AP Statistics
Topic 3
Summary of Bivariate Data
Overview of this Topic
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In Topic 2 we studied the analysis of univariate
data
Now we will study bivariate data
• Observe 2 characteristics on each observational unit
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Graphically display the data
• Scatterplot
• Form, Strength and Direction
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Numerical summary of the data
• Pearson’s Correlation Coefficient,
• Coefficient of Determination, and
• Least Squares Regression Line
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Least square regression lines from data and from
Minitab output
Scatterplots
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Visual display of bivariate data
Describe the scatterplot
•
•
•
•
•
Form: Linear or non-linear
Direction: Positive, negative or random
Strength: Strong, Moderate, Weak
Look for clustering of data
Look for data points that fall far from the body
of data
• Context
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We can incorporate qualitative info into
our scatterplots
Pearson’s Correlation Coefficient
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We’ve collected bivariate data and
created a scatterplot
We’ve described our scatterplot
Now we’re ready to perform some
numerical summaries
The first is Pearson’s Correlation
Coefficient -- r
Pearson’s Correlation Coefficient
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Pearson’s Correlation Coefficient –
more often referred to as the
correlation coefficient – is a statistic.
The Population Correlation Coefficient is
a parameter – 
r is a numerical measure of the
strength of the linear association
between the variables we are studying.
Properties of r
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The value of r does not depend on the unit of
measurement for either variable.
The value of r does not depend on which of the
two variables is considered the independent
variable.
The value of r is between +1 and -1.
r=+1/-1 when data points lie exactly on a
straight line that slopes upward/downward.
The value of r is a measure of the extent to which
the variables are linearly related
Correlation does not imply causation.
Calculating r
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zx z y
 n 1

x  X y  Y 
i
i
n 1
Output of running the LinReg
function on the TI-83/84
Corr program
Fitting a line
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Up to now
•
•
•
•
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We’ve collected data
Created a scatterplot
Described the scatterplot
Summarized linear association with r
Now we’d like to summarize our linear
scatterplot with a straight line
Since we’ll be using this line for
predictions, it now becomes important to
identify an explanatory and response
variable
How do we fit a ‘best line’
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Our line should be a good summary
of our data and will be used as a
prediction tool
We’d like a line that goes through all
our data points
If that’s not possible, at least a line
that is close to all the points
Sketch a Scatterplot
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Sketch a scatterplot for the following data, fit a line that
you think is a best fitting line
M
730
600
740
570
620
720
620
650
760
550
V
790
590
700
530
540
780
640
620
700
610
M
710
640
590
680
770
540
540
520
710
740
V
670
470
490
720
650
590
600
540
560
650
There are many lines that can be chosen that would be a
reasonable fit
A standardized approach is essential so different analysts
working with the same set of data will produce the same
fits
Least Squares Regression Line
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The least squares regression (LSR) is
the most commonly method to find
the best fit line
As the name implies, it is the line
whose summed squared vertical
deviations is the least among all
possible lines
Least Squares Regression Line
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Terminology
LSRL
yˆ  a  bx
Characteristics
•
e
e
i
•
2
i
0
is minimized
• Passes through
( X ,Y )
• close connection between correlation and slope
br
sy
sx
So what’s important here?
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Identify explanatory and response
variables
The form of your model
Determine the coefficients
Interpret the coefficients
So how do we calculate the
coefficients ?
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By hand? What, are you nuts?
We’ll always use the graphing
calculator (or use a MINITAB output)
So let’s do a regression on the
graphing calculator
And you’ll do a LSRL activity for
homework
LSR on TI-83
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Identify the explanatory and response
variables and enter the data into lists
Create your scatterplot
Now calculate your LSRL
• STAT – CALC – LINREG (a+bx) – ENTER
• X-list – comma – Y-list – comma –
VARS – YVARS – Function – Y1 – ENTER -ENTER
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Now print you scatterplot – and the
LSLR will also be graphed
Interpretation of Coefficients
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Form of your LSLR
• yˆ  a  bx define your variables
• Predicted Resp Var = a + b (Explan Var)
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Interpretation of b
• b is the slope of your LSLR
• Basic Algebra I interpretation
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Interpretation of a
• a is the y-intercept
• Basic Algebra I interpretation
• Put your interpretation into the context of the
problem
• Sometimes the a value ‘makes sense’, other times it
doesn’t – no practical interpretation
Assessing the fit of a line
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So far we’ve collected bivariate data
We’ve displayed the data in a
scatterplot
We’ve calculated the correlation
coefficient
We found a best fit line using the
method of least squares regression
Assessing the fit of a line
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Now we want to assess the fit of our
best fit line
Are there any unusual aspects of the
data that we want to address before
we use our line to predict?
How accurate can we expect our
predictions based on the regression
to be?
Residuals
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We’re going to use the residuals
created by the LSR to assess the fit,
and
To give us a sense for how accurate
our predictions will be
What are the residuals?
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What are the residuals?
Where are they stored in our
calculator?
