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## AP®︎/College Statistics

### Course: AP®︎/College Statistics > Unit 5

Lesson 5: Analyzing departures from linearity- R-squared intuition
- R-squared or coefficient of determination
- Standard deviation of residuals or root mean square deviation (RMSD)
- Interpreting computer regression data
- Interpreting computer output for regression
- Impact of removing outliers on regression lines
- Influential points in regression
- Effects of influential points
- Identify influential points
- Transforming nonlinear data
- Worked example of linear regression using transformed data
- Predict with transformed data

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# Influential points in regression

AP.STATS:

DAT‑1 (EU)

, DAT‑1.I (LO)

, DAT‑1.I.1 (EK)

, DAT‑1.I.2 (EK)

, DAT‑1.I.3 (EK)

Outliers and high-leverage points can be influential to different measurements in least-squares regression like the slope, y-intercept, and correlation coefficient (r). Created by Sal Khan.

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## Video transcript

- [Instructor] I'm pretty sure I just tore my calf muscle this morning while sprinting with my son. But the math must not stop, (chuckles) so I'm here to help us think about what we could
call influential points when we're thinking about regressions. And to help us here, I have this tool from BFW Publishing. I encourage you to go here
and use this tool yourself. But 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 the least-squares
line right over there. And you can not only see the line, we can see our correlation coefficient. It's pretty good, 0.8156. It's pretty close to one. So we have a pretty good
fit right over here. But we're gonna think about our points that might influence, or it
might be overly influential, we could say, to different
aspects of this regression line. So one type of influential
point is known as an outlier. And a good way of identifying an outlier is it's a very bad fit to the line, or it has a very large residual. And so if I put a point right
over here, that is an outlier. So what happens when we
have an outlier like that? So before we had a correlation
coefficient of 0.8 something, you put one outlier like that out of, it's now one out 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. And 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, these 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? And something, a point whose X value is further away from the mean X value is considered a high leverage point. And 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. And you can see what I put an outlier, if I put 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. So it's highly influential for everything I just talked about. And 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's 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. And that kills the correlation
coefficient as well. Let's see what happens if
we do things over here. So if I have a high
leverage outlier over here, you see the same thing. A high leverage outliers
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
correlation coefficient, also known as the R value, which is of course the
square root of R squared.