---
title: "The Regression Line and Making Predictions"
description: "Fit the least-squares line through the data, read its gradient and intercept in the units of the problem, and predict only inside the range the data covers."
canonical: https://lightmysky.com/learn/mathematics/the-regression-line-and-making-predictions-mt_tsjctRmkyE
source: https://lightmysky.com/learn/mathematics/the-regression-line-and-making-predictions-mt_tsjctRmkyE.md
retrieved: 2026-09-12
---

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# The Regression Line and Making Predictions

Fit the least-squares line through the data, read its gradient and intercept in the units of the problem, and predict only inside the range the data covers.

Subject: Mathematics · Area: Data & Statistics · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/the-regression-line-and-making-predictions-mt_tsjctRmkyE

## Ready when they can

- Interpret the gradient of a fitted line in the units of the situation
- Predict a value for an x inside the range of the data
- Explain why a prediction well outside that range is not supported

## Lesson: Fitting a line, trusting it wisely

A fitted line reads y equals a plus b x. The intercept a is the predicted y when x hits zero. The gradient b is the predicted change in y for each extra unit of x. Both live in the units of the story, such as marks per hour. The line sums up average drift, and no single pupil must obey it.

**Example.** A class line for marks on hours studied is y equals 5 plus 2 x. For 6 hours, swap in 6 to get 5 plus 12, which is 17 marks. The data cover x from 1 to 8, and 6 sits inside that range, so the value is supported.

**Tip.** Outside the observed range the line guesses into unknown land. Curves and ceilings may lurk there. Kim predicts at x equals 30 and trusts it fully, but 30 sits far past 8. State the value, then judge it: supported inside, unsupported outside.

**Recap.** Read the gradient in story units, predict inside the range, and distrust guesses far outside it.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Lines of Best Fit: Prediction and Its Limits](https://lightmysky.com/learn/mathematics/lines-of-best-fit-prediction-and-its-limits-mt_CnYCx1a0j5)
- [Forms of the Equation of a Straight Line](https://lightmysky.com/learn/mathematics/forms-of-the-equation-of-a-straight-line-mt_dhIIxdFwcp)
- [Correlation and the Product-Moment Coefficient](https://lightmysky.com/learn/mathematics/correlation-and-the-product-moment-coefficient-mt_Rs_Z0P7ELJ)

## Opens up

- [Orthogonal Projection and Least Squares](https://lightmysky.com/learn/mathematics/orthogonal-projection-and-least-squares-mt_kGvRrYJrat)
- [Least-Squares Calibration and the Uncertainty of a Result Read From It](https://lightmysky.com/learn/science/least-squares-calibration-and-the-uncertainty-of-a-result-read-from-it-mt_L08U-Extq_)
- [Linear Regression by Least Squares](https://lightmysky.com/learn/computing/linear-regression-by-least-squares-mt_QRRkpIrwWs)
- [Simple Linear Regression and Inference on the Slope](https://lightmysky.com/learn/mathematics/simple-linear-regression-and-inference-on-the-slope-mt_W2huNTKsMj)