Regression Line Example Regression Line Example
Regression Line Example
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- In the last several videos, we did some fairly hairy
- And you might have even skipped them.
- But we got to a pretty neat result.
- We got to a formula for the slope and y-intercept of the
- best fitting regression line when you measure the error by
- the squared distance to that line.
- And our formula is, and I'll just rewrite it here just so
- we have something neat to look at.
- So the slope of that line is going to be the mean of x's
- times the mean of the y's minus the mean of the xy's.
- And don't worry, this seems really confusing, we're going
- to do an example of this actually in a few seconds.
- Divided by the mean of x squared minus the mean of the
- x squareds.
- And if this looks a little different than what you see in
- your statistics class or your textbook, you might see this
- swapped around.
- If you multiply both the numerator and denominator by
- negative 1, you could see this written as the mean of the
- xy's minus the mean of x times the mean of the y's.
- All of that over the mean of the x squareds minus the mean
- of the x's squared.
- These are obviously the same thing.
- You're just multiplying the numerator and denominator by
- negative 1, which is same thing as multiplying the whole
- thing by 1.
- And of course, whatever you get for m, you can then just
- substitute back in this to get your b.
- Your b is going to be equal to the mean of the
- y's minus your m.
- Let me write that in yellow so it's very clear.
- You solved for the m value.
- Minus m times the mean of the x's.
- And this is all you need.
- So let's actually put that into practice.
- So let's say I have three points, and I'm going to make
- sure that these points aren't colinear.
- Because, otherwise, it wouldn't be interesting.
- So let me draw three points over here.
- Let's say that to one point is the point 1 comma 2.
- So this 1, 2.
- And then we also have the point 2 comma 1.
- And then, let's say we also have the point, let's do
- something a little bit crazy, 4 comma 3.
- So this is 4, 3.
- So those are our three points.
- And what we want to do is find it the best fitting regression
- line, which we suspect is going to look
- something like that.
- We'll see what it actually looks like using our formulas,
- which we have proven.
- So a good place to start is just to calculate these things
- ahead of time, and then to substitute
- them back in the equation.
- So what's the mean of our x's?
- The mean of our x's is going to be 1 plus 2 plus
- 4 divided by 3.
- And what's this going to be?
- 1 plus 2 is 3, plus 4 is 7 divided by 3.
- It is equal to 7/3.
- Now, what is the mean of our y's?
- The mean of our y's is equal to 2 plus 1 plus 3.
- All of that over 3.
- So this is 2 plus 1 is 3.
- Plus 3 is 6.
- Divided by 3 is equal to 2.
- This is 6 divided by 3 is equal to 2.
- Now, what is the mean of our xy's?
- So our first xy over here is 1 times 2.
- Plus 2 times 1 plus 4 times 3.
- And we have three of these xy's.
- So divided by 3.
- So what's this going to be equal to?
- We have 2 plus 2, which is 4.
- 4 plus 12, which is 16.
- So it's going to be 16/3.
- And then the last one we have to calculate is the mean of
- the x squareds.
- So what's the mean of the x squareds?
- The first x squared is just going to be 1 squared.
- Plus this 2 squared, plus this 4 squared.
- And we have three data points again.
- So this is 1 plus 4, which is 5.
- Plus 16.
- Is equal to 21/3, which is equal to 7.
- So that worked out to a pretty neat number.
- So let's actually find our m's and our b's.
- So our slope, our optimal slope for our regression line,
- the mean of the x's is going to be 7/3.
- Times the mean of the y's.
- The mean of the y's is 2.
- Minus the mean of the xy's.
- Well, that's 16/3.
- And then, all of that over the mean of the x's.
- The mean of the x's is 7/3 squared.
- Minus the mean of the x squareds.
- So it's going to be minus this 7 right over here.
- And we just have to do a little bit of mathematics.
- I'm tempted to get out my calculator, but i'll resist
- the temptation.
- It's nice to keep things as fractions.
- Let's see if I can calculate this.
- This is 14/3 minus 16/3.
- All of that over, this is 49/9.
- And then minus 7.
- If I wanted to express that as something over 9, that's the
- same thing as 63/9.
- So in our numerator, we get negative 2/3.
- And then in our denominator, what's 49 minus 63?
- That's negative 14/9.
- And this is the same thing as negative 2/3
- times negative 9/ 14.
- Divide numerator and denominator by 3.
- Well, the negatives are going to cancel out first of all.
- You divide by 3.
- That becomes a 1.
- That becomes a 3.
- Divide by 2.
- Becomes a 1.
- That becomes a 7.
- So our slope is 3/7.
- Not too bad.
- Now, we can go back and figure out our y-intercept.
- So let's figure out our y-intercept using
- this right over here.
- So our y-intercept, b, is going to be equal to the mean
- of the y's, the mean of the y's is 2, minus our slope.
- We just figured out our slope to be 3/7.
- Times the mean of the x's, which is 7/3.
- These just are the reciprocal of each other,
- so they cancel out.
- That just becomes 1.
- So our y-intercept is literally just 2 minus 1.
- So it equals 1.
- So we have the equation for our line.
- Our regression line is going to be y is equal to-- We
- figured out m.
- m is 3/7.
- y is equal to 3/7 x plus, our y-intercept is 1.
- And we are done.
- So let's actually try to graph this.
- So our y-intercept is going to be 1.
- It's going to be right over there.
- And the slope of our line is 3/7.
- So for every 7 we run, we rise 3.
- Or another way to think of it, for every 3.5 we
- run, we rise 1.5.
- So we're going to go 1.5 right over here.
- So this line, if you were to graph it, and obviously I'm
- hand drawing it, so it's not going to be that exact, is
- going to look like that right over there.
- And it actually won't go directly through that line.
- So I don't want to give you that impression.
- So it might look something like this.
- And this line, we have shown, that this formula minimizes
- the squared distances from each of these
- points to that line.
- Anyway, that was, at least in my mind, pretty neat.
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