Why Square the Errors? | Intro to ML - Part 2

Why Linear Regression Squares Errors

The lesson explains why linear regression often squares prediction errors rather than using their absolute values, using a café’s cold-coffee sales as an example.

The café predicts servings from temperature using a straight line:

Predicted servings = W × temperature + B

For one day, the prediction is 55 servings while actual sales are 63, giving an error of 8 servings. Squaring the error gives a cost of 8² = 64.

Squaring makes large errors count disproportionately more. For example, one error of 8 has a squared cost of 64, while two errors of 4 have a combined squared cost of 32.

Finding the Least-Squares Line

Across 56 days, the squared errors are averaged to calculate a loss for each possible line. Each line is represented by its slope and intercept, and the loss is mapped across those parameter values.

The resulting loss surface forms a bowl, with its lowest point at the least-squares line. In the example, the loss is about 26.6, and the line predicts roughly 64 servings at 27°C. A hand-selected line has a loss around 27 and predicts 66 servings.

The absolute-error metric from the previous lesson selects a slightly different line, predicting around 66 servings. Different metrics do not necessarily select the same best line.

Method

1. Choose a candidate line by setting its slope (W) and intercept (B).
2. For each day, compare predicted servings with actual servings.
3. Square each error.
4. Average the squared errors to calculate the line’s loss.
5. Repeat for different values of W and B, then select the line at the lowest point of the loss surface.

The Effect of Unusually Large Errors

Squared error is sensitive to unusually large misses. In the example, a film crew orders 80 coffees on a cold day. That unusual event creates enough squared error to pull the fitted line substantially, while the absolute-error line changes little.

If such days are rare coincidences, absolute error may be more trustworthy.

A Benefit of Squared Error

One benefit of squared error is the shape of its loss surface: the bowl becomes flatter near its minimum. The lesson identifies this flatter bottom as an important property to explore next.

Speakers and Sources Featured

• Narrator/host: Presents the lesson and café examples.
• Viewer question: Referenced by the narrator; asks why errors are squared instead of measured as absolute errors. The viewer is not heard directly.
• Pips Café: The illustrative café and sales data used in the lesson.
• Film crew order: A hypothetical unusually large order used to demonstrate how an outlier can affect least-squares fitting.