Main content

## Statistics and probability

### Unit 5: Lesson 6

More on regression- Squared error of regression line
- Proof (part 1) minimizing squared error to regression line
- Proof (part 2) minimizing squared error to regression line
- Proof (part 3) minimizing squared error to regression line
- Proof (part 4) minimizing squared error to regression line
- Regression line example
- Second regression example
- Calculating R-squared
- Covariance and the regression line

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Squared error of regression line

Introduction to the idea that one can find a line that minimizes the squared distances to the points. Created by Sal Khan.

## Want to join the conversation?

- Why is the error measured VERTICALLY? The shortest distance from a point to a line will be on a perpendicular to that line. If we want to find the "best" fitting line to a set of points then this distance is the one that should be minimized.(151 votes)
- Don't think about it as a problem of geometry, but as one of minimizing error.

In this kind of line fitting, the x-values are assumed to have no error - if your independent variable has been measured with errors then you need a different fitting method.

Since the error for each point is only in its y-value (the dependent measurements), we only examine the vertical distance from each point to the line.(233 votes)

- Beginning at5:20, why do we take the squared error value instead of the actual error value (or its absolute value)?(69 votes)
- If you neither square nor take absolute values, then you can drive the sum of squared errors very low (below zero, all the way to negative infinity if you want) by choosing a line that is very far away from the data.

If you take absolute values of the errors before summing instead of squaring the errors, you still get an interesting problem. It's just much harder to find the optimal settings of the slope and intercept. In particular, it turns into a problem called a "Linear Program", which takes some iteration to solve. That doesn't mean it can't be done--there are many programs that can do it. One of them is Excel's Solver add-in.

There are also some nice theoretical properties that you get if you square the errors. See the Gauss-Markov theorem in particular.(58 votes)

- Why do we square the error? (4:53) I know that it exaggerates/differentiates between large and small errors, but if this is so, why not cube the errors? Or better yet, exponentiate the error to its thirteenth power? Is it because of the Gauss-Markov Theorem that Wikipedia mentions?(23 votes)
- the reason we choose squared error instead of 3rd or 4th power or 26th power of the error is because of the nice shape that squared errors will make when we make a graph of the squared error vs m and b. The graph will make a 3-d parabola with the smallest square error being at our optimally chosen m and b. Since this graph has only 1 minimum value it is really nice since we can always find this minimum, and the minimum will be unique. If we use higher exponents it would be harder to find the minimum value(s), and we could find possibly non unique minimums or only local minimums (values that look good compared to the neighbouring values but not the absolute best). So, in summary we used squared error because it gives us a minimum that is easy to find and is guaranteed to be the only minimum (this guarantees it is the best!).(59 votes)

- Why would somebody be interested in making this line? or what is the underlying idea of creating a Regression Line?(7 votes)
- Here is one reason: if you can find a formula that describes this line, you can predict one value, say y, based on any given x. For example, if I am measuring the average association between height and weight and I can find the equation for the line, then if I am given a height I will be able to say what, on average, is the corresponding weight. Now, this assumes that the sample I used to derive this line is representative of the population to which my prediction applies.(25 votes)

- Is error the same thing as residual?(10 votes)
- Not EXACTLY. The difference is that there is (in theory) a TRUE regression line which we will never know, and then there is the one that we estimate to be the regression line. The difference between the point and the TRUE regression line is your error. The difference between your point and the ESTIMATED regression line is your residual. When we fit a regression line, we make the sum of our residuals equal to 0 but that does not necessarily mean that the sum of our error is 0 (there will always be some error in statistics by its nature).

Here is an example that hopefully won't confuse.. You can see this if you simulate some data in a spreadsheet. If you had a function Y=5+10X as your TRUE regression and say we had 5 different possible inputs for x ( 1,2,3,4,5) now, for x=1, y=15 and for x=2, y=25 etc. Now, if for observation, we had some normally distributed error, and we fitted a line to that data, we might get a line that is reaaallly close to Y=5+10X but we probably would not get that exactly. We might actually get something like Y=5.3+9.5X. Lets say one of our randomly distributed observations were (3,30). the predicted value for Y, according to our regression line, is 5.3+9.5(3)=33.8 but the TRUE regression line (which we won't actually have) would have predicted 35 for Y so, our ERROR is 30-35=5, but our RESIDUAL is 30-33.8=3.8(15 votes)

- Is it just me or does this process look a lot like taking the "variance" of the errors, with the line of best fit being where the errors vary the least?(7 votes)
- I had the same thought, and I would love a video exploring the relationship between the two in depth.(3 votes)

- At around3:00, why is the error at the different points the vertical distance, and not the horizontal distance between the point and the line? Or, for that matter, why not the vertical AND horizontal distances summed up?(4 votes)
- A linear regression model assumes that the relationship between the variables y and x is linear (the measured variable y depends linearly of the input variable x).

Basically, y = mx + b.

A disturbance term (noise) is added (error variable "e").

So, we have y = mx + b + e. So the error is e = y - (mx +b).

So, we try to find m and b (for the line of best fit) that minimize the error, that is the sum of the vertical squared distance Sum(||e||^2) = Sum(||y - (mx +b)||^2).

There are different ways of trying to find a line of best fit. It depends of what x and y represent.

