Chapter 3
Linear Regression and
Correlation
Descriptive Analysis &
Presentation of Two Quantitative Data
Chapter Objectives
• To be able to present two-variables data in
tabular and graphic form
• Display the relationship between two
quantitative variables graphically using a scatter
diagram.
• Calculate and interpret the linear correlation
coefficient.
• Discuss basic idea of fitting the scatter diagram
with a best-fitted line called a linear regression
line.
• Create and interpret the linear regression line.
Terminology
• Data for a single variable is univariate data
• Many or most real world models have
more than one variable … multivariate
data
• In this chapter we will study the relations
between two variables … bivariate data
Bivariate Data
• In many studies, we measure more than
one variable for each individual
• Some examples are
– Rainfall amounts and plant growth
– Exercise and cholesterol levels for a group
of people
– Height and weight for a group of people
Types of Relations
When we have two variables, they could be related in one
of several different ways
– They could be unrelated
– One variable (the input or explanatory or predictor
variable) could be used to explain the other (the
output or response or dependent variable)
– One variable could be thought of as causing the other
variable to change
Note: When two variables are related to each other, one variable may not
cause the change of the other variable. Relation does not always mean
causation.
Lurking Variable
• Sometimes it is not clear which variable is the
explanatory variable and which is the response
variable
• Sometimes the two variables are related without
either one being an explanatory variable
• Sometimes the two variables are both affected
by a third variable, a lurking variable, that had
not been included in the study
Example 1
• An example of a lurking variable
• A researcher studies a group of
elementary school children
– Y = the student’s height
– X = the student’s shoe size
• It is not reasonable to claim that shoe size
causes height to change
• The lurking variable of age affects both of
these two variables
More Examples
• Some other examples
• Rainfall amounts and plant growth
– Explanatory variable – rainfall
– Response variable – plant growth
– Possible lurking variable – amount of sunlight
• Exercise and cholesterol levels
– Explanatory variable – amount of exercise
– Response variable – cholesterol level
– Possible lurking variable – diet
Types of Bivariate Data
Three combinations of variable types:
1. Both variables are qualitative (attribute)
2. One variable is qualitative (attribute) and the
other is quantitative (numerical)
3. Both variables are quantitative (both
numerical)
Two Qualitative Variables
• When bivariate data results from two qualitative
(attribute or categorical) variables, the data is often
arranged on a cross-tabulation or contingency table

Example: A survey was conducted to investigate the
relationship between preferences for television, radio,
or newspaper for national news, and gender. The
results are given in the table below:
Male
Female
TV
280
115
Radio
175
275
NP
305
170
Marginal Totals
• This table, may be extended to display the marginal
totals (or marginals). The total of the marginal totals is
the grand total:
Male
Female
Col. Totals
TV
280
115
395
Radio
175
275
450
NP Row Totals
305
760
170
560
475
1320
Note: Contingency tables often show percentages (relative
frequencies). These percentages are based on the
entire sample or on the subsample (row or column)
classifications.
Percentages Based on the Grand Total
(Entire Sample)
• The previous contingency table may be converted to
percentages of the grand total by dividing each
frequency by the grand total and multiplying by 100
– For example, 175 becomes 13.3%
 175 

