Scatter plots & Association
 Statistics
is about … variation.
 Recognize, quantify and try to
explain variation.
–Variation in contents of cola
cans can be explained, in part,
by the type of cola in the cans.
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Scatter plots & Association
variable – variable
of primary interest.
 Explanatory variable –
variable used to try to explain
variation in the response.
 Response
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Scatter plots & Association
 When
both the response and
the explanatory variables are
quantitative, display them both
in a scatter plot.
 Look for a general pattern of
association.
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Scatter plots & Association
 Example:
Tar (mg) and carbon
monoxide (mg) in cigarettes.
–y, Response: CO (mg).
–x, Explanatory: Tar (mg).
–Cases: 25 brands of cigarettes.
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Scatter plot
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Positive Association
 Above
average values of CO
are associated with above
average values of Tar.
 Below average values of CO
are associated with below
average values of Tar.
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Scatter plots & Association
 Example:
Outside temperature
and amount of natural gas used.
– Response: Natural gas (1000 ft3).
– Explanatory: Outside temperature
(o C).
– Cases: 26 days.
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Negative Association
Gas
10
5
0
-5.0
.0
5.0
Temp
10.0
15.0
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Negative Association
 Above
average values of gas
are associated with below
average temperatures.
 Below average values of gas
are associated with above
average temperatures.
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Correlation
 Linear
Association
– How closely do the points on the
scatter plot represent a straight line?
– The correlation coefficient gives
the direction of the linear
association and quantifies the
strength of the linear association
between two quantitative variables.
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Correlation
 Standardize
y
y y
zy 
sy
 Standardize
x
xx
zx 
sx
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ZxZy > 0
ZxZy > 0
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Correlation Coefficient
z

r
z
x y
n 1
x  x  y  y 

r
s x s y n  1
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Correlation Conditions
 Correlation
applies only to
quantitative variables.
 Correlation measures the
strength of linear association.
 Outliers can distort the value
of the correlation coefficient.
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Correlation Coefficient
 Tar
and CO
z z

r
22.9796

n 1
24
x
r
y
= 0.9575
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Correlation Coefficient
 There
is a strong positive
correlation, linear association,
between the tar content and
carbon monoxide content of
the various cigarette brands.
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JMP
– Multivariate methods
– Multivariate
 Y, Columns
 Analyze
–
–
Tar (mg)
CO (mg)
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Correlation Properties
sign of r indicates the
direction of the association.
 The
value of r is always between
–1 and +1
 Correlation has no units.
 Correlation is not affected by
changes of center or scale.
 The
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Correlation Cautions
 “Correlation”
and “Association”
are different.
–Correlation – specific (linear).
–Association – vague (trend).
 Don’t
correlate categorical
variables.
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Correlation Cautions
 Don’t
confuse correlation with
causation.
– There is a strong positive
correlation between the number of
crimes committed in communities
and the number of 2nd graders in
those communities.
 Beware
of lurking variables.
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