1. Independent Dependent Scatterplot Least Squares
Variable Variable Regression Line
(Trend Line)
Positive Direction Negative Direction Causation Correlation
No Correlation Weak Correlation Moderate Correlation Strong Correlation
(Zero Correlation)
Monotonic Monotonic Form: Linear Form: Exponential
Increasing Decreasing

2. 2 Variable Data
Part 1

3. 2 Variable Data: Independent Variables and Dependent Variables
Two Variables
Most statistical studies look at multiple variables.
Often the studies try to show a relationship between one variable and another
When one variable effects another, one variable will be referred to the explanatory variable and the other as the response variable
Explanatory Variable
“An explanatory variable attempts to explain the observed outcomes.”
Response Variable
A variable that measures an outcome of a study.

4. 2 Variable Data: Independent Variables and Dependent Variables
Example 1
A study looks at smoking and lung cancer.
Which (if any) is the explanatory variable?
Which (if any) is the response variable?
Is smoking a quantitative or categorical variable?
Is lung cancer a quantitative or categorical variable?

5. 2 Variable Data: Independent Variables and Dependent Variables
Example 2
A study looks at cavities and milk drinking.
Which (if any) is the explanatory variable?
Which (if any) is the response variable?
Is cavities a quantitative or categorical variable?
Is milk drinking a quantitative or categorical variable?

6. 2 Variable Data: Independent Variables and Dependent Variables
Example 3
A study looks at rain fall and SAT scores.
Which (if any) is the explanatory variable?
Which (if any) is the response variable?
Is rainfall a quantitative or categorical variable?
Is SAT scores a quantitative or categorical variable?

7. 2 Variable Data: Independent Variables and Dependent Variables
Scatterplots
A good way to try to see if there is a relationship between two quantitative variables is through the use of scatterplot.
On a scatterplot, we usually put what we think might be the explanatory variable on the x-axis and the response variable on the y-axis.

8. 2 Variable Data: Scatterplots and Form
Interpreting Scatterplots
Form: Is there a pattern? linear, parabola, bell shaped
Deviations from pattern: Are there areas where the data conform less to the pattern?
Linear
Parabolic

9. 2 Variable Data: Scatterplots – Direction and Strength
Interpreting Scatterplots
Direction: If linear, is the data positively associated or negatively associated?
Strength: Does the data tightly conform or loosely conform
Moderate
Strength
High Strength

10. 2 Variable Data: Constructing Scatterplots
Scale both axes (make them uniform).
Use a symbol to break from zero if needed.
Label your axes.
Make your plot large enough so that details can be seen.
Use the entire grid if you’re using a grid (not a corner of it)

11. 2 Variable Data: Constructing Scatterplots

12. 2 Variable Data: Constructing Scatterplots

13. 2 Variable Data: Constructing and Interpreting Scatterplots
Form:
Direction:
Strength:

14. 2 Variable Data
Part 2

15. 2 Variable Data: Constructing Least Squares Regression Lines
Linear Regression
Relationship between water and Tomatoes
It would be great to be able to look at multi-variable data and reduce it to a single equation that might help us make predictions
Example:
“What would be the predicted number of Tomatoes Produced if we gave a plant 3.5 Gallons of water?”
Tomato Production
Gallons of Water

16. 2 Variable Data: Constructing Least Squares Regression Lines
Linear Regression
Statisticians use a slightly different version of “slope-intercept” form.
Algebraic form
Statistical form

17. 2 Variable Data: Constructing Least Squares Regression Lines
Linear Regression
Year
1991
1992
1993
1994
1995
1996
Rainfall (inches)
12.3 14
11.6
10.5
9.8
13.2
Crop yield (tons)
43.1
43.0
40.7
38.9
36.2
43.2

18. 2 Variable Data: Correlation
Interpreting r
No Correlation
Strong Negative
Moderate Negative
Weak Negative
Strong Positive
Moderate Positive
Weak Positive -1
-.75
-.5
-.25
.25 .5
.75 1

19. 2 Variable Data
Part 3

20. 2 Variable Data: Correlation
Relationship between ERA and Wins
Is there a “correlation” between a baseball team’s “earned run average” and the number of wins?
Is the association positively associated or negatively associated?
Is the association strong or weak?
The symbol we use for correlation is r
Number of Wins / Season
ERA (Earned Run Average)

21. 2 Variable Data: Correlation
r – “Correlation”
Correlation is the measure of the association between the explanatory and response variables.
The calculation of correlation is based on mean and standard deviation.
Both mean and standard deviation are “not resistant measures.” That is, they cannot resist the influence of outliers. This means that r is also not resistant.

22. 2 Variable Data: Correlation
Facts about correlation:
Both variables need to be quantitative.
Because the data values are standardized, it does not matter which units the variables are in (must be consistent though).
The value of r is unitless (just like z scores are unitless).
Correlation is blind to the relationship between explanatory and response variables.
Even though you may get a r value close to -1 or 1, you may not say that explanatory variable causes the response variable. We will talk about this in detail in the second semester.

