Influential points in regression | AP Statistics | Khan Academy

I'm pretty sure I just tore my calf muscle this morning while sprinting with my son. But the math must not stop, so I'm here to help us think about what we could call influential points when we're thinking about regressions.

To help us here, I have this tool from BFW Publishing. I encourage you to go here and use this tool yourself. What it allows us to do is to draw some points. So just like that, let me draw some points and then fit a least squares line.

So that's a least squares line right over there. You can not only see the line, but we can see our correlation coefficient. It's pretty good: 0.8156. It's pretty close to 1, so we have a pretty good fit right over here.

But we're going to think about points that might influence or might be overly influential, we could say, to different aspects of this regression line.

One type of influential point is known as an outlier. A good way of identifying an outlier is that it's a very bad fit to the line or it has a very large residual. So if I put a point right over here, that is an outlier.

So what happens when we have an outlier like that? Before, we had a correlation coefficient of 0.8 something. You put one outlier like that out of, it's now one of 16 points. It dramatically lowered our correlation coefficient because we have a really large residual right over here.

So an outlier like this has been very influential on the correlation coefficient. It didn't impact the slope of the line a tremendous amount; it did a little bit. Actually, when I put it there, it didn't impact the slope much at all. It does impact the y-intercept a little bit. Actually, when I put it out here, it doesn't impact the y-intercept much at all. If I put it a little bit more to the left, it impacts it a little bit.

But these outliers that are at least close to the mean x value seem to be most relevant in terms of impacting or most influential in terms of the correlation coefficient.

Now, what about an outlier that's further away from the mean x value? Something, a point whose x value is further away from the mean x values, is considered a high leverage point. The way you could think about that is if you imagine this as being some type of a seesaw, somehow pivoted on the mean x value.

Well, if you put a point out here, it looks like it's pivoting down. It's like someone's sitting at this end of the seesaw, and so that's where I think the term "leverage" comes from. You can see, when I put an outlier—a high leverage outlier—out here, that does many things.

It definitely drops the correlation coefficient. It changes the slope and it changes the y-intercept, so it does a lot of things. It's highly influential for everything I just talked about.

Now, if I have a high leverage point that's maybe a little bit less of an outlier, something like this—based on the points that I happen to have—it didn't hurt the correlation coefficient. In fact, in that example, it actually improved it a little bit. But it did change the y-intercept a bit, and it did change the slope a bit, although obviously not as dramatic as when you do something like that, which then kills the correlation coefficient as well.

Let's see what happens if we do things over here. If I have a high leverage outlier over here, you see the same thing: a high leverage outlier seems to influence everything.

If it is a high leverage point that is less of an outlier, actually, once again it improved the correlation coefficient. You could say that it's still influential on the correlation coefficient; in this case, it's improving it. But it's less influential in terms of the slope and the y-intercept, although it is making a difference there.

So I encourage you to play with this. Think about different points—how far they are away from the mean x value, how large of a residual they have, are they an outlier, and how influential they are to the various aspects of a least squares line: the slope, the y-intercept, or the correlation coefficient.

When we're talking about the correlation coefficient, also known as the r value, which is, of course, the square root of r squared.