=

100 13.3
 1320

Male
Female
Col. Totals
TV
21.2
8.7
29.9
Radio
13.3
20.8
34.1
NP Row Totals
23.1
57.6
12.9
42.4
36.0
100.0
Illustration
• These same statistics (numerical values describing
sample results) can be shown in a (side-by-side) bar
graph:
Percentages Based on Grand Total
25
Male
20
Female
15
Percent
10
5
0
TV
Radio
Media
NP
Percentages Based on Row (Column) Totals
• The entries in a contingency table may also be expressed
as percentages of the row (column) totals by dividing each
row (column) entry by that row’s (column’s) total and
multiplying by 100. The entries in the contingency table
below are expressed as percentages of the column totals:
Male
Male
Female
Female
Col.
Col.Totals
Totals
TV
TV Radio
Radio
NP
NP Row
RowTotals
Totals
70.9
70.9 38.9
38.9 64.2
64.2
57.6
57.6
29.1
29.1 61.1
61.1 35.8
35.8
42.4
42.4
100.0
100.0 100.0
100.0 100.0
100.0
100.0
100.0
Note:These statistics may also be displayed in a side-by-side
bar graph
One Qualitative & One Quantitative Variable
1. When bivariate data results from one qualitative and one
quantitative variable, the quantitative values are viewed
as separate samples
2. Each set is identified by levels of the qualitative variable
3. Each sample is described using summary statistics, and
the results are displayed for side-by-side comparison
4. Statistics for comparison: measures of central tendency,
measures of variation, 5-number summary
5. Graphs for comparison: side-by-side stemplot and boxplot
Example
 Example:
A random sample of households from three different
parts of the country was obtained and their electric
bill for June was recorded. The data is given in the
table below:
Northeast
Northeast
23.75
23.75
40.50
40.50
33.65
33.65
31.25
31.25
42.55
42.55
50.60
50.60
37.70
37.70
31.55
31.55
38.85
38.85
21.25
21.25
Midwest
Midwest
34.38
34.38
34.35
34.35
39.15
39.15
37.12
37.12
36.71
36.71
34.39
34.39
35.12
35.12
35.80
35.80
37.24
37.24
40.01
40.01
West
West
54.54
54.54
65.60
65.60
59.78
59.78
45.12
45.12
60.35
60.35
61.53
61.53
52.79
52.79
47.37
47.37
59.64
59.64
37.40
37.40
• The part of the country is a qualitative variable with three levels of
response. The electric bill is a quantitative variable. The electric bills
may be compared with numerical and graphical techniques.
Comparison Using Box-and-Whisker Plots
70
The Monthly Electric Bill
60
50
Electric
Bill
40
30
20
Northeast
Midwest
West
The electric bills in the Northeast tend to be more spread out than
those in the Midwest. The bills in the West tend to be higher than both
those in the Northeast and Midwest.
Descriptive Statistics for Two
Quantitative Variables
Scatter Diagrams and correlation
coefficient
Two Quantitative Variables
• The most useful graph to show the
relationship between two quantitative
variables is the scatter diagram
• Each individual is represented by a point
in the diagram
– The explanatory (X) variable is plotted on the
horizontal scale
– The response (Y) variable is plotted on the
vertical scale
Example
 Example: In a study involving children’s fear related to
being hospitalized, the age and the score each child made
on the Child Medical Fear Scale (CMFS) are given in the
table below:
Age (x )
CMFS (y )
8
9
9 10 11
9
8
9
8 11
31 25 40 27 35 29 25 34 44 19
Age (x )
CMFS (y )
7
6
6
8
9 12 15 13 10 10
28 47 42 37 35 16 12 23 26 36
Construct a scatter diagram for this data
Solution
• age = input variable, CMFS = output variable
Child Medical Fear Scale
50
40
CMFS
30
20
10
6
7
8
9
10
11
Age
12
13
14
15
Another Example
• An example of a scatter diagram
Note: the vertical scale is truncated to illustrate the detail
relation!
Types of Relations
• There are several different types of relations
between two variables
– A relationship is linear when, plotted on a scatter
diagram, the points follow the general pattern of a line
– A relationship is nonlinear when, plotted on a scatter
diagram, the points follow a general pattern, but it is
not a line
– A relationship has no correlation when, plotted on a
scatter diagram, the points do not show any pattern
Linear Correlations
• Linear relations or linear correlations have points
that cluster around a line
• Linear relations can be either positive (the points
slants upwards to the right) or negative (the
points slant downwards to the right)
Positive Correlations
• For positive (linear) correlation
– Above average values of one variable are
associated with above average values of the
other (above/above, the points trend right and
upwards)
– Below average values of one variable are
associated with below average values of the
other (below/below, the points trend left and
downwards)
Example: Positive Correlation
• As x increases, y also increases:
60
50
Output
40
30
20
10
15 20
25
30
35
Input
40
45 50
55
Negative Correlations
• For negative (linear) correlation
– Above average values of one variable are
associated with below average values of the
other (above/below, the points trend right and
downwards)
– Below average values of one variable are
associated with above average values of the
other (below/above, the points trend left and
upwards)
Example: Negative Correlation
• As x increases, y decreases:
95
85
Output
75
65
55
10
15 20
25
30
35
Input
40
45 50
55
Nonlinear Correlations
• Nonlinear relations have points that have a
trend, but not around a line
• The trend has some bend in it
No Correlations
• When two variables are not related
– There is no linear trend
– There is no nonlinear trend
• Changes in values for one variable do not seem to have
any relation with changes in the other
Example: No Correlation
• As x increases, there is no definite shift in y:
55
Output
45
35
10
20
Input
30
Distinction between Nonlinear & No
Correlation
Nonlinear relations and no relations are
very different
– Nonlinear relations are definitely patterns …
just not patterns that look like lines
– No relations are when no patterns appear at
all
Example
• Examples of nonlinear relations
– “Age” and “Height” for people (including both
children and adults)
– “Temperature” and “Comfort level” for people
• Examples of no relations
– “Temperature” and “Closing price of the Dow
Jones Industrials Index” (probably)
– “Age” and “Last digit of telephone number” for
adults
Please Note

