Chapter 6
Scatterplots and Correlation
Chapter 7 Objectives
Scatterplots

Scatterplots

Explanatory and response variables

Interpreting scatterplots

Outliers

Categorical variables in scatterplots
Basic Terminology

Univariate data: 1 variable is measured on each sample unit or
population unit (chapters 2 through 6)
e.g. height of each student in a sample

Bivariate data: 2 variables are measured on each sample unit
or population unit
e.g. height and GPA of each student in a sample; (caution: data
from 2 separate samples is not bivariate data)
Basic Terminology (cont.)

Multivariate data: several variables are measured on each unit in a
sample or population.

For each student in a sample of NCSU students, measure height,
GPA, and distance between NCSU and hometown;

Focus on bivariate data in chapter 7
Same goals with bivariate data that we
had with univariate data

Graphical displays and numerical summaries

Seek overall patterns and deviations from those patterns

Descriptive measures of specific aspects of the data
Here, we have two quantitative
variables for each of 16
students.
1) How many beers they
drank, and
2) Their blood alcohol level
(BAC)
We are interested in the
relationship between the two
variables: How is one affected
by changes in the other one?
Student
Beers
Blood Alcohol
1
5
0.1
2
2
0.03
3
9
0.19
4
7
0.095
5
3
0.07
6
3
0.02
7
4
0.07
8
5
0.085
9
8
0.12
10
3
0.04
11
5
0.06
12
5
0.05
13
6
0.1
14
7
0.09
15
1
0.01
16
4
0.05
Scatterplots

Useful method to graphically describe the relationship between
2 quantitative variables
Scatterplot: Blood Alcohol Content vs Number of Beers
In a scatterplot, one axis is used to represent each of the variables,
and the data are plotted as points on the graph.
Student
Beers
BAC
1
5
0.1
2
2
0.03
3
9
0.19
4
7
0.095
5
3
0.07
6
3
0.02
7
4
0.07
8
5
0.085
9
8
0.12
10
3
0.04
11
5
0.06
12
5
0.05
13
6
0.1
14
7
0.09
15
1
0.01
16
4
0.05
Focus on Three Features of a
Scatterplot
Look for an overall pattern regarding …
1.
Shape - ? Approximately linear, curved, up-and-down?
2.
Direction - ? Positive, negative, none?
3.
Strength - ? Are the points tightly clustered in the particular shape,
or are they spread out?
Blood Alcohol as a function of Number of Beers
… and deviations from the overall
pattern:
Outliers
Blood Alcohol Level (mg/ml)
0.20
0.18

0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
1
2
3
4
5
6
Number of Beers
7
8
9
10
Scatterplot: Fuel Consumption vs Car
Weight. x=car weight, y=fuel cons.

(xi, yi): (3.4, 5.5) (3.8, 5.9) (4.1, 6.5) (2.2, 3.3)
(2.6, 3.6) (2.9, 4.6) (2, 2.9) (2.7, 3.6) (1.9, 3.1) (3.4, 4.9)
FUEL CONSUMP.
(gal/100 miles)
FUEL CONSUMPTION vs CAR WEIGHT
7
6
5
4
3
2
1.5
2.5
3.5
WEIGHT (1000 lbs)
4.5
Explanatory and response variables
A response variable measures or records an outcome of a study. An
explanatory variable explains changes in the response variable.
Typically, the explanatory or independent variable is plotted on the x
axis, and the response or dependent variable is plotted on the y axis.
Blood Alcohol as a function of Number of Beers
Blood Alcohol Level (mg/ml)
0.20
Response
(dependent)
variable:
blood alcohol
content
y
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
x
0
1
2
3
4
5
6
7
8
9
10
Number of Beers
Explanatory (independent) variable:
number of beers
SAT Score vs Proportion of Seniors
Taking SAT 2005
2005 Average SAT Score
2005 SAT Total
1250
IW
IL
1200
1150
NC 74% 1010
1100
1050
1000
DC
950
0%
20%
40%
60%
Percent of Seniors Taking SAT
80%
100%
Some plots don’t have clear explanatory and response variables.
Do calories explain
sodium amounts?
Does percent return on Treasury
bills explain percent return
on common stocks?
Making Scatterplots



Excel:

