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Chapter 7 (part 2)

 Scatterplots, Association, and Correlation

Word of Caution in Correlation: Beware of Outliers

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 Outliers can greatly

 For all n=10 data points,

 For the n=9 clustered data points,

impact correlation.

r = 0.880

r = 0.000

Don’t just remove outliers   It is not appropriate (or ethical) to just remove

a data point because it does not follow the general trend (or the expected trend).

 Start by making sure it is not a mistake.  Often, outliers can show us an interesting phenomenon that leads to further research.

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Perhaps the most important caution about interpreting correlation is: Correlation does not necessarily imply causality.

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Correlation does not imply Causality

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Possible Explanations for a Correlation 1.  The correlation may be a coincidence.

2.  Both correlation variables might be directly influenced by some common underlying cause.

3.  One of the correlated variables may actually be a cause of the other. But note that, even in this case, it may be just one of several causes.

Correlation and Causality (Concepts & Applications)

 For each example that follows, state whether the correlation is most likely due to …

  (Choose which reason is most likely)  Coincidence

 A common underlying cause

 A direct cause (i.e. changing X causes a change in Y)

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Correlation and Causality (Concepts & Applications)

 Greater damage was observed when more firefighters were at a fire.

  (Which reason is most likely?)  Coincidence

 A common underlying cause

 A direct cause (i.e. changing X causes a change in Y)

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Number of firefighters at scene

fire

dam

age

(dol

lars

$)

Correlation and Causality (Concepts & Applications)

 There is a positive correlation between practice time and skill among piano players; that is, those who practice more tend to be more skilled.

  (Which reason is most likely?)  Coincidence

 A common underlying cause

 A direct cause (i.e. changing X causes a change in Y) 8

Correlation and Causality (Concepts & Applications)

 Amount of ice cream bought at the beach, and the number of swimmers requiring help from the lifeguards was found to be positively correlated.

  (Which reason is most likely?)  Coincidence

 A common underlying cause

 A direct cause (i.e. changing X causes a change in Y) 9

Example: SAT score vs. public $$$ spent on education

  In setting public policy, we may often hear something like…

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“State spending on education is positively correlated with SAT scores and therefore we should increase our state’s spending on education.”

Example: SAT score vs. public $$$ spent on education  Data was taken from the 1997 Digest of

Education Statistics, an annual publication of the U.S. Department of Education.

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800

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SAT score vs. money spent on students by state

Expenditures per Pupil in $1000s

Tot

al S

AT

sco

re

Are you surprised by the relationship in this plot? What could be going on here?

r = −0.3805

Example: SAT score vs. public $$$ spent on education  First, just looking at this scatterplot, would we

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SAT score vs. money spent on students by state

Expenditures per Pupil in $1000s

Tot

al S

AT

sco

re

interpret it as “Increasing expenditures causes a decrease in SAT scores?”

NO.

r = −0.3805

Simpson’s Paradox

 A statistical relationship between two variables can be reversed by including additional factors in the analysis.

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SAT score vs. money spent on students by state

Expenditures per Pupil in $1000s

Tot

al S

AT

sco

re

Alaska

Iowa

Minnesota

New Jersey

North Dakota

Utah Wisconsin

Connecticut

Idaho

New York

South Carolina

Example: SAT score vs. public $$$ spent on education   Let’s ask… do ALL students in these states

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take the SAT? It turns out that the answer is ‘no’ and this REALLY matters…

r = −0.3805

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SAT score vs. money spent on students by state

Expenditures per Pupil in $1000s

Tot

al S

AT

sco

re

% of students taking SAT<9%9-10%11%-20%21%-40%41%-60%61%-69%70%-81%

Example: SAT score vs. public $$$ spent on education

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Does the percent of students

taking the SAT in a

state help explain the paradox?

YES!

Low percentage

high percentage

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SAT score vs. money spent on students by state

Expenditures per Pupil in $1000s

Tot

al S

AT

sco

re

% of students taking SAT<9%9-10%11%-20%21%-40%41%-60%61%-69%70%-81%

Example: SAT score vs. public $$$ spent on education

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Within states with the same %

taking the SAT, we

actually see a positive

relationship!!

Low percentage

high percentage

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When only a few students take the SAT in a state, who are these students? (best students)

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900

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SAT score vs. money spent on students by state

Expenditures per Pupil in $1000s

Tot

al S

AT

sco

re

% of students taking SAT<9%9-10%11%-20%21%-40%41%-60%61%-69%70%-81%

Example: SAT score vs. public $$$ spent on education

If you have ALL students in a state taking the SAT, then you won’t be grabbing just the ‘good students’… and the average SAT will be lower compared to states with a small percentage of their ‘best’ students taking the SAT (is this a fair state-to-state comparison?)

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Within similar states (i.e. similar % taking the SAT) we see that spending more $$$ is associated with higher SAT scores.

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900

1000

1100

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SAT score vs. money spent on students by state

Expenditures per Pupil in $1000s

Tot

al S

AT

sco

re

% of students taking SAT<9%9-10%11%-20%21%-40%41%-60%61%-69%70%-81%

Example: SAT score vs. public $$$ spent on education

Simpson’s Paradox

 A statistical relationship between two variables can be reversed by including additional factors in the analysis.

 By including the variable called “% of students taking the SAT”, we saw a reversal of the relationship shown in the original scatterplot between SAT score and $$$ spent per student.

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Interpret correlation with caution

 Remember that correlation is a simple summary of a sometimes complex situation.

 Scatterplots are useful, but they do have limitations when many variables impact each other in a complex manner.

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The SAT score and expenditure information was a modification of material available at:

www.stat.ucla.edu/labs/pdflabs/sat.pdf

Food for thought… Should we reward school teachers based on student’s standardized test scores?

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0200

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Test Scores vs. teaching ability and effort

Teacher's genuine ability and effort

Stu

dent

's n

atio

nal t

est s

core

s What might this scatterplot look like? If students score high, was the teacher good? If students score low, was the teacher bad?