Perfect positive correlation: all the points lie along a
line with positive slope

Perfect negative correlation: all the points lie along a
line with negative slope

If the points lie along a horizontal or vertical line: no
correlation

If the points exhibit some other nonlinear pattern:
nonlinear relationship

Need some way to measure the strength of correlation
Linear Correlation Coefficient
Measure of Linear Correlation
• The linear correlation coefficient is a measure of the
strength of linear relation between two quantitative
variables
• The sample correlation coefficient “r” is
r =

( xi  x ) ( y i  y )
sx
sy
n 1
X , Y , Sx , S y are the sample means and sample variances
Note:
of the two variables X and Y.
Properties of Linear Correlation
Coefficients
Some properties of the linear correlation
coefficient
– r is a unitless measure (so that r would be the same
for a data set whether x and y are measured in feet,
inches, meters etc.)
– r is always between –1 and +1.


r = -1 : perfect negative correlation
r = +1: perfect positive correlation
– Positive values of r correspond to positive relations
– Negative values of r correspond to negative relations
Various Expressions for r
There are other equivalent expressions for
the linear correlation r as shown below:
(x

r=
 x )( y  y )
(n  1) S x S y
r=
 ( x  x )( y  y )
 ( x  x )  ( y  y)
2
2
However, it is much easier to compute r using the
short-cut formula shown on the next slide.
Short-Cut Formula for r
SS( xy)
r=
SS( x)SS( y)
SS( x ) = “sum of squ ares for x”=  x 2 
SS( y ) = “sum of squ ares for y”=  y 2 
( x)
2
n
( y)
2
n
SS( xy ) = “sum of squ ares for xy”=  xy  
x y
n
Example
The table below presents the weight (in thousands of
 Example:
pounds) x and the gasoline mileage (miles per gallon) y for ten different
automobiles. Find the linear correlation coefficient:
x
Sum
Sum
Sum
y
x
2
2.5
2.5
2.5
3.0
3.0
3.0
4.0
4.0
4.0
3.5
3.5
3.5
2.7
2.7
2.7
4.5
4.5
4.5
3.8
3.8
3.8
2.9
2.9
2.9
5.0
5.0
5.0
2.2
2.2
2.2
34.1
34.1
34.1
40
40
40
43
43
43
30
30
30
35
35
35
42
42
42
19
19
19
32
32
32
39
39
39
15
15
15
14
14
14
309
309
309
6.25
6.25
6.25
9.00
9.00
9.00
16.00
16.00
16.00
12.25
12.25
12.25
7.29
7.29
7.29
20.25
20.25
20.25
14.44
14.44
14.44
8.41
8.41
8.41
25.00
25.00
25.00
4.84
4.84
4.84
123.73
123.73
123.73
x
y
 x2
y2
1600
1600
1600
1849
1849
1849
900
900
900
1225
1225
1225
1764
1764
1764
361
361
361
1024
1024
1024
1521
1521
1521
225
225
225
196
196
196
10665
10665
10665
 y2
xy
100.0
100.0
100.0
129.0
129.0
129.0
120.0
120.0
120.0
122.5
122.5
122.5
113.4
113.4
113.4
85.5
85.5
85.5
121.6
121.6
121.6
113.1
113.1
113.1
75.0
75.0
75.0
30.8
30.8
30.8
1010.9
1010.9
1010.9
 xy
Completing the Calculation for r
SS( x) = 
x)
(

