Math foundations
Least Squares: Why Squared Error Picks One Best Line
The idea
Many lines can pass near your points. Least squares picks the one that minimizes the sum of squared residuals. Squaring penalizes big misses more than small ones and keeps the math differentiable enough to solve in closed form.
Least squares answers: Which line makes the total squared prediction error smallest?
Example: total squared error as you move the line
Each point contributes (actual − predicted)². Least squares minimizes the sum.
Your SSE
0.08
Minimum SSE
0.08
Total squared error = 0.08. This is the minimum for these points. Least squares picks the line where squared residuals sum to the smallest value.
The math
Sum of squared errors
Each point contributes its vertical distance to the line, squared.
Objective
The best-fit line from the linear models explorer is the SSE minimizer for those points.
Sensitivity
One bad week moves the line more than one modest miss. Robust methods use absolute error when outliers dominate.
A simple application
When you report a trend line, you are implicitly reporting the SSE-minimizing line unless you chose another loss on purpose. Pair with overfitting and holdout posts before you trust SSE on training data alone.