Exploring Two-Variable Data

Least-Squares Regression

Fitting a regression line to bivariate data and interpreting slope and intercept.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Scatterplots & Correlation

Fitting a Line to the Car Value Data

The car age/value data from the last lesson showed an extremely strong negative linear relationship (r ≈ −0.998). Before reading on: with a relationship this strong, would a single specific line — one exact equation — be able to predict a car's value from its age fairly reliably?

Definition — Least-Squares Regression Line

The least-squares regression line ŷ = b₀ + b₁x is the line that minimizes the sum of squared prediction errors (residuals) across all the data points. Its slope, b₁ = Σ(x−x̄)(y−ȳ) / Σ(x−x̄)², reuses the same building blocks as the correlation coefficient — regression and correlation are closely related calculations.

b_1 = \frac{\sum (x-\bar{x})(y-\bar{y})}{\sum (x-\bar{x})^2}

Worked Example — Fitting and Interpreting the Regression Line

Using Σ(x−x̄)(y−ȳ) = −46 and Σ(x−x̄)² = 17.5 from the previous lesson: b₁ = −46/17.5 ≈ −2.63. The intercept: b₀ = ȳ − b₁x̄ ≈ 15.33 − (−2.63)(3.5) ≈ 24.53. The regression line: ŷ = 24.53 − 2.63x. The slope means: each additional year of age is associated with a resale value about $2,630 lower, on average. The intercept, $24,530, would be the predicted value for a brand-new (age 0) car — a modest extrapolation just past the observed data (which starts at age 1), reasonable here but still worth noting as an assumption rather than an observed value.

Worked Example — Checking the Fit

At age 1, the line predicts ŷ = 24.53 − 2.63(1) ≈ 21.90, versus the actual value of 22 — a residual of about 0.10. At age 6, the line predicts ŷ ≈ 24.53 − 2.63(6) ≈ 8.75, versus the actual value of 9 — a residual of about 0.25. Both predictions land very close to the actual values, consistent with the extremely strong correlation found in the previous lesson.

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Domain & range

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x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

Small residuals across every data point (as seen here) confirm a strong linear fit — if even one or two residuals were unusually large while the rest were small, that would be worth investigating rather than ignoring.

Common Mistakes

Using the regression line to predict a car's value at an age far beyond the observed data, like age 30.
A regression line is only trustworthy within (or very close to) the range of x actually observed — predicting age 30 assumes the same linear pattern continues indefinitely, which the data can't confirm and which isn't realistic for a car's value anyway (it can't go negative).

Key Takeaways

The least-squares regression line minimizes the sum of squared residuals, and its slope shares the same underlying calculation as the correlation coefficient.
Slope and intercept have specific, context-based interpretations tied to the variables' actual units.
Checking residuals at a few points is a quick way to confirm a line's fit matches what the correlation coefficient already suggests.

Summary

This closes the study of two-variable data. The next unit turns to how data should be collected in the first place, so conclusions drawn from it can be trusted.

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