1. Ch. 10 – Scatterplots, Association and Correlation (Day 1)

2. Scatterplots
So far, all of our analysis has looked at one variable at a time
In this chapter, we will look at the relationship between two variables
If the variables are quantitative, we can do this by starting with a graph called a scatterplot

3. Scatterplots
Ex Use the following data to examine the relationship between the amount of fertilizer (lbs per acre) used on plots of land in a particular farming region and the number of bushels per acre of grain produced.
Fertilizer
Bushels 31 43 33 44 40 45 50 54 59 53 68 80 63

4. THINK: How will we draw the graph?
To decide which variable will go on which axis, think about what you are trying to learn
Do the variables have an explanatory/response relationship?
In this case, we are wondering how the amount of fertilizer used affects the amount of grain produced
Fertilizer is the explanatory variable
Bushels produced is the response variable
In a scatterplot, the explanatory variable goes on the x-axis and the response variable goes on the y-axis
If we aren’t looking at this type of relationship for the variables, you can use either axis

5. SHOW: Draw the scatterplot
Bushels
Lbs of Fertilizer 60 55 50 45 40
Don’t forget about labels and scale!

6. TELL: What does a scatterplot show us?
In most of our previous graphs, we were looking for center, shape, and spread of a single quantitative variable
This time we are looking at the relationship between two quantitative variables
If the two variables seem related, this is referred to as an association
Specifically, we are looking at the form, direction and strength of the association

7. Form: Is it linear?
Our eventual goal is to create a model for the data
In order to decide which calculations to use, we need to first look at the form (shape) the pattern follows
A scatterplot has a linear form if a straight line could be used to describe it reasonably well
For now, we will simply describe form as linear or nonlinear
Linear
Nonlinear

8. Direction: Positive, Negative or No Association?
Once we decide that the form is linear, we now turn to direction
If y increases as x increases, this is a positive association
If y decreases as x increases, this is a negative association
Positive association
Negative association
No association

9. Strength: Strong, Moderate, Weak?
The last thing we should address is the strength of the relationship
The conclusions we draw about strength are highly subjective, especially if they are based strictly on looking at the scatterplot
Strong association
Moderate association
Weak association

11. Correlation Coefficient
r = correlation coefficient for linear relationships
Measures the strength and direction of a linear relationship between two quantitative variables

12. Calculating r r = .9782 Fertilizer Bushels 31 43 33 44 40 45 50 54 59
Lbs of Fertilizer 60 55 50 45 40
Fertilizer
Bushels 31 43 33 44 40 45 50 54 59 53 68 80 63
r = .9782

13. What does r tell us? Close to +1 = strong, positive linear association
Close to -1 = strong, negative linear association
Close to 0 = weak or no linear association
r = 1 or r = -1 means a perfect linear correlation

14. Properties of r r is a number between -1 and 1
Since r is based on z-scores, it is not affected by shifting or re-scaling, and it has no units
The correlation of x with y is the same as the correlation of y with x (it doesn’t matter which variable is used as x or y – the correlation stays the same)
Remember that r only works for linear associations of quantitative variables
r is very sensitive to outliers – be careful!
Even though we have this numerical calculation, strength is still subjective – a value such as 0.68 that is considered strong for one set of data might be considered weak for another

15. Outliers A scatterplot can also show us outliers
In this context, an outlier is a point which doesn’t seem to fit within the pattern formed by the rest of the data

16. Homework
Pg. 542 # 12, 14, 16 Directions: Make a scatterplot of the data. Calculate the correlation coefficient and interpret what this means.