Scatterplots,
Association, and
Correlation
Chapter 7
Looking at Scatterplots
• Scatterplots may be the most common and
most effective display for data.
o In a scatterplot, you can see patterns, trends,
relationships, and even the occasional
extraordinary value sitting apart from the
others.
• Scatterplots are the best way to start observing
the relationship and the ideal way to picture
associations between two quantitative
variables.
Slide 7 - 2
Looking at Scatterplots
• Relationships between variables are often at the
heart of what we’d like to learn from data:
o Does education level effect income?
o Does the cost of an athletic shoe indicate the
quality of the shoe?
o Do students learn better with technology?
Looking at Scatterplots
• When looking at scatterplots, we will look for
direction, form, strength, and unusual features.
• Direction:
o A pattern that runs from the upper left to the
lower right is said to have a negative direction.
o A trend running the other way has a positive
direction.
Slide 7 - 4
Looking at Scatterplots
Can the NOAA predict where a hurricane will go?
• The figure shows a
negative direction
between the year
since 1970 and the
and the prediction
errors made by
NOAA.
• As the years have
passed, the
predictions have
improved (errors
have decreased).
Slide 7 - 5
Direction
• A pattern that runs from upper left to lower
right is said to be negative.
Direction
• A pattern that runs from lower left to upper
right is said to be positive.
Looking at Scatterplots
• The example in
the text shows a
negative
association
between central
pressure and
maximum wind
speed
• As the central
pressure
increases, the
maximum wind
speed decreases.
Slide 7 - 8
Form
If there is a straight
line (linear)
relationship, it will
appear as a cloud or
swarm of points
stretched out in a
generally consistent,
straight form.
Slide 7 - 9
Form
If the relationship isn’t straight, but curves
gently, while still increasing or decreasing
steadily, we can often find ways to make it
more nearly straight.
Slide 7 - 10
Form
o If the relationship curves sharply,
the methods of this book cannot really
help us.
Slide 7 - 11
Strength
At one extreme, the points appear to follow
a single stream (whether straight, curved, or
bending all over the place).
Slide 7 - 12
Strength
• At the other extreme, the points appear as a vague
cloud with no discernible trend or pattern:
• Note: we will quantify the amount of scatter soon.
Slide 7 - 13
Unusual Features
• Look for the unexpected.
• Often the most interesting thing to see in a
scatterplot is the thing you never thought to look
for.
• One example of such a surprise is an outlier
standing away from the overall pattern of the
scatterplot.
• Clusters or subgroups should also raise questions.
o They may be a clue that you should split the
data into subgroups rather than looking at it all
together.
Slide 7 - 14
What do you think the scatterplot
would look like?
• Shoe size and grade point average?
• The scatterplot is likely to be randomly scattered.
• There is no association between shoe size and
grade point average.
• Time for a mile run and age?
• The very young will probably have very high
times. Older people will probably have very high
times. Run times are likely to be lowest for people
in their late teens and early twenties. The
association is likely to be moderate and curved
with no dominant direction.
• Drug dosage and pain relief?
• The association is likely to be positive, strong,
and curved. Assuming that the drug is an
effective pain reliever, the degree of pain
relief is will increase. Eventually , the
association is likely to level off until no further
pain relief is possible since the pain is gone.
• Age of car and cost of repairs?
• The association is positive, moderate, and
linear. As cars get older, they usually require
more repairs.
Roles for Variables
• It is important to determine which of the two
quantitative variables goes on the x-axis and
which on the y-axis.
• This determination is made based on the roles
played by the variables.
• When the roles are clear, the explanatory or
predictor variable goes on the x-axis, and the
response variable (variable of interest) goes on
the y-axis.
Slide 7 - 17
Roles for Variables
• The roles that we choose for variables are more
about how we think about them rather than
about the variables themselves.
• Just placing a variable on the x-axis doesn’t
necessarily mean that it explains or predicts
anything. And the variable on the y-axis may not
respond to it in any way.
Slide 7 - 18
Example
• Suppose you were to collect data for each pair
of variables listed below. You want to make a
scatterplot. Which variable would you use as the
explanatory variable and which is the response
variable? Why? Discuss the likely direction, form,
and strength.
• A. When climbing a mountain: altitude,
temperature
• Explanatory – altitude; Response – temperature
• To predict temperature based on altitude.
Scatterplot: negative, possibly straight, moderate
• People: Age, grip strength
• Explanatory – age; Response – grip strength
• To predict grip strength based on age.
Scatterplot: curved down, moderate. Very
young and elderly would have less grip
strength than adults
• Drivers: blood alcohol level, reaction time
• Explanatory – blood alcohol level; Response
– reaction time and other way around 
• To predict reaction time based on blood
alcohol level. Scatterplot: positive,
nonlinear, moderately strong
Correlation
• Data collected from students in Statistics classes
included their heights (in inches) and weights (in
pounds):
• Here we see a positive association and a fairly
straight form, although there seems to be
a high outlier.
Slide 7 - 21
Correlation
• How strong is the association between weight
and height of Statistics students?
• If we had to put a number on the strength, we
would not want it to depend on the units we
used.
• A scatterplot of heights
(in centimeters) and
weights (in kilograms)
doesn’t change the
shape of the pattern:
Slide 7 - 22
Correlation
• Since the units don’t
matter, why not
remove them
altogether?
• We could standardize
both variables and
write the coordinates
of a point as (zx, zy).
