Точки распределения, остатки модели и прогнозы
Введено| English | Значение |
|---|---|
| line of best fit/laɪn ɒv best fɪt/ | A line chosen to summarise the overall relationship between two variables in a scatter plot. |
| residual/rɪˈsɪdʒuːəl/ | остаток |
| extrapolation/ekˈstræpəleɪʃn/ | экстраполяция |
Predicting from a scatterplot
- A scatterplot shows hours studied ($x$) and test score ($y$) for 30 students, with the line of best fit 最佳拟合线 $y = 4.5x + 52$.
- The SAT asks you to interpret that line, measure how far a point is from it, and judge whether a prediction is safe.
Interpret the line of best fit
- Slope 4.5: each extra hour of study is associated with about 4.5 more points on average. Intercept 52: the predicted score for 0 hours.
- "Associated with" matters: a scatterplot shows a relationship, not proof that one thing causes the other.
Scatterplot with the line y = 4.5x + 52 and a residual 残差 of +6 at 6 hours
For houses, the line of best fit for price (thousand $) against floor area (m²) is$P = 2.1A + 35$. What does 2.1 mean?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
The slope is the average change in price per extra square metre.
Residuals
- Residual $=$ actual $-$ predicted. A student who studied 6 hours and scored 85: predicted $4.5(6) + 52 = 79$, residual $= 85 - 79 = 6$.
- A positive residual means the point is above the line; negative means below.
A student studied 4 hours and scored 65. Predicted: $4.5(4) + 52 = 70$. Residual: $65 - 70 = -5$ (5 points below the line).
Above, on or below the line?
Use the sign of a residual to place a point relative to the line.
The line of best fit is $y = 3x + 20$. A data point is $(10, 46)$. What is its residual?
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Predicted $3(10) + 20 = 50$; residual $46 - 50 = -4$.
A point below the line of best fit has a ____ residual.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Actual is less than predicted.
Interpolation versus extrapolation 外推
- Predictions inside the range of the data (interpolation) are more reliable than predictions far outside it (extrapolation). If the data cover 0–10 hours, predicting 40 hours (a score of 232) is meaningless.
- With equal steps in $x$, constant differences in $y$ suggest a linear model; constant ratios suggest an exponential model.
If data cover temperatures from 10 °C to 30 °C, a prediction for 80 °C from the same line is reliable.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
That is extrapolation far outside the data.
Traps
- Residual sign: compute actual minus predicted, not the other way round.
- Correlation as causation: the line describes an association in this data, not a cause.
A strong positive association in a scatterplot proves that the x-variable causes the y-variable to increase.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Association is not causation.
Recap
- Slope = average change in $y$ per unit of $x$; intercept = predicted $y$ at $x = 0$.
- Residual = actual − predicted; trust predictions inside the data range.