Scatter graphs, correlation and cautious predictions · Foundation
| English | 中文 | Pinyin |
|---|---|---|
| correlation/ˌkɒrɪˈleɪʃn/ | 相关性 | xiāng guān xìng |
Taller students may tend to have larger shoes. A plotted association does not prove that changing someone’s height would directly change their shoe size.
- Taller students may tend to have larger shoes. A plotted association does not prove that changing someone’s height would directly change their shoe size.
- This lesson studies correlation 相关性: A pattern of association between two measured variables.
Choose the mathematical structure
- Plot paired observations as points, with one variable on each axis. Positive correlation rises, negative falls and no correlation has no clear trend. Strong/weak describes how tightly points follow a trend, not its steepness. Draw an estimated line of best fit through the middle of the pattern, balancing points around it; it need not pass through the origin. Interpolation stays within observed inputs; extrapolation goes outside them.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines correlation?
A pattern of association between two measured variables.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Observed study hours 1,2,3,4,5 with scores 44,49,53,58,61 show a positive trend. An estimated line y=40+4.5x predicts 53.5 at x=3 and 56.2 at x=3.6. These inputs are inside 1–5, so the predictions interpolate. At x=10 the line gives 85, but this extrapolation may fail if gains flatten or the group differs. Sleep hours and tiredness may show a negative correlation; an almost horizontal cloud with no ordered pattern may show little correlation. A shared cause such as prior preparation can affect both study time and score; the plot alone does not establish causation.
Scatter graphs, correlation and cautious predictions
Plot paired observations as points, with one variable on each axis
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Use estimated y=40+4.5x to predict y at x=3.
40+4.5×3=53.5.
Test a tempting shortcut
- Do not join scatter points in observation order as though they formed a time series. A steep line need not imply strong correlation. An extrapolated formula value is a prediction whose assumptions need scrutiny.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A positive correlation proves that one variable causes the other. This claim is false. Explain which definition or assumption it violates.
Predict at x=10.
40+4.5×10=85, with extrapolation risk.
A positive correlation proves that one variable causes the other.
Do not join scatter points in observation order as though they formed a time series. A steep line need not imply strong correlation. An extrapolated formula value is a prediction whose assumptions need scrutiny.
Interpret a new situation
- AQA S6 Foundation includes correlation, estimated best-fit lines, predictions and dangers of extrapolation. State the observed input range, association direction/strength and the limits of a causal claim.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the width of the observed x range 1–5.
5-1=4; prediction at x=10 lies outside.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.6. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A pattern of association between two measured variables. Choose the relationship, show the method, check its assumptions and interpret the result.