In text: see p. 169-170
Statcrunch
 In the left panel of our class webpage
http://www.stat.ncsu.edu/people/reiland/courses/st311/ click on
Student Resources, in “Statcrunch Instructional Videos” see
“Scatterplots and Regression” instructional video; in “Many
Statcrunch Instructional Videos” see videos 15 and 47.
TI calculator:
 In the left panel of our class web page click on Student
Resources, under “Graphing Calculators, Online Calculations”
click on “TI Graphing Calculator Guide”; see p. 7-9.
Form and direction of an association
Linear
No relationship
Nonlinear
Positive association: High values of one variable tend to occur together
with high values of the other variable.
Negative association: High values of one variable tend to occur together
with low values of the other variable.
No relationship: X and Y vary independently. Knowing X tells you nothing
about Y.
One way to think about this is to remember the following:
The equation for this line is y = 5.
x is not involved.
Strength of the association
The strength of the relationship between the two variables can be
seen by how much variation, or scatter, there is around the main form.
With a strong relationship, you
can get a pretty good estimate
of y if you know x.
With a weak relationship, for any
x you might get a wide range of
y values.
This is a weak relationship. For a
particular state median household
income, you can’t predict the state
per capita income very well.
This is a very strong relationship.
The daily amount of gas consumed
can be predicted quite accurately for
a given temperature value.
How to scale a scatterplot
Same data in all four plots
Using an inappropriate
scale for a scatterplot
can give an incorrect
impression.
Both variables should be
given a similar amount of
space:
• Plot roughly square
• Points should occupy all
the plot space (no blank
space)
Outliers
An outlier is a data value that has a very low probability of occurrence
(i.e., it is unusual or unexpected).
In a scatterplot, outliers are points that fall outside of the overall pattern
of the relationship.
Not an outlier:
Outliers
The upper right-hand point here is
not an outlier of the relationship—It
is what you would expect for this
many beers given the linear
relationship between beers/weight
and blood alcohol.
This point is not in line with the
others, so it is an outlier of the
relationship.
IQ score and
Grade point average
a) Describe in words what this
plot shows.
b) Describe the direction,
shape, and strength. Are
there outliers?
c) What is the deal with these
people?
Categorical variables in scatterplots
Often, things are not simple and one-dimensional. We need to group
the data into categories to reveal trends.
What may look like a positive linear
relationship is in fact a series of
negative linear associations.
Plotting different habitats in different
colors allows us to make that
important distinction.
Comparison of men and women
racing records over time.
Each group shows a very strong
negative linear relationship that
would not be apparent without the
gender categorization.
Relationship between lean body mass
and metabolic rate in men and women.
Both men and women follow the same
positive linear trend, but women show
a stronger association. As a group,
males typically have larger values for
both variables.
Correlation
Objectives
Correlation

The correlation coefficient “r”

r does not distinguish x and y

r has no units

r ranges from -1 to +1

Influential points
The correlation coefficient "r"
The correlation coefficient is a measure of the direction and strength of
the linear relationship between 2 quantitative variables. It is calculated
using the mean and the standard deviation of both the x and y variables.
Time to swim: x = 35, sx = 0.7
Pulse rate: y = 140 sy = 9.5
Correlation can only be used to
describe quantitative variables.
Categorical variables don’t have
means and standard deviations.
Part of the calculation
involves finding z, the
standardized score we used
when working with the
normal distribution.
You DON'T want to do this by hand.
Make sure you learn how to use
your calculator!
Calculating Correlation



Excel:

In text: see p. 169 (bottom of second column in Excel section)
Statcrunch
 In the left panel of our class webpage
http://www.stat.ncsu.edu/people/reiland/courses/st311/ click on
Student Resources, in “Many Statcrunch Instructional Videos”
see videos 18 and 22.
TI calculator:
 In the left panel of our class webpage click on Student
Resources; under “Graphing Calculators, Online Calculations”,
either
 click on TI Graphing Calculator Guide and see p. 8, or
 Click on Online Graphing Calculator Tutorials
Example: calculating correlation


(x1, y1), (x2, y2), (x3, y3)
(1, 3) (1.5, 6) (2.5, 8)
x  1.67, y  5.67, sx  .76, s y  2.52
r
11.67  35.67   1.51.67  6 5.67    2.51.67 85.67 
(31)(.76)(2.52)
 .9538
Properties of Correlation



r is a measure of the strength of the linear relationship between x
and y.
No units [like demand elasticity in economics (-infinity, 0)]
-1 < r < 1
Values of r and scatterplots
r near +1
r near -1
y
r near 0
r near 0
y
x
x
Properties (cont.) r has no unit
Changing the units of variables does
not change the correlation coefficient
"r", because we get rid of all our units
when we standardize (get z-scores).
r = -0.75
z-score plot is the same
for both plots
r = -0.75
Properties (cont.)
r ranges from
-1 to+1
"r" quantifies the strength
and direction of a linear
relationship between 2
quantitative variables.
Strength: how closely the points
follow a straight line.
Direction: is positive when
individuals with higher X values
tend to have higher values of Y.
Properties of Correlation (cont.)

