1. Scatterplots and Correlation
AP STATISTICS
Chapter 3:
Scatterplots and Correlation

5. Size and abundance of carnivores

6. Basically r tells us the strength and direction of two variables

7. Properties of the Linear Correlation Coefficient r
2. Value of r does not change if all values of either variable are converted to a different scale
3. Interchanging all x and y values will not change r
4. r measures strength of a linear relationship
5. No linear relationship does not imply no relationship at all. There is a possibility of a non-linear relationship.

8. non-linear relationship
Title 50
100
150
200
250 1 2 3 4 5 6 7 8
Distance
(feet)
Time (seconds)

9. Abundance of carnivores
Correlation tells us about strength (scatter) and direction of the linear relationship between two quantitative variables.
R does not change when we change units of measurement. R is not resistant and is strongly affected by outliers
In addition, we would like to have a numerical description of how both variables vary together. For instance, is one variable increasing faster than the other one? And we would like to make predictions based on that numerical description.

11. Interpreting Correlation
A perfect straight line
tilting up to the right
r = 0
No overall tilt
No relationship?
Not so no linear one
r = – 1
A perfect straight line tilting down to the right X Y X Y X Y X Y X Y X Y

12. Nonlinear Correlation
No Correlation
Nonlinear Correlation
r = – 0.471
r = 0.089
r = 0.395

13. Examples Internet Site Ratings Mortgage Rates & Fees
Correlation is r = 0.964
Very strong positive association
(since r is close to 1)
Internet Site Ratings 30 60 90
100
200
Pages per person
Minutes per person
eBay
Yahoo!
MSN
Linear relationship
Straight line with scatter
Increasing relationship
Tilts up and to the right positive
Mortgage Rates & Fees
Correlation is r = – 0.890
Strong negative association
Linear relationship
Straight line with scatter
Decreasing relationship
Tilts down and to the right
5.0%
5.5%
6.0% 0% 1% 2% 3% 4%
Loan fee
Interest rate

14. Example Is there momentum? Correlation is r = 0.11 No relationship?
If the market was up yesterday, is it more likely to be up today?
Or is each day’s performance independent?
Correlation is r = 0.11
A weak relationship?
No relationship?
Tilt is neither
up nor down
The Stock Market

15. Example Maximizing Yield 120 130 140 150 160 500 600 700 800 900
A nonlinear relationship
Not a straight line: A curved relationship
Correlation r = –
r suggests no relationship but in reality there is a relationship it is just NOT linear
But relationship is strong
It tilts neither up nor down
Maximizing Yield
120
130
140
150
160
500
600
700
800
900
Temperature
Yield of process

16. Example unequal variability Correlation r = 0.820
Variability is stabilized by taking logarithms (lower right
Correlation r = 0.820
investment
investment 15 20
Log of investment
Log of miles
1,000
2,000
Investment
($millions)
Circuit miles
(millions)

17. All the same R value
However, making the scatterplots shows us that the correlation/ regression analysis is not appropriate for all data sets.
Just one very influential point and a series of other points all with the same x value; a redesign is due here…
Obvious nonlinear relationship; regression inappropriate.
One point deviates from the (highly linear) pattern of the other points; it requires examination before a regression can be done
Moderate linear association; regression OK.

18. Always plot your data!
The correlations all give r ≈ 0.816, and the regression lines are all approximately = x. For all four sets, we would predict = 8 when x = 10. The previous slides data.

19. We want the line as close as possible to the points.

20. So, we need the line that will minimize the sum of the squares of the vertical distances.

21. The regression line BAC levels
The least-squares regression line is the unique line such that the sum of the squared vertical (y) distances between the data points and the line is the smallest possible.
BAC levels
Distances between the points and line are squared so all are positive values. This is done so that distances can be properly added

22. Properties LSRL
The least-squares regression line can be shown to have this equation:
is the predicted y value (y hat)
b is the slope
a is the y-intercept
"a" is in units of y
"b" is in units of y/units of x

23. How to:
First we calculate the slope of the line, b, from statistics we already know:
r is the correlation
sy is the standard deviation of the response variable y
sx is the the standard deviation of the explanatory variable x
Once we know b, the slope, we can calculate a, the y-intercept:
where x and y are the sample means of the x and y variables
This means that we don’t have to calculate a lot of squared distances to find the least-squares regression line for a data set. We can instead rely on the equation.

