Reading the Model: Slope, Intercept and Correlation

In this lesson you will: State what the slope of a fitted model means in the units of the situation; State what the intercept means, and decide whether it means anything at all; Interpret a correlation coefficient computed with technology, and say what it cannot tell you.

The slope is a rate, and it carries two units

Chapter 9 taught you to read the m in y=mx+b as a rate of change in context. Nothing about that changes when the line comes from a scatter plot. The only new word is about, because a fitted line describes a trend rather than a rule.

The bike-share model is y=19.79x-487.16, with x in degrees Fahrenheit and y in rides. So the slope says: for each additional degree of daily high temperature, the model predicts about 19.8 more rides. Ten degrees warmer, about 198 more rides.

The intercept is the value at zero — if zero is a real possibility

The intercept is what the model predicts when x=0. Whether that prediction means anything depends entirely on whether x=0 makes sense and on how far it sits from the data.

For the used cars, y=-1607x+20107 gives $20,107 at age zero: the model's estimate for a brand-new car of that model. The youngest car in the data is one year old, so this is a short step beyond the evidence, and the number is at least the right kind of thing.

For the bike share, the intercept is -487 rides at 0^. Negative rides do not exist, and 0^ degrees is 52 degrees colder than the coldest day in the data. Here the intercept is a number the line needs in order to sit in the right place, and nothing more.

One number for direction and tightness: the correlation coefficient

Describing strength as "fairly strong" works until two people disagree. The correlation coefficient, written r, replaces the adjective with a number between -1 and 1. Its sign gives the direction. Its distance from zero gives the tightness.

You compute r with technology — the same regression screen that produced the line reports it. Ohio's standard says so explicitly, and no by-hand formula is expected of you.

For the bike data r=0.99: positive and very tight. For the used cars r=-0.99: the dots hug a falling line just as closely. For sleep against quiz score, r=0.60: a real upward drift in a wide band. For shoe size against the history quiz, r=0.01: nothing.

Moving together is not the same as one causing the other

A shop near Lake Erie tracked bottles of sunscreen sold and visits to the beach's first-aid tent over twelve summer days. The calculator reports r=0.997, about as tight as real data get. Nobody believes sunscreen causes scrapes and stings.

What both variables answer to is a third one: how hot and how busy the day was. Crowded days sell more sunscreen and also produce more minor injuries, so the two rise together without either touching the other.

When two variables move together, at least four explanations are in play, and a correlation alone cannot choose between them.

Common mistake — Treating a large r as proof of cause

The sunscreen data give r=0.997. Written up carelessly, that becomes "sunscreen sales drive first-aid visits," and the sentence sounds authoritative because it has a number in it.

The number is real and the association is real. What is invented is the arrow between them. A correlation coefficient is computed from two columns of numbers; nothing in that computation knows which column came first or what else was going on that day.

This is not a technicality. Newspapers, advertisements and study guides run this error constantly, and Ohio asks you to catch it.

How to catch it: Ask two questions before you accept any causal claim. First: did anyone actually change x on purpose and watch y respond, or were both simply observed? Second: can you name a third variable that would push both up together? If the answer to the first is no and to the second is yes, you have an association and nothing more — and the third variable you named is the sentence that earns the credit.