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Chapter 4 Scatterplots and Correlation
Chapter 4 Scatterplots
and Correlation
Explanatory and Response Variables● A response variable measures an outcome of a study (Dependent Variable)
● An explanatory variable may explain or influence changes in a response variable (Independent variable)
Scatterplots● Shows the
relationship between two quantitative variables measured on the same individuals
● Always plot the explanatory variable on the x-axis and response variable on the y-axis
Associations● We can say two variables are positively associated when above-average values
of one tend to accompany above-average values of the other and below-average values of one tend to accompany above below-values of the other
● We can say two variables are negatively associated when above-average
values of one tend to accompany below-average values of the other and vice versa
Examples of Different Types of Association
Scatterplot Example #1Use the given data to make a scatter plot of the weight and height of each member of a basketball team.
Scatterplot Example #1 (Answer)
Scatterplot Example #2Use the given data to make a table of the weight and height of each member of a soccer team.
Scatterplot Example #2 (Answer)
Strength of a Correlation
Association VS Correlation● Technically, association refers to any relationship between
two variables, whereas correlation is often used to refer only to a linear relationship between two variables.
Scatterplot StationsGet into groups of 5.
At each station, describe whether the situation is positively, negatively, or not associated and why. Then, make a conclusion about the data.
Warm-Up 09/28Which types of variables do scatterplots compare?
How To Add Categorical Variables to a Scatterplot
More Examples
Correlation (AKA Regression Coefficient)● The correlation of a graph measures the direction and strength of the linear
relationship between two quantitative variables● We denote correlation as r
Interpreting Correlation ● For calculating r, it makes no difference which variable you use for x and y● Changing the units of x or y also makes no difference when calculating r● Correlation Coefficient, r is between -1 and 1
1 ≤ r ≤ 1● r > 0 Best-fit (regression) line has positive slope
variables are positively linearly correlated● r < 0 Best-fit (regression) line has negative slope
variables are negatively linearly correlated
Interpreting Correlation (Cont.)● When r = 0, no correlation● r values near +1 or 1 indicates strong linear relationship● r values near 0 indicates weak linear relationship
You Try!
r = 0.85
Were you close?
Predicting r From Pictures
Calculating r by HandEX:
Hours Spent
Studying
0.5 0.5 1 1 1.5 2 2 2 3
Test Scores
72 90 88 90 90 87 90 95 95
Let’s revisit this example...EX:
Hours Spent
Studying
0.5 0.5 1 1 1.5 2 2 2 3
Test Scores
72 90 88 90 90 87 90 95 95
r ≈ 0.61
Warm-Up 10/03What does an r value of 0.9 mean for a set of data? 0.19? -0.9? -0.19? 0? Provide sketches of each scatterplot.
Warm-Up 10/04Think about the relationship between the amount of time a student spends studying and the grades they get. Predict what the value of r would be for this situation and what that implies.
Now suppose that I survey 100 students at Peninsula and get r = 0.1. What does this imply for the study, and why might this be surprising? What could explain why r is this value?
“Whodunnit?”
Found at the Scene of The Crime“Mr. Buda,
I took something from you when you weren’t looking. I bet you can’t guess who I am.
Sincerely,
Your Favorite Student”
Stolen ItemsItem Footprint Size (Inches)
Morty Action Figure 9.875
Sora Action Figure 11.125
Panther 10.75
Graphing Calc 9.5
Concluding questions...1. What if the person left a footprint of size 9 inches? 9.5 inches? 10 inches?
2. How would you find someone’s shoe size (using our graph) if we knew their height?
3. Do you think this is an accurate way to find the height of suspects? Explain.
Chapter 5 Regression
Warm-Up 10/08
Given: x̄ = 6.3, ȳ= 40.8, sx= 0.87, s
y=12.8,
r = -0.4
Find: a and b
Pirate Lab
Additional Instructions● Next door is testing
● Finish #1 by the end of class
● Class ends at 1:47
Beware of extrapolation● (Avoid) extrapolation- The use of a regression line to
predict far outside the range of values of the explanatory variable
● Example: Suppose you have data on how children grow
between the ages of 3 and 8. Using your regression line equation, you get that at 25 the child will grow to be 8 ft tall.
Correlation does NOT imply causation...
● Lurking variables are variables that are not among the explanatory or response variables, yet may influence the interpretation of relationships among those variables
Find the lurking
variable
Example #1Cities with more schools have more prisons. Therefore, being educated causes you to go to jail.
Lurking Variable: Cities with more schools have a higher population.
Example #2Students who eat Cheetos die. Therefore, Cheetos kill.
Lurking Variable: All people eventually pass away.
Example #3Students failed their exams the year Mr. Kuykendall became principal. Therefore, having Mr. Kuykendall as principal causes students to fail.
Lurking Variable: Perhaps the students did not study as much as they should have or the material was tough, etc.
Example #4Students scored highly on their exams the year Mr. Kuykendall became principal. Therefore, having Mr. Kuykendall as principal causes students to succeed.
Lurking Variable: Perhaps the students studied or the material was not as challenging, etc.
Warm-Up 10/12Jimmy notices every time the seatbelt sign turns on during his flight, the plane begins to shake violently. He sure would like the pilot to stop doing that. What lurking variable could explain what is happening?
R2
● The fraction of the variation in the values of y covered by the regression line
● For example, a scatterplot with r = -0.7786 has a strong negative correlation. Then, r2 = (-0.7786)2 = 0.6062. Therefore, 61% of the data is “covered” by the regression line
Influential Observations● An observation is influential if removing it would markedly change the result of
a calculation. In scatterplots, outliers are influential and greatly influence the regression line
