Скачать или смотреть Why We Square the Error: The Core Principle of Linear Regression

In linear regression, we square the differences between the predicted values and the actual data points (the "errors") for one primary reason: to penalize large errors more severely than small ones. This is known as the Ordinary Least Squares (OLS) method.

Here’s a simple breakdown of why squaring is so effective:

It Makes Errors Positive: Squaring ensures all error values are positive, so a point 2 units above the line doesn't cancel out a point 2 units below it.

It Heavily Penalizes Outliers: A large error (e.g., 10 units off) becomes 100 when squared, forcing the model to work hard to avoid such big mistakes.

It's Mathematically "Nice": The squared error function is smooth and differentiable, making it easy to find the unique "best-fit" line using calculus.

While alternative methods (like using absolute value) exist, the squared error is the standard because it provides a unique, stable, and computationally efficient solution that works best for data without extreme outliers.