--- title: "Lines of Best Fit: Prediction and Its Limits" description: "Use a line of best fit to predict inside the range of the data, say why predicting far outside it is unsafe, and separate correlation from cause." canonical: https://lightmysky.com/learn/mathematics/lines-of-best-fit-prediction-and-its-limits-mt_CnYCx1a0j5 source: https://lightmysky.com/learn/mathematics/lines-of-best-fit-prediction-and-its-limits-mt_CnYCx1a0j5.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Lines of Best Fit: Prediction and Its Limits Use a line of best fit to predict inside the range of the data, say why predicting far outside it is unsafe, and separate correlation from cause. Subject: Mathematics · Area: Data & Statistics · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/lines-of-best-fit-prediction-and-its-limits-mt_CnYCx1a0j5 ## Ready when they can - Predict a value from the line and state the range the prediction is valid over - Explain what can go wrong when the line is extended past the data - Give an example of two correlated quantities with no causal link ## Lesson: How Far a Line May Be Trusted Last stop you cut a slice out of a histogram and called it an estimate, because the chart only knew what it had been told. A line of best fit has the same manners. Ten Wren Lane runners logged their training runs for the month before the race and their finishing time. The fewest was 2 runs, the most was 14, and the cloud leans down: more training, faster time. The line through the middle is time = 76 - 2 × runs. Those two end values matter as much as the line, because they mark where it has evidence. *(drawing: Four of the ten pairs, training runs first and minutes second. Every runner in the log did between 2 and 14 training runs.)* Inside that stretch the line does an honest job. A runner who did 7 training runs is not in the log, but 7 sits between 2 and 14, surrounded by runners on both sides. Put 7 into the line: 76 - 2 × 7 = 62 minutes. That is a reading, not a guess, because the line was drawn through people who trained a little more and a little less than that. Predicting inside the range of the data is called interpolation, and it is the safe half of the job. How tightly the points sit around the line is a separate question: wide scatter makes the reading rough. *(drawing: Between the two marks the line is a reading. Everywhere else along this axis, it is the line talking about people nobody measured.)* Now push it. A runner asks what 40 training runs would give: 76 - 2 × 40 = -4 minutes, a race finished before it started. Ask for 25 and it offers 26 minutes, faster than anyone has run this course. The arithmetic is faultless. The line has no idea that legs tire, that a 10k has a floor, or that nobody in the log trained more than 14 times. Predicting outside the range is called extrapolation, and the further out you go the less the data has to say. **Example.** The reason is that a straight line is a local description. Real relationships usually bend. The first few training runs change a great deal, the tenth changes less, and the fortieth might cause injury. Between 2 and 14 runs that bend is too gentle to see, so a straight line fits well. Out at 40 the bend is the whole story, and the line keeps going straight past it. One more limit, and it bites inside the range too. The kiosk log shows ice creams sold and wasp stings reported rising together all summer. Neither causes the other; hot days drive both. The running log has its own version: runners who bought new shoes finished faster, but buying shoes is what someone deep in training does anyway. Before saying one thing causes another, ask what else moves with both. **Recap.** A line of best fit predicts safely between the smallest and largest values the data actually covers, which is called interpolation. Push it beyond that range and you are extrapolating: the arithmetic still works, but the line is describing people nobody measured, and real relationships bend where a straight line cannot. Always state the range a prediction is good for. And two quantities moving together is not the same as one causing the other, so ask what else might be moving both. ## Practice 33 questions on this page, each with its working shown. ## Needs first - [Scatter Graphs & Correlation](https://lightmysky.com/learn/mathematics/scatter-graphs-and-correlation-mt_8atyuvPUZc) ## Opens up - [The Regression Line and Making Predictions](https://lightmysky.com/learn/mathematics/the-regression-line-and-making-predictions-mt_tsjctRmkyE) - [Straightening Data with Logarithmic Graphs](https://lightmysky.com/learn/mathematics/straightening-data-with-logarithmic-graphs-mt_TVar2jxRMI)