• Here is a scatterplot of
the standardized
weights and heights:
Slide 7 - 23
Correlation
• Note that the underlying linear pattern seems
steeper in the standardized plot than in the
original scatterplot.
• That’s because we made the scales of the axes
the same.
• Equal scaling gives a neutral way of drawing the
scatterplot and a fairer impression of the strength
of the association.
Slide 7 - 24
Correlation
• Some points (those in
green) strengthen the
impression of a positive
association between
height and weight.
• Other points (those in red)
tend to weaken the
positive association.
• Points with z-scores of
zero (those in blue)
don’t vote either way.
Slide 7 - 25
Correlation
• The correlation coefficient (r) gives us a numerical
measurement of the strength of the linear
relationship between the explanatory and
response variables.
zz

r
x y
n 1
Slide 7 - 26
Correlation
• For the students’ heights and weights, the
correlation is 0.644.
• What does this mean in terms of strength? We’ll
address this shortly.
Slide 7 - 27
TI Directions
• See directions on page 145 to enter points
for a scatterplot
• Pass out example
Correlation Conditions
• Correlation measures the strength of the
linear association between two quantitative
variables.
• Before you use correlation, you must check
several conditions:
o Quantitative Variables Condition
o Straight Enough Condition
o Outlier Condition
Slide 7 - 29
Quantitative Variables Condition:
o Correlation applies only to quantitative
variables.
o Don’t apply correlation to categorical
data masquerading as quantitative.
o Check that you know the variables’ units
and what they measure.
Slide 7 - 30
Straight Enough Condition:
• You can calculate a correlation coefficient
for any pair of variables.
• But correlation measures the strength only of
the linear association, and will be misleading
if the relationship is not linear.
Slide 7 - 31
Outlier Condition:
• Outliers can distort the correlation dramatically.
• An outlier can make an otherwise small
correlation look big or hide a large correlation.
• It can even give an otherwise positive
association a negative correlation coefficient
(and vice versa).
• When you see an outlier, it’s often a good idea
to report the correlations with and without the
point.
Slide 7 - 32
Checkpoint exercise
• Your statistics teacher tells you the correlation
between the scores on exam one and exam 2
was 0.75.
• Before answering any questions about the
correlation, what would you like to see and
how?
• Scores are quantitative. Check if the straight
enough condition and outlier condition are
satisfied by looking at scatterplots.
• If she adds 10 points to each exam one
score, how will it change the correlation?
• It won’t change.
• If she standardizes both scores how will this
affect the correlation?
• It won’t change
• In general, if someone does poorly on exam
1, are they likely to do poorly or well on
exam 2? Why?
• They are more likely to do poorly. The
positive correlation means that low scores
on exam 1 are associated with low scores
on exam 2.
• If someone does poorly on exam 1, will they
definitely do poorly on exam 2 as well?
• No. The general association is positive, but
individual performances may vary.
Correlation Properties
• The sign of a correlation coefficient gives the
direction of the association.
• Correlation is always between –1 and +1.
o Correlation can be exactly equal to –1 or +1,
but these values are unusual in real data
because they mean that all the data points
fall exactly on a single straight line.
o A correlation near zero corresponds to a weak
linear association.
Slide 7 - 36
Correlation Properties
• Correlation treats x and y symmetrically:
o The correlation of x with y is the same as the
correlation of y with x.
• Correlation has no units.
• Correlation is not affected by changes in the
center or scale of either variable.
o Correlation depends only on the z-scores, and
they are unaffected by changes in center or
scale.
Slide 7 - 37
Correlation Properties
• Correlation measures the strength of the linear
association between the two variables.
o Variables can have a strong association but
still have a small correlation if the association
isn’t linear.
• Correlation is sensitive to outliers. A single outlying
value can make a small correlation large or
make a large one small.
Slide 7 - 38
How strong is strong?
• There is NO agreement on characterizations such
as “weak”, “moderate” or “strong.”
• To use these words adds a value judgment to the
to the numerical summary that correlation
provides. What is weak in one context may be
strong in another.
• Tell the correlation and show the scatterplot,
others can judge for themselves.
Correlation ≠ Causation
• Whenever we have a strong correlation, it is
tempting to explain it by imagining that the
predictor variable has caused the response to
help.
• Scatterplots and correlation coefficients never
prove causation.
• A hidden variable that stands behind a
relationship and determines it by simultaneously
affecting the other two variables is called a
lurking variable.
Slide 7 - 40
Correlation Tables
• It is common in some fields to compute the
correlations between each pair of variables in a
collection of variables and arrange these
correlations in a table.
Slide 7 - 41
• Be careful with correlation tables!
• Presenting the correlations without the checks
for linearity and outliers, the table risks showing
truly small correlations have been inflated by
outliers, truly large correlations that are hidden
by outliers, and correlations of any size that
may be meaningless because the form is not
linear.
How to find the correlation coefficient
• Directions
p. 151
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Straightening Scatterplots P. 166 #35
• Create scatterplot and describe the association.
(Direction, form, strength!)
• The association between position number of
each planet and its distance from the sun is very
strong, positive, and curved.
• Why would you not want to talk about the
correlation between planet position and
distance from the sun?
• It’s not linear. Correlation is a measure of
the degree of linear association between
two variables.
• Make a scatterplot showing the logarithm of
the distance vs. position. What is better
about this scatterplot?
• Log(L2)storL3
• Still shows a curve but it is straight enough
that correlation may now be used.
p. 164
• #19