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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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# Transforming nonlinear data

AP.STATS:

DAT‑1 (EU)

, DAT‑1.J (LO)

, DAT‑1.J.1 (EK)

Use logarithms to transform nonlinear data into a linear relationship so we can use least-squares regression methods.

## Want to join the conversation?

- Thank You for the awesome video. However I was interested to know if we can or should log transform long term temperature anomaly data.? Thank you(4 votes)
- What does this mean?(2 votes)
- Why do we transform the y coordinates of the points in our function by *log_(10)*? Why don't we consider using a different log with a base not equal to 10?(2 votes)
- I have a scatter plot where data is in shape of inverted-U, what kind of transformation does it need. Please help.(1 vote)
- When would be best to transform with the base 10 log versus the base e natural log? Aren’t they both “exponential relationships”?(1 vote)

## Video transcript

- [Instructor] 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 you can input it into a computer to try to develop a
linear regression model to try to minimize the sum
of the square 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 dataset. 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. So 10 to the what power
is equal to 2,307.23. 10 to the 3.36 power is equal to 2,307.23. 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's different ways you can do it. In this case, the value
taking the log of y, and thinking about 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 can imagine a line that
looks something like this, it would fit the data quite well. And the reason why you might wanna 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 wanna 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.