23. 2 Variable Data: Finding the Equation, r, and 𝑟 2
Using the TI-83 to calculate r
Procedure:
1. Put study Hours into L1
Put Grade on Test into L2
2. Graph a scatterplot
3. In CATELOG, select DiagnosticOn
4. Under Stat and Calc select LinReg (a+bx)
Find r (correlation) for this data set:
Sussy
Mike
Tommy
Sarah
Study Hours 5 3 7 1
Grade on Test
90%
85%
93%
82%
This procedure is super important. It gives you correlation (r) and will also dump the Linear Regression line into Y1 for you so that you can see it on your graph.
r = .995
Strong Positive Correlation

24. 2 Variable Data: Finding the Equation, r, and 𝑟 2
Other facts about least-square regression
Make sure you know which is the explanatory (x) variable and which is the response (y) variable.
-Switching them gets a different regression line.
Regression line always goes through the point (x-bar, y-bar)
The coefficient of correlation (r) explains the strength of the linear relationship
R-squared is known as the “coefficient of determination”
The square of the correlation (r2) is the variation in the values of y that is explained by x.
___%(r2) of the variation of ______ (y) is explained by _____ (x).
Facts about r2

25. 2 Variable Data: Finding the Equation, r, and 𝑟 2
Checking for Understanding
Find the line of best fit and describe the association between the variables:
Age
Height (inches) 7 51 11 56 15 61 8 50 6 49 12 53 10 63

26. 2 Variable Data
Part 4

27. r2 “coefficient of explanation”
In the regression of ERA vs. WINS, we find a r2 value of .4512
We say “45% of the variation in WINS can be explained by ERA”

28. Outliers and Influential Data
An outlier is an observation outside the overall pattern
If an observation is influential it has a large effect on the regression line. Removing the observation changes the line.

29. Residuals
It is important to note that the observed value almost never match the predicted values exactly
The difference between the observed value and predicted has a special name: residual
Predicted Value ( ): 5.3 ERA Wins
Residuals are negative when the observed value is below the predicted value
Residuals are positive when the observed value is higher than the predicted value
Residual: =
Observed Value (y): 5.3 ERA 43 Wins
Residual = Observed - Predicted

30. Residual Plots
You can plot the residuals to see if the there is any trends with the quality of the predictive model
This residual shows no tendencies. It is equally bad throughout.
This suggests that the original relationship is linear.

31. Not Linear because the residual plot is curved:
Linear because the residual plot is well distributed:

32. Bear Age 4 7 12 10 6 9 kP
145
166
221
201
153
182
Use your calculator to make a scatter plot
Find r and r2
3. Plot the trend line
4. Find the residual for the bear with age 6.
5. Plot a Residual Plot and verify the residual for age 6 p.bear
6. Is a linear trend the most appropriate? How do you know?

33. 2 Variable Data
Chapter Summary

34. Concepts to Remember:
To make a scatterplot, and thus, work with correlation and least squares regression, requires that both variables be quantitative.
You assign the explanatory variable to x and the response variable to y.
* Keep in mind, however, that in some circumstances you might be asked to assign them in the reverse order simply for the sake of doing a problem. It is also possible that there is no clear explanatory or response variable (ie, Rain and Income) but you can still do least squares regression and try to calculate correlation!
Correlation (r) is unitless and measures the strength and direction of a relationship between two variables. Negative or positive indicates direction while the number value indicates the strength to which the points are conforming to a line.
r2 is also unitless and will always be positive.
r2 means “The percent of variation of y that is due to x.” This applies to residuals!
Every least squares regression line will pass through the mean of x and the mean of y.
The least squares regression line is a model which we use to predict outcomes.

35. Interpreting r No Correlation -1 -.75 -.5 -.25 .25 .5 .75 1
Strong Negative
Moderate Negative
Weak Negative
Strong Positive
Moderate Positive
Weak Positive -1
-.75
-.5
-.25
.25 .5
.75 1

36. Concepts to Remember:
Least Squares Regression is also called Linear Regression and uses the form y = a + bx.
A least squares regression line minimizes the area of the squares of the residuals.
Thus the name “least squares!”
A cloudlike residual plot is ideal because it shows the points are scattered randomly above and below the least squares regression line. A curved residual plot indicates the original data was not linear and the model is invalid.
Just as the mean and standard deviation are not resistant, meaning they cannot resist the pull of outliers, correlation (r) and the least squares regression line are not resistant. Outliers, also called influential observations, will cause r to approach zero and will change the slope (b) and intercept (a) of the least squares regression line.

37. Monotonic Functions move in 1 direction as it’s argument (x) increases
Monotonic Increasing Functions
preserves the order of data: if a > b, then f(a) > f(b) [ie, trends upward]
Monotonic Decreasing Functions
reverses the order of the data: if a > b, then f(a) < f(b) [ie, trends downward]