24. BEWARE !!! Not all calculators and software use the same convention:
We use this one:
Ti-89 and Ti-83
give both
Some use instead:
Make sure you know what YOUR calculator gives you for a and b before you answer homework or exam questions.

25. The y-intercept
Sometimes the y-intercept is not biologically possible. Here we have negative blood alcohol content, which makes no sense…
But the negative value is appropriate for the equation of the regression line.
There is a lot of scatter in the data and the line is just an estimate.
BAC

26. The equation completely describes the regression line.
To plot the regression line, you only need to plug two x values into the equation, get y, and draw the line that goes through those two points.
Hint: The regression line always passes through the mean of x and y
Thus:
The points you use for drawing the regression line are derived from the equation.
They are NOT points from your sample data (except by pure coincidence).
Regression examines the distance of all points from the line in the y direction only

27. Correlation and regression
The correlation is a measure of spread (scatter) in both the x and y directions in the linear relationship
In regression we examine the variation in the response variable (y) given change in the explanatory variable (x)

28. Equations
The general form for a regression line is

29. Coefficient of determination, r2
It measures the fraction (or percent) of the variation in y that is explained by the least squares regression line of y on x. (Memorize this sentence!)
Basically, this helps us interpret r in a more understandable way since r2 is a percent.

30. Coefficient of determination, r2
r2, the coefficient of determination, is the square of the correlation coefficient r
r2 represents the percentage of the variance in y (vertical scatter from the regression line) that can be explained by changes in x.

31. Here the change in x only explains 76% of the change in y
Here the change in x only explains 76% of the change in y. The rest of the change in y (the vertical scatter, shown as red arrows) must be explained by something other than x.
r = 0.87
r2 = 0.76
Changes in x explain 100% of the variations in y.
y can be entirely predicted for any given value of x.
r = 0
r2 = 0
Changes in x explain 0% of the variations in y.
The value(s) y takes is (are) entirely independent of what value x takes.

32. There is quite some variation in BAC for the same number of beers drunk. A person’s blood volume is a factor in the equation that was overlooked here.
We changed the number of beers to the number of beers/weight of a person in pounds.
r =0.9
r2 =0.81
In the first plot, number of beers only explains 49% of the variation in blood alcohol content.
But number of beers/weight explains 81% of the variation in blood alcohol content.
Additional factors contribute to variations in BAC among individuals (like maybe some genetic ability to process alcohol).

33. Grade performance If class attendance explains 16% of the variation in grades, what is the correlation between percent of classes attended and grade?
1. We need to make an assumption: Attendance and grades are positively correlated. So r will be positive too.
2. r2 = 0.16, so r = +√0.16 = A weak correlation.

34. Residuals
The distances from each point to the least-squares regression line give us potentially useful information about the contribution of individual data points to the overall pattern of scatter.
These distances are called “residuals.” The sum of these residuals is always 0
Points above the line have a positive residual.
Points below the line have a negative residual ^
Predicted y
Observed y

35. Residual plots
Residuals are the distances between y-observed and y-predicted. We plot them in a residual plot.
If residuals are scattered randomly around 0, chances are your data
fit a linear model, were normally distributed, and you didn’t have outliers

36. The x-axis in a residual plot is the same as on the scatterplot.
The line on both plots is the regression line.
Only the y-axis is different

37. Residuals are randomly scattered—good!
A curved pattern—means the relationship
you are looking at is not linear but does not mean there is no relationship.
A change in variability across plot is a warning sign. You need to find out why it is and remember that predictions made in areas of larger variability will not be as good.

38. Child 19 is an outlier of the relationship.
Outliers and influential points
Outlier: An observation that lies outside the overall pattern of observations.
“Influential individual”: An observation that markedly changes the regression if removed. This is often an outlier on the x-axis.
Child 19 = outlier in y direction
Child 18 = outlier in x direction
Child 19 is an outlier of the relationship.
Child 18 is only an outlier in the x direction and thus might be an influential point.

39. Example: Cost and Quantity
An outlier is visible
A disaster (a fire at the factory)
High cost, but few produced
outlier
Outlier removed
Cost vs. Number Produced
Cost vs. Number Produced
10,000 20 40 60
Number produced
Cost
3,000
4,000
5,000 20 30 40 50
Number produced
Cost

40. Outlier in y-direction
Are these points influential?
Influential

41. OUTLIERS IN TERMS OF REGRESSION:
Observations with large (in absolute value) residuals.
Observations falling f a r from the regression line while not following the pattern of the relationship apparent in the others
Residual=actual-fitted Residual = y-y(hat)

42. INFLUENTIAL POINTS ARE:
Points whose removal would greatly affect the association of two variables
Points whose removal would significantly change the slope of an LSR line
Points with a large moment (i.e they are far away from the rest of the data.)
Usually outliers in the x direction.

