Fitting a Line, and Using It to Predict

In this lesson you will: Draw a line that fits a scatter plot and find its equation from two points on that line; Obtain a line of best fit with a calculator's regression feature; Predict with a fitted model, and judge whether the prediction sits inside the reach of the data.

Draw the line first, read the equation off it second

A bike-share system in Columbus recorded the daily high temperature and the number of rides for ten days. The cloud rises and it is close to straight, so a line is a fair summary of it.

Lay a ruler so that roughly half the dots sit above it and half below, with the misses on each side about the same size. Then pick two points on your line — not two data points — and use the Chapter 8 slope formula. Here the line passes through (55,600) and (75,1000).

What the calculator's line adds

Two people fitting by eye will draw two slightly different lines, and both can be defensible. Statistics settles the argument with one rule: choose the line for which the total of how far off the dots sit is as small as it can be. That line is called the line of best fit, and every graphing calculator computes it. Chapter 10 has the keystrokes.

For the bike data the calculator reports y=19.79x-487.16. Set that beside the line fitted by eye, y=20x-500, and the predictions differ by a couple of rides across the whole range of the data. Fitting by eye is not a lesser method; it is the same idea done by hand.

Predicting inside the data, and beyond it

A prediction at 64^is safe. The data run from 52^ degrees to 78^ degrees, so 64 sits among readings you have, and the model gives 20(64)-500=780 rides.

A prediction at 20^ degrees is a different animal. The model returns 20(20)-500=-100 rides, which is impossible. Nothing went wrong with the algebra. The line was built from warm-weather days and knows nothing about February, so it kept marching downward past zero because that is what lines do.

Predicting inside the range of the data is interpolation, and the model was built for it. Predicting outside is extrapolation, and the further out you go the less the data have to say.

Common mistake — Running the model far past the data that built it

A model is a summary of the rows you fed it. It has no opinion about anything else, and it cannot warn you when you have left the neighborhood.

Ruth uses the bike-share line to plan for a 20^ degrees day and gets -100 rides. The negative number at least announces itself. The dangerous case is the prediction that comes back looking perfectly reasonable — 1400 rides at 95^ degrees — while the real world quietly stops cooperating, because above about 90^ degrees people ride less, not more.

The further outside the data you go, the more the answer depends on the straightness of a line nobody ever checked out there.

How to catch it: Write the smallest and largest x in the data beside every prediction you make. If your x falls outside that interval, label the answer as an extrapolation and say in one sentence what could break down. On a test, a prediction that is flagged and explained earns credit; the same number reported as fact does not.