Linear Regression & Error Bowl

Linear Regression & Error Bowl connects statistics and optimization in a split 3D studio. On the left board, six draggable scatter points—including a high-leverage orange outlier—are fitted by a trial line y = mx + b. Attached to each point is a literal red square whose side equals the vertical residual |yᵢ − (mxᵢ + b)|, making its area equal to the squared error eᵢ². On the right stage, the total area of all squares is plotted as a 3D paraboloid bowl SSE(m, b) over slope and intercept parameter space. Students can drag the parameter marker across the bowl or toggle Snap to Optimal OLS to watch the marker settle into the unique global minimum (m̂, b̂) while monitoring r and R².

• Adjust the Trial slope m and Trial intercept b sliders or drag the gold ball on the 3D bowl to shrink the red residual squares as much as possible.
• Turn on Snap to Optimal OLS and verify that the gold parameter ball lands precisely on the green OLS Minimum marker at the bottom of the error bowl.
• Drag the orange outlier point sharply downward on the scatter plot and watch how strongly a single outlier pulls the optimal slope m̂ and lowers R².
• Line up all six scatter points into a nearly straight line and watch Correlation r approach 1.000 and SSE approach zero.
• Toggle Show residual squares on and off to compare vertical residuals eᵢ against their squared areas eᵢ².

1. 1Explain why Ordinary Least Squares (OLS) minimizes the sum of squared vertical residuals Σ (yᵢ − ŷᵢ)² rather than signed residuals.
2. 2Connect the algebraic regression parameters (slope m and intercept b) to coordinates on a convex 3D loss surface SSE(m, b).
3. 3Evaluate how outliers and high-leverage points distort the least-squares regression line and reduce the coefficient of determination R².
4. 4Interpret correlation r and R² as measures of linear association and explained variance.

Questions teachers ask