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Interpreting Intercept and Slope
Common Core: 8.SP.3 S.ID.7

Interpreting Intercept and Slope

Author: Katherine Williams
Description:

Interpret the slope of a regression line.

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This tutorial covers how to interpret slope and intercept. The slope of a regression line can be represented by either b1 or m. So if you have y equals mx plus b, or y equals b sub 0 plus b sub 1 x. In either case, this m and this b sub 1 are representing the slope. And the slope is telling us for each unit change in x, so as x changes by 1 foot or 1 yard or 1 day or 1 deciliter-- whatever it is that x is changing by-- y increases or decreases by whatever number that slope is.

Now on the other hand, we have the intercept of the regression line. So the intercept is represented by this b here or this b sub 0 here. And the intercept is telling us what's the value of y if x is 0. Sometimes when you're trying to interpret that, it's not going to make sense. The x equals 0 is something crazy, or the x-values are so far away from 0 and all of your data points that to interpret it for 0 doesn't make any sense. Let's look at two examples.

Here in example 1, we're talking about humans. And we're saying that y is the glucose level, and x is the age. So this is our slope. And so it's telling us that as the age increases, so if the age is measured in years, as the age increases by one year, the glucose level increases by 0.5563. Now over here-- actually, we'll talk about intercept here.

The intercept is this plus b part over here-- the 60.082. That's telling us the y-value when x is 0. So it's telling us for someone who's 0 years old-- so for a baby, someone's who's just born-- you would have a glucose level of 60.082. In this case, that makes sense. You can't have someone who's 0 years old, and they would have a glucose level. And after that for every year, it goes up by the slope. It goes up by 0.5563.

In this next example we're talking about iPads. The iPads are saying the y is the cost, and the x is the megabytes. So here this slope is telling you how much the cost goes up when the megabytes increase. So it's essentially saying, what's the price of each megabyte? And each megabyte costs 4.17. Now you might not be able to buy an iPad with 52 megabytes or, sorry, gigabytes, depending on what we're talking about of data, however, you can interpret the slope that way, that each additional megabyte is costing you $4.17.

And the intercept is this 433. So it's telling you what the cost of some iPad with 0 megabytes would be. Now that doesn't necessarily make sense. You probably wouldn't buy one with 0 megabytes because you need the memory in order to do anything.

| it is telling you kind of the base price, what all the components cost before you start adding memory in. So that 433 doesn't necessarily make sense. You wouldn't want to buy an iPad without megabytes on it. But it can be interpreted still as the cost for the basis for all of the hardware and the development, the software-- things like that.

One important thing to remember here is that when interpreting slope, we're really looking at the average change. So here, this 0.5563, that's the average increase in glucose level for one unit increase in age. And the same thing over here in our iPad example, this 4.17 slope is the average increase in cost in the y for one increase in x in a megabyte increase.

So again, it's just the average change. It's not the exact change. So every time you increase by 1 megabyte, the cost doesn't necessarily increase by this exact 4.17. If we had a perfectly linear model, then, yes, that would be the same at every level. It would always increase by 4.17. But that's pretty rare in a real world model. So again, thinking about the average change for slope. So this has been your tutorial on interpreting slope and y-intercept.

Terms to Know
Slope of a least-squares regression line

The amount by which the response variable increases or decreases, on average, when the explanatory variable increases by one.

y-intercept of a least-squares regression line

The predicted value of the response variable when the explanatory variable is zero.