Assessing the fit of a line
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We create and inspect the residual
plot for our regression
The residual plot is a scatterplot of
the (x, residual) pairs
What are we looking for?
• Isolated points or patterns in the
residual plot indicate potential problems
Why the residual plot?
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Why do we inspect the residual plot
and not just the scatterplot of our
original data?
Sometimes it’s easier to see
curvature or problem points in a
residual plot.
Heights and Weights of American
Women example
To summarize …
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We determine the appropriateness of
our least squares regression line by
inspection of the residual plot
We look for patterns and unusual
values
RANDOMNESS = GOOD
PATTERNS/UNUSUAL VALUES = BAD
Assess accuracy of line
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Once we determine that our LSR line
is an appropriate summary of our
data, we want to assess the accuracy
of predictions
• Are predictions made with the additional
variable better than predictions without
knowledge of the second variable?
Two numerical measures
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The two numerical measures we use in
this part of our assessment are
2
r
• Coefficient of Determination
• Standard deviation about the regression line
Coefficient of Determination
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The Coefficient of Determination provides the
proportion of variability that is explained by
the LSRL – that is the proportion of variation
attributed to the linear association between
the two variables.
It is one of the outputs when we run our
regression.
0  r 1
2
Let’s digress …
Let’s say …
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Let’s use our post knee surgery
range of motion data
Let’s say we wanted to predict range
of motion – but only had the range
of motion data – not the ages of the
patients
How?
Now let’s say …
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You also have additional information
– the ages of the patients
We’ve already seen that there is a
linear association between age and
range of motion
When we fit a line to this bivariate
data, how do the residuals compare?
SSR vs SST
Coefficient of Determination
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If we look at the ratio of SSR to SST,
can you interpret this?
Now subtract this value from 1 – now
give me an interpretation of this
value
It’s equal to R-sq – Coefficient of
Determination
Standard Deviation about the LSRL
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The CoD measures the extent of
variation about the best fit line to the
overall variation in y.
A high R-sq value does not promise
that the deviations from the line are
small in an absolute sense.
The Standard Deviation about the
LSRL – se – is the typical amount an
observation deviates from the LSRL
Standard Deviation about the LSRL
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This value is analogous to the standard
deviation
s
 x  X 
n 1
SSR
se 
n2
Minitab Output
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Minitab is a desktop statistics software
package
You are responsible for reading and using
Minitab output
Things you can be expected to perform:
•
•
•
•
Determine
Determine
Determine
Determine
LSR equation
se
R-sq
R-sq from ANOVA table
Non-linear Relationships and
Transformations
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If our data has a linear pattern in a
scatterplot, our approach is to
summarize the data as a line using the
method of least squares
What if our data does not have a linear
pattern – as depicted in the scatterplot
or the residual plot
Look at the ‘Rice Paddy’ data (L1, L2)
Rice Paddy
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Is the time between flowering and
harvesting related to the yield of a paddy?
Time Btwn
Yield
Time Btwn
Yield
16
18
20
22
24
26
28
30
2508 2518 3304 3423 3057 3190 3500 3883
32
34
36
38
40
42
44
46
3823 3646 3708 3333 3517 3214 3103 2776
Nonlinear Relationships
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We have 2 choices
• We can fit a nonlinear regression
• Transform the data
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Ex: Nonlinear regression for the ‘Rice
Paddy’ data, inspect the residuals, R2
Transforming the Data
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An alternative is to transform the
data so the transformed data
scatterplot is linear,
And then perform a linear regression
We then ‘back-transform’ our data
when we use the LSRL for
predictions.
Let’s look at an example
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Ex: River Velocity and Distance from
Shore
Is it reasonable to fit a line to this
data?
If we wanted to transform our data
• Which variable to transform ?
• What type of transformation ?
River Water Velocity
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As fans of white-water rafting know, a
river flows more slowly close to its banks.
Let’s look at the relationship between
water velocity (cm/sec) and distance from
shore (m).
Distance
.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
Velocity
22
23.18
25.48
25.25
27.15
27.83
28.49
28.18
28.50
28.63
Transformations
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Which variable to transform ?
• Explanatory and/or Response variable
• Transform and then redraw scatterplot
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Which transformation to use ?
• Use the transformation ladder and the shape
of your scatterplot
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Transform the data, inspect the scatterplot
and follow the steps for LSR
Use your LSR to make a prediction
• What is the velocity of the river 9 m from the
bank?
Let’s do another example
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In the previous problem, we only
transformed the explanatory variable.
Consequently the ‘back tranformation’ was
easy.
Let’s do another example where we
transform both variables, run a regression
and then make a prediction.
Tortilla Chips: Relationship between
Frying Time and Moisture Content
Tortilla Chips
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No tortilla chip lover likes soggy chips, so it’s important to
find characteristics of the production process that produce
chips with an appealing texture. The following data on
frying time (sec) and moisture content (%) are given below
Frying Time
5
10
15
20
25
30
45
60
Moisture
Content
16.3
9.7
8.1
4.2
3.4
2.9
1.9
1.3