2
x 
n
SS( y ) =  y
SS( xy) = 
r=
2
2
y)
(


n
2
(34.1) 2
= 123.73 
= 7.449
10
(309) 2
= 10665 
= 1116.9
10
x y
(34.1)(309)

xy 
= 1010.9 
= 42.79
SS ( xy )
=
SS ( x )SS ( y )
n
10
 42.79
( 7.449 )(1116 .9 )
= 0.47
Please Note

r is usually rounded to the nearest hundredth

r close to 0: little or no linear correlation

As the magnitude of r increases, towards -1 or +1,
there is an increasingly stronger linear correlation
between the two variables

We’ll also learn to obtain the linear correlation
coefficient from the graphing calculator.
Positive Correlation Coefficients
• Examples of positive correlation
Strong Positive
r = .8
Moderate Positive
r = .5
Very Weak
r = .1
• In general, if the correlation is visible to the
eye, then it is likely to be strong
Negative Correlation Coefficients
• Examples of negative correlation
Strong Negative
r = –.8
Moderate Negative
r = –.5
Very Weak
r = –.1
• In general, if the correlation is visible to the
eye, then it is likely to be strong
Nonlinear versus No Correlation
• Nonlinear correlation and no correlation
Nonlinear Relation
No Relation
• Both sets of variables have r = 0.1, but the
difference is that the nonlinear relation
shows a clear pattern
Interpret the Linear Correlation
Coefficients
• Correlation is not causation!
• Just because two variables are correlated does
not mean that one causes the other to change
• There is a strong correlation between shoe sizes
and vocabulary sizes for grade school children
– Clearly larger shoe sizes do not cause larger
vocabularies
– Clearly larger vocabularies do not cause larger shoe
sizes
• Often lurking variables result in confounding
How to Determine a Linear
Correlation?
• How large does the correlation coefficient
have to be before we can say that there is
a relation?
• We’re not quite ready to answer that
question
Summary
• Correlation between two variables can be described with
both visual and numeric methods
• Visual methods
– Scatter diagrams
– Analogous to histograms for single variables
• Numeric methods
– Linear correlation coefficient
– Analogous to mean and variance for single variables
• Care should be taken in the interpretation of linear
correlation (nonlinearity and causation)
Linear Regression Line
Learning Objectives
• Find the regression line to fit the data and use
the line to make predictions
• Interpret the slope and the y-intercept of the
regression line
• Compute the sum of squared residuals
Regression Analysis
• Regression analysis finds the equation of
the line that best describes the relationship
between two variables
• One use of this equation: to make predictions
Best Fitted Line
• If we have two variables X and Y which tend to be
linearly correlated, we often would like to model the
relation with a line that best fits to the data.
• Draw a line through the scatter diagram
• We want to find the line that “best” describes the linear
relationship … the regression line
Residuals
• One difference between math and stat is
that statistics assumes that the
measurements are not exact, that there is
an error or residual
• The formula for the residual is always
Residual = Observed – Predicted
• This relationship is not just for this chapter
… it is the general way of defining error in
statistics
What is a Residual?
• Here shows a residual on the scatter
The residual
The regression line
diagram
The observed value y
The predicted value y
The x value of interest
Example
• For example, say that we want to predict a value
of y for a specific value of x
– Assume that we are using y = 10 x + 25 as our model
– To predict the value of y when x = 3, the model gives
us y = 10  3 + 25 = 55, or a predicted value of 55
– Assume the actual value of y for x = 3 is equal to 50
– The actual value is 50, the predicted value is 55, so
the residual (or error) is 50 – 55 = –5
Method of Least Squares
•
•
We want to minimize the prediction errors or residuals, but we need
to define what this means
We use the method of least-squares which involves the following 3
steps:
1. We consider a possible linear model to fit the data
2. We calculate the residual for each point
3. We add up the squares of the residuals ( We square all of the residuals
to avoid the cancellation of positive residuals and negative residuals,
since some observed values are under predicted, some of the
observed valued are over predicted by the proposed linear model.)
•
The line that has the smallest overall residuals ( i.e. the sum of all
the squares of the residuals) is called the least-squares regression
line or simply the regression line which is the best-fitted line to the
data.
Method of Least Squares
• Assume the equation of the best-fitting line:
yˆ =b0 b1 x
Where
yˆ
(called, y hat) denotes the predicted value of
• Least squares method:
Find the constants b0 and b1 such that the sum
2
2
ˆ
(
y

