1. Chapter 10: Re-expressing Data (Get it Straight)
Jami Copeland
Shriya Varma
Semester Project

2. Straightening Relationships
In order to compare two variables using a linear regression model, the relationship between them must be linear
To re-express data you can use square roots, reciprocals or logarithms

3. Goals of Re-expression
Make the distribution of a variable more symmetric
Make the spread of several groups (as seen in side-by-side boxplots) more alike
Make the form of a scatterplot more nearly linear
Make the scatter in a scatterplot spread out evenly rather than following a fan shape.

4. The Ladder Of Powers
The Ladder of Powers helps to decide how to re-express data
It offers a range of options and when each option should be used
It also orders the effects of the re-expressions, from weakest to strongest

5. The Ladder of Powers Power Name Comment 2 Square of data values
Try with unimodal distributions that are skewed to the left. 1
Raw data
Data with positive and negative values and no bounds are less likely to benefit from re-expression.
1/2
Square root of data values
Counts often benefit from a square root re-expression.
“0”
We’ll use logarithms here
Measurements that cannot be negative often benefit from a log re-expression.
-1/2
Reciprocal square root
An uncommon re-expression, but sometimes useful. -1
The reciprocal of the data
Ratios of two quantities (e.g., mph) often benefit from a reciprocal.

6. Plan B: Attack of the Logarithms
If a stronger re-expression is needed, you can use Logarithms
When none of the data is zero or negative, logarithms can be used in different combinations

7. Plan B: Attack of the Logarithims
Model Name
X-axis
Y-axis
Comment
Exponential x
log(y)
This model is the “0’ power in the ladder approach, useful for values that grow by percentage increases.
Logarithmic
Log(x) y
A wide range of x-values, or a scatterplot descending rapidly at the left but leveling off toward the right, may benefit from trying this model.
Power
Log(y)
The Goldilocks model: When one of the ladder’s powers is too big and the next is too small, this one may be just right.

8. Problem 13: Planet distances and order
Let’s look again at the pattern in the locations of the planets in our solar system seen in the given table.
Use re-expressed data to create a model for the distance from the sun based on the planet’s position
There is some debate among astronomers as to whether Pluto is truly a planet or actually a large member of the Kuiper Belt of comets and other icy bodies. Does your model suggest that Pluto may not belong in the planet group? Explain.

9. Problem 13 Planet Position Number Distance from Sun Mercury 1 36 Venus2 67
Earth 3 93
Mars 4
142
Jupiter 5
484
Saturn 6
887
Uranus 7
1784
Neptune 8
2796
Pluto 9
3666

10. Problem 13a
Re-express the “Distance from the sun” data using logarithms and your calculator to find the new distances, log(y).

11. Problem 13a Planet Position Number Distance from Sun Log(y) Mercury 136
1.5563
Venus 2 67
1.8261
Earth 3 93
1.9685
Mars 4
142
2.1523
Jupiter 5
484
2.6848
Saturn 6
887
2.9479
Uranus 7
1784
3.2514
Neptune 8
2796
3.4465
Pluto 9
3666
3.5642

12. Problem 13b
This problem is just asking whether Pluto should be counted as a planet.
The linear model predicts that Pluto will be 5741 million miles away, while the data shows it is only 3666 million miles away.
This means it doesn’t fit very well and supports the claim that Pluto doesn’t behave like a planet.

13. Problem 15: Quaoar Planet
Caltech astronomers discovered a new large body orbiting around the Sun, a billion miles beyond Jupiter named Quaoar. Quaoar orbits at a distance of about 4 billion miles. It is classified as a member of the Kuiper Belt instead of a planet. There are many reasons for suspecting that Pluto is unlike other planets.

14. Problem 15: Quaoar Planet
Omit Pluto from your count of planets, and consider Quaoar as a candidate for the new planet.
Based on its position, how does Quaoar’s distance from the sun compare with the prediction made by your model?
Refit the model using Quaoar’s distance and position in the model instead of Pluto’s. Now how well does your model predict the re-expressed distance and position?

15. Problem 15a
To solve this, you find the predicted distance from the re-expressed linear regression model. The predicted distance is Pluto’s distance is Quaoar’s is Quaoar is therefore a better fit on the model.

16. Problem 15b
To refit the model, you replace Pluto’s data in the original chart with Quaoar's data. This makes the R^2 value go up to 99.5% which means it is a better fit.