Transforming nonlinear data | More on regression | AP Statistics | Khan Academy

So we have some data here that we can plot on a scatter plot that looks something like that.

And so the next question, given that we've been talking a lot about lines of regression or regression lines, is can we fit a regression line to this?

Well, if we try to, we might get something that looks like this or maybe something that looks like this.

I'm just eyeballing it, obviously. We could input it into a computer to try to develop a linear regression model to try to minimize the sum of the squared distances from the points to the line, but you can see it's pretty difficult.

And some of you might be saying, well, this looks more like some type of an exponential, so maybe we could fit an exponential to it.

So it could look something like that, and you wouldn't be wrong.

But there is a way that we can apply our tools of linear regression to this data set.

And the way we can is instead of plotting x versus y, we can think about x versus the logarithm of y.

So this is the exact same data set. You see the x values are the same, but for the y values, I just took the log base 10 of all of these.

10 to the what power is equal to 2307.23?

10 to the 3.36 power is equal to 2307.23, and I did that for all of these data points.

I did it on a spreadsheet, and if you were to plot all of these, something neat happens.

All of a sudden, when we're plotting x versus the log of y or the log of y versus x, all of a sudden it looks linear.

Now be clear, the true relationship between x and y is not linear; it looks like some type of an exponential relationship.

But the value of transforming the data—and there are different ways you can do it—in this case, the value of taking the log of y and thinking about it that way is now we can use our tools of linear regression.

Because this data set, you could actually fit a linear regression line to this quite well.

You could imagine a line that looks something like this; it would fit the data quite well.

And the reason why you might want to do this versus trying to fit an exponential is because we've already developed so many tools around linear regression and hypothesis testing around the slope and confidence intervals, and so this might be the direction you want to go at.

And what's neat is once you fit a linear regression, it's not difficult to mathematically unwind from your linear model back to an exponential one.

So the big takeaway here is that the tools of linear regression can be useful even when the underlying relationship between x and y are non-linear.

And the way that we do that is by transforming the data; here we took a logarithm of the y's, and that helped us see a more linear relationship of log y versus x.