y
)
=
(
y

(
b

b
x
))


0
1
of the overall prediction errors is as small as possible
y
Illustration
• Observed and predicted values of y:
y
^y = b0  b1 x
 ( x, y)
y  ^y
 ( x , ^y )
^y
y
x
Linear Regression Line
• The equation for the regression line is given by
yˆ =b0 b1 x
– Yˆ denotes the predicted value for the response
variable.
– b1 is the slope of the least-squares regression line
– b0 is the y-intercept of the least-squares regression
line
Note: Different textbooks may use different notations for the slope
and the intercept.
Find the Equation of a Linear
Regression Line
• The equation is determined by:
b0: y-intercept
b1: slope
• Values that satisfy the least squares criterion:
( x  x )( y  y ) SS( xy )

b1 =
=
2
SS( x )
 ( x  x)
y  (b1   x )

b0 =
= y  (b1  x)
n
Example
 Example: A recent article measured the job satisfaction of
subjects with a 14-question survey. The data
below
represents the job satisfaction scores, y, and the
salaries, x, for a sample of similar individuals:
x
y
31
17
33
20
22
13
24
15
35
18
29
17
23
12
37
21
1) Draw a scatter diagram for this data
2) Find the equation of the line of best fit (i.e., regression line)
Finding b1 & b0
• Preliminary calculations needed to find b1 and b0:
x
23
23
23
23
31
31
31
31
33
33
33
33
22
22
22
22
24
24
24
24
35
35
35
35
29
29
29
29
37
37
37
37
234
234
234
234
x
y
xy
x2
12
12
12
12 529
529
529
529 276
276
276
276
17
17
17
17 961
961
961
961 527
527
527
527
20
20
20
20 1089
1089
1089
1089 660
660
660
660
13
13
13
13 484
484
484
484 286
286
286
286
15
15
15
15 576
576
576
576 360
360
360
360
18
18
18
18 1225
1225
1225
1225 630
630
630
630
17
17
17
17 841
841
841
841 493
493
493
493
21
21
21
21 1369
1369
1369
1369 777
777
777
777
133
133
133
133 7074
7074
7074
7074 4009
4009
4009
4009
y
 x2
 xy
Linear Regression Line
x)
 234 
(

x 
= 7074 
= 229.5
2
SS( x ) = 
2
b1 =
b0
 8 


n
SS( xy ) = 
2
x y
(234)(133) 


xy 
= 4009 
= 118.75

n
8

SS( xy ) = 118.75 =
0.5174
SS( x )
229.5
y  (b1   x ) 133  (0. 5174)(234)