For example, if both x and y are observations with errors and x and y have equal variances then what is called the "Deming regression" measured the deviations perpendicularly to the line of best fit.(6 votes)

- is there a lesson on nonlinear regressions?(3 votes)
- No, there isn't. However, if you have data which exhibits a nonlinear relationship, then more often than not you're best course of action is to
*transform*the data itself and proceed with a linear regression.(4 votes)

- Why are you interested in the squared error of the line as opposed to just the error of the line?(5 votes)
- the reason we choose squared error instead of 3rd or 4th power or 26th power of the error is because of the nice shape that squared errors will make when we make a graph of the squared error vs m and b. The graph will make a 3-d parabola with the smallest square error being at our optimally chosen m and b. Since this graph has only 1 minimum value it is really nice since we can always find this minimum, and the minimum will be unique. If we use higher exponents it would be harder to find the minimum value(s), and we could find possibly non unique minimums or only local minimums (values that look good compared to the neighbouring values but not the absolute best). So, in summary we used squared error because it gives us a minimum that is easy to find and is guaranteed to be the only minimum (this guarantees it is the best!).(2 votes)

- why is this called linear "regression"? What does regression mean?(4 votes)
- The term "regression" was used by Francis Galton in his 1886 paper "Regression towards mediocrity in hereditary stature". To my knowledge he only used the term in the context of regression toward the mean. The term was then adopted by others to get more or less the meaning it has today as a general statistical method.(5 votes)

## Video transcript

In the next few videos I'm going
to embark on something that will just result in
a formula that's pretty straightforward to apply. And in most statistics classes,
you'll just see that end product. But I actually want to show
how to get there. But I just want to warn
you right now. It's going to be a lot
of hairy math, most of it hairy algebra. And then we're actually going
have to do a little bit of calculus near the end. We're going to have to do a
few partial derivatives. So if any of that sounds
daunting, or sounds like something that will discourage
you in some way, you don't have to watch it. You could skip to the end and
just get the formula that we're going to derive. But I, at least, find it
pretty satisfying to actually derive it. So what we're going to think
about here is, let's say we have n points on a
coordinate plane. And they all don't have to
be in the first quadrant. But just for simplicity of
visualization, I'll draw them all in the first quadarant. So let's say I have this
point right over here. Let me do them in different
colors. And that coordinate is x1, y1. And then let's say I have
another point over here. The coordinates there
are x2, y2. And then I can keep
adding points. And I could keep drawing them. We'd just have a
ton of points. There and there and there. And we go all the way
to the nth point. Maybe it's over here. And we're just going to
call that xn, yn. So we have n points here. I haven't drawn all of
the actual points. But what I want to do is find
a line that minimizes the squared distances to these
different points. So let's think about it. Let's visualize that
line for a second. So there's going to
be some line. And I'm going to try to draw a
line that kind of approximates what these points are doing. So let me draw this line here. So maybe the line might look
something like this. I'm going to try my best
to approximate it. Actually, let me draw it
little bit different. Maybe it looks something
like that. I don't even know what it
looks like right now. And what we want to do is
minimize this squared error from each of these points
to the line. So let's think about
what that means. So if the equation of this line
right here is y is equal to mx plus b. And this just comes straight
out of Algebra 1. This is the slope on the line,
and this is the y-intercept. This is actually
the point 0, b. What I want to do, and that's
what the the topic of the next few videos are going to be, I
want to find an m and a b. So I want to find these two
things that define this line. So that it minimizes
the squared error. So let me define what
the error even is. So for each of these points, the
error between it and the line is the vertical distance. So this right here we
can call error one. And then this right here
would be error two. It would be the vertical
distance between that point and the line. Or you can think of it as the y
value of this point and the y value of the line. And you just keep going all
the way to the endpoint between the y value of
this point and the y value of the line. So this error right here, error
one, if you think about it, it is this value right
here, this y value. It's equal to y1 minus
this y value. Well what's this y value
going to be? Well over here we have
x is equal to x1. And this point is the
point m x1 plus b. You take x1 into this equation
of the line and you're going to get this point
right over here. So that's literally going to
be equal to m x1 plus b. That's that first error. And we can keep doing it
with all the points. This error right over here
is going to be y2 minus m x2 plus b. And then this point right
here is m x2 plus b. The value when you take
x2 into this line. And we keep going all the
way to our nth point. This error right here is going
to be yn minus m xn plus b. Now, so if we wanted to just
take the straight up sum of the errors, we could just
some these things up. But what we want to do is a
minimize the square of the error between each of these
points, each of these n points on the line. So let me define the squared
error against this line as being equal to the sum of
these squared errors. So this error right here, or
error one we could call it, is y1 minus m x1 plus b. And we're going to square it. So this is the error
one squared. And we're going to go to
error two squared. Error two squared is y2
minus m x2 plus b. And then we're going to
square that error. And then we keep going, we're
going to go n spaces, or n points I should say. We keep going all the way
to this nth error. The nth error is going to
be yn minus m xn plus b. And then we're going
to square it. So this is the squared
error of the line. And over the next few videos, is
I want to find the m and b that minimizes the squared error
of this line right here. So if you viewed this as the
best metric for how good a fit a line is, we're going to try to
find the best fitting line for these points. And I'll continue in
the next video. Because I find that with these
very hairy math problems, it's good to kind of just deliver
one concept at a time. And it also minimizes my probability of making a mistake.