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Chapter 4. Scatterplots and Correlation. Explanatory and Response Variables. Studying the relationship between two variables. Measuring both variables on the same individuals. a response variable measures an outcome of a study

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chapter 4

Chapter 4

Scatterplots and Correlation

Chapter 4

explanatory and response variables
Explanatory and Response Variables
  • Studying the relationship between two variables.
  • Measuring both variables on the same individuals.
    • a response variable measures an outcome of a study
    • an explanatory variable explains or influences changes in a response variable
    • sometimes there is no distinction

Chapter 4

case study
Case study

♦ In a study to determine whether surgery

or chemotherapy results in higher survival

rates for some types of cancer.

♦ Whether or not the patient survived is one

variable, and whether they received

surgery or chemotherapy is the other variable.

♦ Which is the explanatory variable and

which is the response variable?

Chapter 4

scatterplot
Scatterplot
  • Graphs the relationship between two quantitative (numerical) variables measured on the same individuals.
  • Usually plot the explanatory variable on the horizontal (x) axis and plot the response variable on the vertical (y) axis.

Chapter 4

slide5

Scatterplot

Relationship between mean SAT verbal score and percent of high school grads taking SAT

Chapter 4

scatterplot1
Scatterplot
  • Look for overall pattern and deviations from this pattern
  • Describe pattern by form, direction, and strength of the relationship
  • Look for outliers

Chapter 4

linear relationship
Linear Relationship

Some relationships are such that the points of a scatterplot tend to fall along a straight line -- linear relationship

Chapter 4

direction
Direction
  • Positive association

◙ A positive value for the correlation implies a positive association.

◙ large values of X tend to be associated with large values of Y and small values of X tend to be associated with small values of Y.

  • Negative association

◙ A negative value for the correlation implies a negative or inverse association

◙ large values of X tend to be associated with small values of Y and vice versa.

Chapter 4

examples
Examples

From a scatterplot of college students, there is a positive associationbetween verbal SAT score and GPA.

For used cars, there is a negative association between the age of the car and the selling price.

Chapter 4

measuring strength direction of a linear relationship
Measuring Strength & Directionof a Linear Relationship
  • How closely does a straight line (non-horizontal) fit the points of a scatterplot?
  • The correlation coefficient (often referred to as just correlation): r
    • measure of the strength of the relationship: the stronger the relationship, the larger the magnitude of r.
    • measure of the direction of the relationship: positive r indicates a positive relationship, negative r indicates a negative relationship.

Chapter 4

correlation coefficient
Correlation Coefficient
  • special values for r :
    • a perfect positive linear relationship would have r = +1
    • a perfect negative linear relationship would have r = -1
    • if there is no linear relationship, or if the scatterplot points are best fit by a horizontal line, then r = 0
    • Note: r must be between -1 and +1, inclusive
  • both variables must be quantitative; no distinction between response and explanatory variables
  • r has no units; does not change when measurement units are changed (ex: ft. or in.)

Chapter 4

examples of correlations1
Examples of Correlations
  • Husband’s versus Wife’s ages
      • r = .94
  • Husband’s versus Wife’s heights
      • r = .36
  • Professional Golfer’s Putting Success: Distance of putt in feet versus percent success
      • r = -.94

Chapter 4

not all relationships are linear miles per gallon versus speed
Not all Relationships are Linear Miles per Gallon versus Speed
  • Linear relationship?
  • Correlation is close to zero.

Chapter 4

not all relationships are linear miles per gallon versus speed1
Not all Relationships are Linear Miles per Gallon versus Speed
  • Curved relationship.
  • Correlation is misleading.

Chapter 4

problems with correlations
Problems with Correlations
  • Outliers can inflate or deflate correlations (see next slide)
  • Groups combined inappropriately may mask relationships (a third variable)
    • groups may have different relationships when separated

Chapter 4

outliers and correlation
Outliers and Correlation

A

B

For each scatterplot above, how does the outlier affect the correlation?

A: outlier decreases the correlation

B: outlier increases the correlation

Chapter 4

correlation calculation
Correlation Calculation
  • Suppose we have data on variables X and Y for n individuals:

x1, x2, … , xnand y1, y2, … , yn

  • Each variable has a mean and std dev:

Chapter 4

case study1
Case Study

Per Capita Gross Domestic Product

and Average Life Expectancy for

Countries in Western Europe

Chapter 4

case study2
Case Study

Chapter 4

case study3
Case Study

Chapter 4

case study4
Case Study

Chapter 4

summary
Summary
  • Explanatory variable Vs responsible variable
  • Scatterplot: display the relationship between two quantitative variables measured on the same individuals
  • Overall pattern of scatterplot showing:

☻ Direction ☻Form ☻ Strength

♦ Correlation coefficientr:Measures only straight-line

linear relationship

♦ R indicate the direction of a linear relationship by sign

R > 0, positive association R< 0, negative association

♦ The range of R:-1 ≤ R ≤ 1,

Chapter 4

scatterplots
Scatterplots

Which of the following scatterplots displays the stronger linear relationship?

  • Plot A
  • Plot B
  • Same for both
correlation
Correlation

If two quantitative variables, X and Y, have a correlation coefficient r = 0.80, which graph could be a

scatterplot of the two variables?

  • Plot A
  • Plot B
  • Plot C

Plot A ? Plot B? Plot C?

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