=
=
= 14902
.
n
8
. 0. 517 x
Solution 1) Equation of the line of best fit: ^y = 149
Scatter Diagram
Solution 2)
Job Satisfaction Survey
22
21
20
19
18
Job
Satisfaction
17
16
15
14
13
12
21
23
25
27
29
Salary
31
33
35
37
Please Note

Keep at least three extra decimal places while doing the
calculations to ensure an accurate answer

When rounding off the calculated values of b0 and b1,
always keep at least two significant digits in the final
answer

The slope b1 represents the predicted change in y per unit increase in
x

The y-intercept is the value of y where the line of best fit intersects the
y-axis. That is, it is the predicted value of y when x is zero.

The line of best fit will always pass through the point
( x, y)
Please Note
• Finding the values of b1 and b0 is a very
tedious process
• We should also know to use Graphing
calculator for this
• Finding the coefficients b1 and b0 is only
the first step of a regression analysis
– We need to interpret the slope b1
– We need to interpret the y-intercept b0
Making Predictions
1. One of the main purposes for obtaining a regression
equation is for making predictions
2. For a given value of x, we can predict a value of ^y
3. The regression equation should be used only to cover the
sample domain on the input variable. You can estimate
values outside the domain interval, but use caution and use
values close to the domain interval.
4. Use current data. A sample taken in 1987 should not be
used to make predictions in 1999.
Interpret the Slope
• Interpreting the slope b1
– The slope is sometimes defined as as
Rise
Run
– The slope is also sometimes defined as as
Change in y
Change in x
• The slope relates changes in y to changes
in x
Interpret the Slope
• For example, if b1 = 4
– If x increases by 1, then y will increase by 4
– If x decreases by 1, then y will decrease by 4
– A positive linear relationship
• For example, if b1 = –7
– If x increases by 1, then y will decrease by 7
– If x decreases by 1, then y will increase by 7
– A negative linear relationship
Example
• For example, say that a researcher studies the
population in a town (which is the y or response variable)
in each year (which is the x or predictor variable)
– To simplify the calculations, years are measured from 1900 (i.e.
x = 55 is the year 1955)
• The model used is
y = 300 x + 12,000
• A slope of 300 means that the model predicts that, on
the average, the population increases by 300 per year.
• An intercept of 12,000 means that the model predicts
that the town had a population of 12,000 in the year
1900 (i.e. when x = 0)
Interpret the y-intercept
• Interpreting the y-intercept b0
• Sometimes b0 has an interpretation, and
sometimes not
– If 0 is a reasonable value for x, then b0 can be
interpreted as the value of y when x is 0
– If 0 is not a reasonable value for x, then b0 does not
have an interpretation
• In general, we should not use the model for
values of x that are much larger or much smaller
than the observed values of x included (that is, it
may be invalid to predict y for x values lying
outside the range of the observed x.)
Summary
• Summarize two quantitative data
– Scatter diagrams
– Correlation coefficients
• Linear models of correlation
– Least-squares regression line
– Prediction
Obtain Linear Correlation Coefficient and
Regression Line Equation from TI Calculator
1. Turn on the diagnostic tool: CATALOG[2nd 0]  DiagnosticOn 
ENTER  ENTER
2. Enter the data: STAT  EDIT. Enter the x-variable data into L1 and
the corresponding y-variable data into L2
3. Obtain regression line and the linear correlation r: STAT  CALC 
4:LinReg(ax+b)  ENTER  L1, L2, Y1 (Notice: to enter Y1, use
VARS  Y-VARS  1:Function  1:Y1  ENTER). (The screen will also show r2.
Just ignore it.)
4. Display the scatter diagram and the fitted regression line:
Zoom  9:ZoomStat  TRACE (press up or down arrow keys to
move the cursor to the regression line. Now, you can trace the
points along the line by pressing the right or left arrow keys.
While the cursor is on the regression line, you can also enter a
number, the screen will show the predicted value of y for the x
value you just entered.)