Scatter diagrams and lines of best fit
Plot paired observations, describe correlation and use a sensible straight-line estimate within the data range.
Core and Extended.
Before you begin
- Plot coordinate pairs.
- Read linear scales.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What do rising scatter points usually suggest?
What you will learn
- Identify positive, negative or zero correlation.
- Draw a single line of best fit by eye and assess prediction limits.
Every point represents one paired observation
A scatter diagram links two measurements taken on the same individual or object. Plot each (x,y) pair on labelled axes. Points rising overall as x increases suggest positive correlation; falling points suggest negative correlation; no clear trend suggests zero or little correlation.
Correlation is a pattern of association, not proof of cause. A few outliers can affect the apparent pattern, and a small sample may be unreliable. Do not join the dots in observation order: that creates a different graphic and hides the point-cloud relationship.
A best-fit line summarises a roughly linear trend
For a roughly linear cloud, draw one ruled straight line by eye across the data range, balancing points approximately above and below. The line need not pass through any particular point or the origin. It should represent the whole trend rather than connect the first and last observations.
Use the line for an estimated y at a chosen x. Interpolation within the observed range is generally better supported than extrapolation far beyond it, where the relationship may change. If the pattern is not roughly linear, a straight-line prediction may be inappropriate. State that a prediction is an estimate.
Pause and explain
Which description fits a best-fit line drawn by eye?
Worked example
A cloud for x = 1 to 5 is roughly summarised by y = 2x + 1. Estimate y when x = 4.
Show the worked solution
- Four lies inside the observed x range, so this is interpolation.
- The line gives y = 2×4 + 1 = 9.
- Nine is an estimate from the trend, not necessarily an observed point.
Answer Estimated y = 9.
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Describe a cloud in which larger x tends to give smaller y.
Give me a hint
Look at the overall trend.
Compare my reasoning
- The cloud slopes down as x increases.
- The association is negative correlation.
- Do not infer a causal relationship from that description.
Negative correlation.
Look for these in your work
- Used the overall pattern.
- Separated correlation from cause.
A fitted line is y = 3x + 2 for observed x from 2 to 8. Estimate y at x = 6 and classify the prediction.
Give me a hint
Check the observed range first.
Compare my reasoning
- Six lies within two to eight.
- The line gives 3×6+2 = 20.
- This is interpolation and remains an estimate.
Estimated 20; interpolation.
Look for these in your work
- Substituted into the line.
- Identified the prediction type.
Why is using the same line at x = 50 less reliable when the data only range from 2 to 8?
Give me a hint
The trend has not been observed there.
Compare my reasoning
- Fifty is far outside the sample's x range.
- The relationship may change or physical limits may apply.
- It is distant extrapolation with weak direct support.
It is far extrapolation; the observed trend may not continue.
Look for these in your work
- Compared with the actual range.
- Explained the uncertainty.
Common mistakes
- Joining scatter points.
- Forcing the line through the origin.
- Treating far extrapolation as certain.
What can you explain now?
Must a line of best fit pass through two chosen data points?
Compare with the explanation
No.
It should summarise the whole cloud rather than be forced through arbitrary observations.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, sketch positive and negative clouds and mark one interpolated and one extrapolated estimate.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources