Slope, intercept and prediction

What you will be able to do

• find the slope $b = r\,\dfrac{s_y}{s_x}$ and the intercept $a = \bar{y} - b\bar{x}$;
• say what the slope and the intercept mean, with their units;
• predict inside the data, and explain the danger of extrapolation.

A running club

A made-up running club asks seven members how far they train each week and times them over $5$ km.

The pattern falls: members who train more run faster. $r = -0.9942$.

The recipe

The least-squares line $\hat{y} = a + bx$ has

Here $s_x$ and $s_y$ are the standard deviations. We use the sample ones, with $n - 1$, but the ratio $\dfrac{s_y}{s_x}$ is the same with $n$. The second formula says that the line always passes through the point of the means $(\bar{x}, \bar{y})$.

Why $r$ times $\dfrac{s_y}{s_x}$? The ratio turns a step in $x$ into the matching step in $y$. The factor $r$ shrinks it when the pattern is loose. If $r = 0$, the slope is $0$ and the best prediction is simply $\bar{y}$.

For the club, $\bar{x} = 27.14$ km, $\bar{y} = 27.79$ minutes, $s_x = 14.10$ km and $s_y = 2.797$ minutes. So

• $b = -0.9942 \times \dfrac{2.797}{14.10} = -0.1972$ minutes per km;
• $a = 27.79 - (-0.1972) \times 27.14 = 33.14$ minutes.

The line is $\hat{y} = 33.14 - 0.1972x$.

Reading the slope and the intercept

The slope has the unit of $y$ per unit of $x$. Here: each extra kilometre of training a week goes with a race time about $0.197$ minutes (about $12$ seconds) shorter, on average. Ten more kilometres go with about $2$ minutes less.

The intercept is the predicted $y$ when $x = 0$: a member who does not train at all is predicted to take $33.14$ minutes. Is that sensible? The nearest data is at $10$ km, so it is a guess just outside the data. Often the intercept has no real meaning. For the canteen, $\hat{y} = -9.13 + 2.432x$ predicts $-9.13$ drinks at $0\ ^\circ\text{C}$, which is impossible. It only fixes the height of the line.

Predicting, and its limits

To predict, put a value of $x$ into the line. For $35$ km a week, $\hat{y} = 33.14 - 0.1972 \times 35 = 26.24$ minutes. That is interpolation: $35$ km lies inside the measured range, $10$ to $50$ km.

For $150$ km a week, the line predicts $3.55$ minutes for $5$ km, faster than any human has ever run. That is extrapolation, a prediction outside the range of the data. The line was fitted only between $10$ and $50$ km. Beyond that, nothing says that the pattern stays straight. Here it clearly does not: race times level off, they cannot fall forever.

For the canteen, $27\ ^\circ\text{C}$ gives $\hat{y} = -9.13 + 2.432 \times 27 = 56.5$ drinks. That is inside the data, and fair to use.

A last warning. The line predicts $y$ from $x$. To predict the training distance from a race time, you need another line, with the roles swapped. The Maths course on Correlation and Regression works with both lines.

Your turn: new numbers

Try it in Python

The program builds the line of the running club from $r$, the two standard deviations and the means, and then predicts twice.

The usual mistakes

• Turning the ratio upside down. The slope is $r$ times $s_y$ over $s_x$.
• Forgetting the units of the slope. Say "minutes per km", not just $-0.197$.
• Trusting every intercept. If $x = 0$ is far from the data, the intercept is only the height of the line.
• Extrapolating. A line fitted between $10$ and $50$ km says nothing reliable about $150$ km.

Remember these

• $b = r\,\dfrac{s_y}{s_x}$ and $a = \bar{y} - b\bar{x}$.
• The line passes through $(\bar{x}, \bar{y})$.
• The slope is the change in $\hat{y}$ for each extra unit of $x$.
• Predict inside the data (interpolation). Be very careful outside it (extrapolation).

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. For some data, $r = 0.9$, $s_x = 3\ ^\circ\text{C}$ and $s_y = 6$ drinks. Find the slope of the least-squares line, with its unit.

2. A least-squares line has slope $2$, and the means are $\bar{x} = 10$ and $\bar{y} = 30$. Find the intercept.

3. What does the slope $-0.1972$ minutes per km of the running club mean in words?

4. The canteen line is $\hat{y} = -9.13 + 2.432x$, from days of $18$ to $32\ ^\circ\text{C}$. Would you use it for a day of $45\ ^\circ\text{C}$?

5. What is the least-squares line when $r = 0$?

References

• OpenStax, Introductory Statistics, Chapter 12 (Linear Regression and Correlation), the sections on the regression equation and on prediction.