How does vehicle weight relate to fuel economy?
A heavier car may require more fuel to move. We can examine whether this pattern appears in real measurements.
Higher mpg means a car travels farther on one gallon.
The built-in R dataset mtcars contains measurements for 32 automobiles from the 1974 Motor Trend US magazine.
These are historical observations, not a random sample of today’s cars. Engine design, horsepower, and other characteristics may also matter.
Source: R’s datasets::mtcars documentation. No external data file is needed.
The downward pattern supports a negative association. The gold line summarizes the trend, while the points show variation around it.
For car \(i\), let \(x_i\) be weight and \(y_i\) be mpg:
\[y_i = \beta_0 + \beta_1 x_i + \epsilon_i.\]
The fitted line is \(\widehat y_i=b_0+b_1x_i\).
A zero-weight car is outside the data, so the intercept is a mathematical starting point rather than a practical prediction.
Least squares chooses the coefficients that minimize squared residuals:
\[\mathrm{SSE}=\sum_{i=1}^{n}(y_i-\widehat y_i)^2.\]
\[b_1=\frac{\sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)} {\sum_{i=1}^{n}(x_i-\bar x)^2},\qquad b_0=\bar y-b_1\bar x.\]
A residual is the observed mpg minus the fitted mpg. Squaring prevents positive and negative errors from canceling and gives larger errors more weight.
library(ggplot2) library(plotly) data(mtcars) fit <- lm(mpg ~ wt, data = mtcars) summary(fit) ggplot(mtcars, aes(x = wt, y = mpg)) + geom_point() + geom_smooth(method = "lm", se = FALSE) + labs(x = "Weight (1,000 lb)", y = "Fuel economy (mpg)")
lm() fits the regression. summary() reports its coefficients and fit. The line in the earlier chart uses the same regression model.
\[\widehat{\mathrm{mpg}}=37.29-5.34\times\mathrm{weight}.\]
An additional 1,000 lb is associated with approximately 5.34 fewer mpg in the fitted model.
At 3,000 lb, the fitted mean is 21.3 mpg. At 4,000 lb, it is 15.9 mpg.
\(R^2 = 0.753\): the fitted line accounts for about 75.3% of the observed variation in mpg in this dataset.
This describes association. It does not prove that changing weight alone causes this difference.
Cars above zero outperform the fitted line. Several high fitted-mpg cars have positive residuals, suggesting possible curvature. A straight line is useful, but imperfect.
Hover to identify a car. Click a legend entry to hide or show a cylinder group. The groups overlap, but heavier cars tend to have more cylinders.
In these 32 cars, heavier vehicles generally have lower fuel economy.
A useful next step would compare models that also include horsepower or cylinders, then evaluate predictions on new data.
Use this example to understand regression, rather than to predict today’s cars or the effect of modifying a specific car.
datasets::mtcars, Motor Trend Car Road Tests. Documentation credits Henderson and Velleman (1981), Biometrics, 37, 391–411. See ?mtcars in R.All analysis and plots are generated from the Rmd file. The dataset is included with R. No supporting image or CSV files are required.