Calculating R-Squared Calculating R-Squared to see how well a regression line fits data
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- In the last video, we were able to find the equation for
- the regression line for these four data points.
- What I want to do in this video is figure out the r
- squared for these data points.
- Figure out how good this line fits the data.
- Or even better, figure out the percentage-- which is really
- the same thing-- of the variation of these data
- points, especially the variation in y, that is due
- to, or that can be explained by variation in x.
- And to do that, I'm actually going to get
- a spreadsheet out.
- I've actually tried to do this with a calculator and it's
- much harder.
- So hopefully this doesn't confuse you too much to use a
- And I'm a make a couple of columns here.
- And spreadsheets actually have functions that'll do all of
- this automatically, but I really want to do it so that
- you could do it by hand if you had to.
- So I'm going to make a couple of columns here.
- This is going to be my x column.
- This is going to be my y column.
- This is going to be the column-- I'll call this y
- star-- this'll be the y value that our line predicts based
- on our x value.
- This is going to be the error with the line.
- Let me caught it the squared error with the line.
- I don't want us to take up too much space.
- And then the next one, I'm going to have the squared
- variation for that y value from the mean y.
- And I think these columns by themselves will be enough for
- us to do everything.
- So let's first put all the data points in.
- So we had negative 2 comma negative 3.
- That was one data point.
- Negative 1 comma negative 1.
- And we had 1 comma 2.
- Then we have 4 comma 3.
- Now, what does our line predict?
- Well our line says, you give me an x value, I'm going to
- tell you what y value I'll predict.
- So when x is equal to negative 2, the y value on the line is
- going to be the slope.
- So this is going to be equal to 41 divided by
- 42 times our x value.
- And I just selected that cell.
- And just a little bit of a primer on spreadsheets, I'm
- selecting the cell D2.
- I was able to just move my cursor over and select that.
- But that tells me the x value.
- Minus 5/21.
- Minus 5 divided by 21.
- Just like that.
- So just to be clear of what we're even doing.
- This y star here, I got negative 2.19.
- That tells us at this point right over
- here is negative 2.19.
- So when we figure out the error, we're going to figure
- out the distance between negative 3, that's our y
- value, and negative 2.19.
- So let's do that.
- So the error is just going to be equal to our y value.
- That's cell E2.
- Minus the value that our line would predict.
- So just that value is the actual error.
- But we want to square it.
- And then, the next thing we want to do
- is the squared distance.
- so this is equal to the squared distance of our y
- value from the y's mean.
- So what's the mean of the y's?
- Mean of the y's is 1/4.
- So minus 0.25, is the same thing is 1/4.
- And we also want to square that.
- Now, this is what's fun about spreadsheets.
- I can apply those formulas to every row now.
- And notice, what it did when I did that.
- Now all of a sudden, this is the y value that my line would
- predict, it's now using this x value and
- sticking it over here.
- It's now figuring out the square distance from the line
- using what the line would predict and using the
- y value, this one.
- And then does the same thing over here.
- It's figures out the squared distance of this y
- value from the mean.
- So what is the total squared error with the line?
- So let me just sum this up.
- The total squared error with the line is 2.73.
- And then the total variation from the mean, squared
- distances from the mean of the y, are 22.75.
- So let me be very clear what this is.
- So let me write these numbers down.
- I'll write it up here so we can keep looking at this
- actual graph.
- So are squared error versus our line, our total squared
- error, we just computed to be 2.74.
- I rounded a little bit.
- And what that is, is you take each of these data points'
- vertical distance to the line.
- So this distance squared, plus this distance squared, plus
- this distance squared, plus this distance squared.
- That's all we just calculated on Excel.
- And that total squared variation to the line is 2.74.
- Or total squared error with the line.
- And then the other number we figured out was the total
- distance from the mean.
- So the mean here is y is equal to 1/4.
- So that's going to be right over here.
- This is 1/2.
- So right over here.
- So this is our mean y value.
- Or the central tendency for our y values.
- And so what we calculated next was the total error, the
- squared error, from the means of our y values.
- That's what we calculated over here in the spreadsheet.
- You see in the formula.
- It is this number, E2, minus 0.25, which is the mean of our
- y's squared.
- That's exactly what we calculated.
- We calculated for each of the y values.
- And then we summed them all up.
- It's 22.75.
- It is equal to 22.75.
- So this is essentially the error that the
- line does not explain.
- This is the total error, this is the total
- variation of the numbers.
- So if you wanted to know the percentage of the total
- variation that is not explained by the line, you
- could take this number divided by this number.
- So 2.74 over 22.75.
- This tells us the percentage of total variation not
- explained by the line or by the variation in x.
- And so what is this number going to be?
- I can just use Excel for this.
- So I'm just going to divide this number divided by this
- number right over there.
- I get 0.12.
- So this is equal to 0.12.
- Or another way to think about it is 12% of the total
- variation is not explained by the variation in x.
- The total squared distance between each of the points or
- their kind of spread, their variation, is not explain by
- the variation in x.
- So if you want the amount that is explained by the variance
- in x, you just subtract that from 1.
- So let me write it right over here.
- So we have our r squared, which is the percent of the
- total variation that is explained by x, is going to be
- 1 the minus that 0.12 that we just calculated.
- Which is going to be 0.88.
- So our r squared here is 0.88.
- It's very, very close to 1.
- The highest number it can be is 1.
- So what this tells us, or a way to interpret this, is that
- 88% of the total variation of these y values is explained by
- the line or by the variation in x.
- And you can see that it looks like a pretty good fit.
- Each of these aren't too far.
- Each of these points are definitely much closer to the
- line than they are to the mean line.
- In fact, all of them are closer to our actual line than
- to the mean.
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