lm() — Linear regression

Draws the best straight line through a cloud of points.

lm(y ~ x)

Key idea

• Linear regression fits a straight line, written y = a + b*x, through the dots. (e.g. y = 2 + 3x)
• The slope b is how much y changes for each one-unit rise in x. (e.g. b = 3)
• The intercept a is the predicted y when x is 0. (e.g. a = 2)
• You can use the line to predict y at a new x value. (e.g. x = 4 -> y = 14)
• It assumes the relationship is roughly a straight line.

Definition

$$\hat{y}=a+bx,b=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$$

$x$

the predictor (what you know)

$\hat{y}$

the predicted value

$a$

the intercept

$b$

the slope

$$b=\frac{\sum (x-\bar{x})(y-\bar{y})}{\sum (x-\bar{x}{)}^2}\approx 4.6,\hat{y}\approx 45.2+4.6x$$

Where you'd use it

• From past data, lm might fit the line score ≈ 45 + 4.6 × hours. To predict a student who studies 6 hours, you plug in 6 and get about 73, turning study time into an expected exam score.
• Fitting ice-cream sales against temperature gives a line whose slope says each extra degree adds, say, 20 sales; plug in tomorrow’s forecast of 30 °C to estimate the day’s sales.
• For houses, lm learns price ≈ a + b × floor area; feeding in a 100 m² home returns a predicted price, but trusting it for a 1000 m² mansion far outside your data would be risky extrapolation.

Frequently asked questions

How do I read the slope?

The change in y for a one-unit increase in x.

Is a high R² enough?

No — check the residual plots, and remember correlation isn’t causation.