Scatterplots, model residuals and predictions
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| residual/rɪˈsɪdʒuːəl/ | 残差 | cán chà |
| extrapolation/ekˈstræpəleɪʃn/ | 外推 | wài tuī |
A decision before an answer
- A line predicting 18 when the observed value is 21 has a residual of +3. It does not pass through every observation.
- Your goal: Interpret slope and intercept of a fitted model in context.
Read the relationship
- A scatterplot pairs two quantitative variables, one point per paired observation. A fitted line summarises a trend rather than connecting each point. In a model y=a+bx, b predicts the change in y for one-unit increase in x; its units are y-units per x-unit. The intercept a predicts y when x=0, which may lie outside the data range.
- Calculate a residual and distinguish interpolation from extrapolation.
Model predicts 30; observation is 26. Residual:
26-30=-4.
Use the defining rule
- A residual is observed y minus predicted y. Positive means the observation lies above the model; negative means below. Substitute the observed x into the model before subtracting. A large residual marks a poor prediction for that point, not automatically an error in measurement or proof that the trend is absent.
- Choose linear or exponential patterns from equal-step data.
Observed x ranges from 2 to 8. Predicting at x=5 is:
5 is inside the observed range.
Check the conditions
- Interpolation predicts inside the observed x-range; extrapolation goes beyond it and needs stronger caution because the relationship may change. Use a model’s contextual domain as well as its algebraic form. A negative predicted travel time, for example, can reveal an unjustified extrapolation rather than a physically possible outcome.
- Choose linear or exponential patterns from equal-step data.
For study time x hours and predicted practice score y=12+3x, the slope is 3 score points/hour. At x=2 the prediction is 18. If the observed score is 21, residual=21-18=3. With x-values observed from 1 to 5, predicting at x=3 interpolates and x=12 extrapolates. Outputs 5,10,20,40 at equal input steps suggest factor-two exponential growth.
For y=4+2x, prediction at x=7 is ____.
4+2·7=18.
Apply the task format
- For equal input steps, nearly constant output differences suggest a linear model; nearly constant output ratios suggest an exponential model. Scatter can make either pattern approximate. Association does not identify a cause: both variables may reflect a third factor. A fitted trend is evidence about prediction, not by itself about intervention.
- Choose linear or exponential patterns from equal-step data.
Keep observed minus predicted in that order. Do not turn a model slope into a causal treatment effect.
Which answer fits this case?
Interpret slope and intercept of a fitted model in context
A strong scatterplot association alone proves x causes y.
Confounding or other explanations can produce association.
Keep the distinctions
- residual 残差 — Observed response minus the model’s predicted response.
- extrapolation 外推 — Prediction outside the range of observed inputs.
- Interpret slope and intercept of a fitted model in context.
- Calculate a residual and distinguish interpolation from extrapolation.
- Choose linear or exponential patterns from equal-step data.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.