Transcript Slide 1

LECTURE UNIT 7
Understanding Relationships
Among Variables
7.1-7.2 Scatterplots and correlation
7.3 -7.4 Fitting a straight line to bivariate data
Objectives (Lecture Unit 7.1)
Scatterplots

Scatterplots

Explanatory and response variables

Interpreting scatterplots

Outliers

Categorical variables in scatterplots
7.1 Basic Terminology

Univariate data: 1 variable is measured on each sample unit or
population unit (lecture unit 2)
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 lecture unit 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
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


Excel:
 In text: see p. 179-180
Statcrunch
On our course web page under Online Statistics
Resources, in “Statcrunch Instructional Videos” see
“Scatterplots and Regression” instructional video
TI calculator:
 Our course web page: under Online Resources for
Students, click on “Statistics on TI graphing
Calculators”; 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.