Interpreting Slope and Intercept
A regression line is only useful once you can read it. In
\hat y = a + b x,
the two numbers a and b each carry a plain
meaning in the units of the data — and together they let you turn any
x into a prediction.
The slope b
The slope b is the change in the predicted
\hat y for a one-unit increase in x.
If a line predicts exam score from hours studied as
\hat y = 50 + 3x, the slope 3 says: each
extra hour of study is associated with 3 more points on the predicted score. A
negative slope means \hat y falls as x rises.
The intercept a
The intercept a is the predicted \hat y
when x = 0 — where the line crosses the vertical axis.
In \hat y = 50 + 3x it predicts a score of
50 for someone who studied zero hours.
But beware: x = 0 is often outside the range of the data,
or physically meaningless. A line fitted to adult heights and weights might report a "weight at
height zero" — a number with no sensible interpretation. Reading the intercept then is
extrapolation, and should be done with care or not at all.
Predict by plugging in
To predict, substitute a value of x into the line. Set the slope and
intercept, then slide the prediction point along the
x-axis: the marker rides the line and the readout shows
\hat y = a + b x at that x. Notice how a
steeper slope makes the prediction climb faster, and the intercept shifts the whole line up or
down.
- Slope b: the change in \hat y per one-unit increase in x.
- Intercept a: the predicted \hat y when x = 0 — meaningless if x = 0 is far outside the data.
- Predict by plugging an x into \hat y = a + b x.