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Characteristics of Scatterplots
• Form
• Direction
• Strength
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FORM
• Linear
• Curvilinear
• Clustering
• Outliers
• Other patterns
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DIRECTION
• POSITIVE• Large values of X are associated with large values of Y,
and small values of X are associated with small values of Y.
• For example, IQ and SAT.
• NEGATIVE• Large values of one variable are associated with small
values of the other variable
• For example, SPEED and ACCURACY.
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STRENGTH If the points do not fall along a straight line, then
there is NO linear association. If the points fall nearly along a straight line, then
there is a STRONG linear association. If the points fall exactly along a straight line, then
there is a PERFECT linear association.
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y
x
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Responsevariable
Explanatory variable
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Dependentvariable
(DV)
Independent variable (IV)
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Dinosaur-bone example
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How to calculate Pearson’s r
z-score z-score product of
Specimen Femur Humerus Femur Humerus z-scores
A 38 41 -1.53 -1.57 2.41B 56 63 -0.17 -0.19 0.03C 59 70 0.06 0.25 0.02D 64 72 0.44 0.38 0.17E 74 84 1.20 1.13 1.36
Mean 58.20 66.00 0.00 0.00 0.99St Dev 13.20 15.89 1.00 1.00
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How to calculate Pearson’s r
z-score z-score product of
Specimen Femur Humerus Femur Humerus z-scores
A 38 41 -1.53 -1.57 2.41B 56 63 -0.17 -0.19 0.03C 59 70 0.06 0.25 0.02D 64 72 0.44 0.38 0.17E 74 84 1.20 1.13 1.36
Mean 58.20 66.00 0.00 0.00 0.99St Dev 13.20 15.89 1.00 1.00
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How to calculate Pearson’s r
z-score z-score product of
Specimen Femur Humerus Femur Humerus z-scores
A 38 41 -1.53 -1.57 2.41B 56 63 -0.17 -0.19 0.03C 59 70 0.06 0.25 0.02D 64 72 0.44 0.38 0.17E 74 84 1.20 1.13 1.36
Mean 58.20 66.00 0.00 0.00 0.99St Dev 13.20 15.89 1.00 1.00
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How to calculate Pearson’s r
z-score z-score product of
Specimen Femur Humerus Femur Humerus z-scores
A 38 41 -1.53 -1.57 2.41B 56 63 -0.17 -0.19 0.03C 59 70 0.06 0.25 0.02D 64 72 0.44 0.38 0.17E 74 84 1.20 1.13 1.36
Mean 58.20 66.00 0.00 0.00 0.99St Dev 13.20 15.89 1.00 1.00
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The Pearson product-moment correlation coefficient
r = 1
n -1zx i
zy i
i1
n
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The Pearson product-moment correlation coefficient
Convert the X variable to z scores Convert the Y variable to z scores Multiply each pair of z scores Add up the products and divide by n-1
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The range of the correlation coefficient
-1 0 +1perfect
negativelinear
relationship
nolinear
relationship
perfectpositivelinear
relationship
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Outliers and influential cases
• An outlier is a case which does not follow the overall pattern of the others
• An influential case is one which draws the regression line toward its point in the scatterplot.
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Example 2.18
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0 10 20 30 40 50
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Example 2.18
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Age at first word, in months
Gese
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ESTABLISHING CAUSATION
Associationdoes not implycausation
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When there is association between X and Y
• Perhaps X causes Y
• Perhaps Y causes X
• Perhaps some third variable causes both X and Y
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COMMON RESPONSE Two variables might be associated because
they share a common cause.• For example, SAT scores and College Grades
are highly associated, but probably not because scoring well on the SAT causes a student to get high grades in college.
• Being a good student, etc., would be the common cause of the SATs and the grades.
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CONFOUNDING For example, there is a strong positive
association between Number of Years of Education and Annual Income.
• In part, getting more education allows people to get better, higher-paying jobs.
• But these variables are confounded with others, such as socio-economic status (SES).
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ESTABLISHING CAUSATION The best way to establish that X causes Y is
to have a controlled experiment, in which X is varied by the experimenter and the effects on Y can be seen.
But experimentation is not always possible.
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ESTABLISHING CAUSATION The association is strong
The association is consistent Stronger treatments are associated with stronger
responses
The alleged cause precedes the effect in time
The alleged cause is plausible