Slope of a regression line

In $\hat{y} = a + bx$ the slope $b$ is the change in the predicted response for a one-unit increase in the explanatory variable. The word predicted is load-bearing: $b$ describes how the line moves, not what happens to any individual. The slope also carries units, always units of $y$ per unit of $x$, so a bare number is an incomplete answer.

Six students study 1, 2, 3, 4, 5 and 6 hours and score 60, 72, 65, 78, 71 and 86. The least-squares line is $\hat{y} = 58 + 4x$, and $b = 4$ points per hour. Said in context: for each additional hour of study, the model predicts a score about 4 points higher. It comes from $b = r \cdot \frac{s_y}{s_x}$, here $0.8107 \times \frac{9.2304}{1.8708}$, which is why $b$ and the correlation $r$ always share a sign but rarely share a size.

The interpretation that loses credit is "each extra hour of study raises a student's score by 4 points." It makes two claims the line cannot support. Raises is causal, and these are observational data where a student who studies more may differ in other ways. And it promises 4 points to an individual, while the six real students sit as much as 7 points off the line. Predicted change, not guaranteed change.

The slope is not symmetric in the two variables, which surprises people. Regress hours on score instead of score on hours and the slope is 0.1643 hours per point, not the reciprocal $1/4 = 0.25$. The two slopes multiply to $4 \times 0.1643 = 0.657$, which is $r^2$. The correlation, by contrast, is 0.8107 whichever variable you call explanatory. Rescaling changes the slope too: measure study time in minutes and $b$ becomes $4/60$ points per minute while $r$ does not budge.