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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

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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


Chapter 4

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 valuesof 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


Examples of relationships

Examples of Relationships

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 correlations

Examples of Correlations

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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