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Chapter 4

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Chapter 4

Scatterplots and Correlation

Chapter 4

- 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

♦ 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

- 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.

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Scatterplot

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

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

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

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- 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.

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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

Chapter 4

- 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

- 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.)

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- Husband’s versus Wife’s ages
- r = .94

- r = .36

- r = -.94

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- Linear relationship?
- Correlation is close to zero.

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- Curved relationship.
- Correlation is misleading.

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- 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

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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

- 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:

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Per Capita Gross Domestic Product

and Average Life Expectancy for

Countries in Western Europe

Chapter 4

Chapter 4

Chapter 4

Chapter 4

- 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

Which of the following scatterplots displays the stronger linear relationship?

- Plot A
- Plot B
- Same for both

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?