, click and drag any point to a new location. The line and residuals recalculate live as you drag.

What do r and R-squared mean?

r is the correlation coefficient (-1 to 1). R-squared = r² is the proportion of y-variance explained by x; e.g., 0.8 means 80% explained.

What are the dashed vertical lines?

They are residuals: the vertical distance from each point to the regression line. Toggle them off with the checkbox if desired.

Is this only for y vs x regression?

Yes, it models y as a linear function of x. For multiple regression or other curves, more advanced tools are needed.