r = -1 only if y = a + bx with slope b<0

r = +1 only if y = a + bx with slope b>0
10
20
y = 11 - x
8
y = 1 + 2x
r=1
r = -1
15
6
Y
10
y
4
5
2
0
0
0
2
4
6
x
8
10
0
2
4
6
X
8
10
Properties (cont.) High correlation
does not imply cause and effect
CARROTS: Hidden terror in the produce
department at your neighborhood grocery



Everyone who ate carrots in 1920, if they
are still alive, has severely wrinkled skin!!!
Everyone who ate carrots in 1865 is now
dead!!!
45 of 50 17 yr olds arrested in Raleigh for
juvenile delinquency had eaten carrots in
the 2 weeks prior to their arrest !!!
Properties (cont.) Cause and Effect

There is a strong positive correlation between the monetary damage
caused by structural fires and the number of firemen present at the
fire. (More firemen-more damage)

Improper training? Will no firemen present result in the least amount
of damage?
Properties (cont.) Cause and Effect
(1,2) (24,75) (1,0) (18,59) (9,9) (3,7) (5,35) (20,46) (1,0)
(3,2) (22,57)
x = fouls committed by player;
y = points scored by same player

r measures the strength of
the linear relationship
between x and y; it does not
indicate cause and effect
The correlation is due to a third “lurking”
variable – playing time
(x, y) = (fouls, points)
correlation
r = .935
Points

80
70
60
50
40
30
20
10
0
0
5
10
15
Fouls
20
25
30
Properties (cont.) r
does not distinguish x & y
The correlation coefficient, r, treats
x and y symmetrically.
r = -0.75
r = -0.75
"Time to swim" is the explanatory variable here, and belongs on the x axis.
However, in either plot r is the same (r=-0.75).
Outliers
Correlations are calculated using
means and standard deviations,
and thus are NOT resistant to
outliers.
Just moving one point away from the
general trend here decreases the
correlation from -0.91 to -0.75
PROPERTIES (Summary)

r is a measure of the strength of the linear relationship between x and y.

No units [like demand elasticity in economics (-infinity, 0)]

-1 < r < 1

r = -1 only if y = a + bx with slope b<0

r = +1 only if y = a + bx with slope b>0

correlation does not imply causation

r does not distinguish between x and y

r can be affected by outliers
Correlation: Fuel Consumption vs Car
Weight
FUEL CONSUMP.
(gal/100 miles)
FUEL CONSUMPTION vs CAR WEIGHT
r = .9766
7
6
5
4
3
2
1.5
2.5
3.5
WEIGHT (1000 lbs)
4.5
SAT Score vs Proportion of Seniors
Taking SAT
88-89 SAT vs % Seniors Taking SAT
r = -.868
88-89 SAT State Avg.
IW
ND
1075
1025
975
88-89 SAT
925
875
SC
825
0
20
40
DC
NC
60
% Seniors that Took SAT
80
Standardization:
Allows us to compare
correlations between data
sets where variables are
measured in different units
or when variables are
different.
For instance, we might
want to compare the
correlation between [swim
time and pulse], with the
correlation between [swim
time and breathing rate].
When variability in one
or both variables
decreases, the
correlation coefficient
gets stronger
( closer to +1 or -1).
Correlation only describes linear relationships
No matter how strong the association,
r does not describe curved relationships.
Note: You can sometimes transform a non-linear association to a linear form,
for instance by taking the logarithm. You can then calculate a correlation using
the transformed data.
Review examples
1) What is the explanatory variable?
Describe the form, direction and strength
of the relationship?
Estimate r.
r = 0.94
(in 1000’s)
2) If women always marry men 2 years older
than themselves, what is the correlation of the
ages between husband and wife?
r=1
ageman = agewoman + 2
equation for a straight line
Thought quiz on correlation
1. Why is there no distinction between explanatory and response
variable in correlation?
2. Why do both variables have to be quantitative?
3. How does changing the units of one variable affect a correlation?
4. What is the effect of outliers on
correlations?
5. Why doesn’t a tight fit to a horizontal line
imply a strong correlation?