43. The two graphs below show the same data – the one on the right with the removal of the green data point. As you can see, the removal of this point significantly affects the slope of the regression line. This is an influential point!

44. !!!REMEMBER!!! An observation does NOT have to be an Outlier to be an
Influential Point!!
Nor does an observation need
to be an Influential Point in order
to be an Outlier!!

45. Given the five-number summary {8 21 35 43 77}, which of the following is correct?
A. There are no outliers
B. There are at least two outliers
C. There is not enough data to make any conclusion
D. There is exactly one outlier
E. There is at least one outlier

46. The correct answer is E The five number summary gives you
{Min Q1 Median Q3 Max}
The IQR is calculated by Q3-Q1
So, the IQR for the given data is 43-21=22
An outlier for this data would be:
>Q3+1.5*IQR or <Q1-1.5*IQR 
>43+(22*1.5)=76 or <21-(22*1.5)=-12
Since the max is 77, there must be at least one outlier in this data set, but we cannot conclude how many outliers without more data.

47. Given the following scatterplot and residual plot
Given the following scatterplot and residual plot. Which of the following is true about the yellow data point?
I. It is an influential point
II. It is an outlier with respect to the regression model
III. It appears to be an outlier in the x direction
A. I only
B. I and II
C. I and III
D. None of the above
E. All of the above

48. The correct answer is c
I. Because this point has a large moment and is far from the rest of the data, it is an influential point. If this point was removed, the slope of the line would markedly change.
II. This point is not an outlier with respect to the model because as you can see in the residual plot, it does not have a large residual (It follows the regression pattern of the data).
III. By looking at both the scatterplot and the residual plot, you can see that the yellow point is an outlier in the x direction (far right of the rest of the data).

49. Example: Salary and Experience
Salary vs. Years Experience
For n = 6 employees
Linear (straight line) relationship
Increasing relationship
higher salary generally goes with higher experience
Correlation r =
Mary earns $55,000
per year, and has
20 years of experience
Salary and Experience
Experience 15 10 20 5
Salary 30 35 55 22 40 27 20 30 40 50 60 10
Experience
Salary ($thousand)

50. The Least-Squares Line Y=a+bX
Summarizes bivariate data: Predicts Y from X
with smallest errors (in vertical direction, for Y axis)
Intercept is salary (at 0 years of experience)
Slope is salary (for each additional year of experience, on average)
Salary 10 20 30 40 50 60
Experience (X)
Salary (Y)
Salary = Experience

51. Predicted Values and Residuals
Predicted Value comes from Least-Squares Line
For example, Mary (with 20 years of experience)
has predicted salary (20) = 48.8
So does anyone with 20 years of experience
Residual is actual Y minus predicted Y
Mary’s residual is 55 – 48.8 = 6.2
She earns about $6,200 more than the predicted salary for a person with 20 years of experience
A person who earns less than predicted will have a negative residual

52. Predicted and Residual (continued)
Predicted salaries are about 6.52 (i.e., $6,520) away from actual salaries
Mary’s residual is 6.2 10 20 30 40 50 60
Experience
Salary
Mary earns 55 thousand
Mary’s predicted value is 48.8

53. Making predictions: interpolation
The equation of the least-squares regression allows you to predict y for any x within the range studied. This is called interpolating.
Nobody in the study drank 6.5 beers, but by finding the value of from the regression line for x = 6.5, we would expect a blood alcohol content of mg/ml.

54. Deaths
There is a positive linear relationship between the number of powerboats registered and the number of manatee deaths. The least-squares regression line has for equation:
Thus, if we were to limit the number of powerboat registrations to 500,000, what could we expect for the number of manatee deaths? Roughly 21 manatees

55. Extrapolation
Extrapolation is the use of a regression line for predictions outside the range of x values used to obtain the line.
This can be a very stupid thing to do, as seen here.

56. Caution with regression
Do not use a regression on inappropriate data.
Pattern in the residuals
Presence of large outliers Use residual plots for help.
Clumped data falsely appearing linear
Recognize when the correlation/regression is performed on averages.
A relationship, however strong, does not itself imply causation.
Beware of lurking variables.
Avoid extrapolating (going beyond interpolation).