correlation_regression.ppt

Correlation
Finding the relationship between two
quantitative variables without being able to
infer causal relationships
Correlation is a statistical technique used to
determine the degree to which two
variables are related

Scatter plots
The pattern of data is indicative of the type of
relationship between your two variables:
 positive relationship
 negative relationship
 no relationship

Negative relationship
Reliability
Age of Car

Correlation Coefficient
Statistic showing the degree of relation
between two variables

Simple Correlation coefficient (r)
 It is also called Pearson's correlation
or product moment correlation
coefficient.
 It measures the nature and strength
between two variables of
the quantitative type.

The sign of r denotes the nature of
association
while the value of r denotes the
strength of association.

 If the sign is +ve this means the relation
is direct (an increase in one variable is
associated with an increase in the
other variable and a decrease in one
variable is associated with a
decrease in the other variable).
 While if the sign is -ve this means an
inverse or indirect relationship (which
means an increase in one variable is
associated with a decrease in the other).

 The value of r ranges between ( -1) and ( +1)
 The value of r denotes the strength of the
association as illustrated
by the following diagram.
-1 1
0
-0.25
-0.75 0.75
0.25
strong strong
intermediate intermediate
weak weak
no relation
perfect
correlation
perfect
correlation
Direct
indirect

If r = Zero this means no association or
correlation between the two variables.
If 0 < r < 0.25 = weak correlation.
If 0.25 ≤ r < 0.75 = intermediate correlation.
If 0.75 ≤ r < 1 = strong correlation.
If r = l = perfect correlation.





















 
 
  
n
y)
(
y
.
n
x)
(
x
n
y
x
xy
r
2
2
2
2
How to compute the simple correlation
coefficient (r)

Example:
A sample of 6 children was selected, data about their
age in years and weight in kilograms was recorded as
shown in the following table . It is required to find the
correlation between age and weight.
Weight
(Kg)
Age
(years)
serial
No
12
7
1
8
6
2
12
8
3
10
5
4
11
6
5
13
9
6

Y2
X2
xy
Weight
(Kg)
(y)
Age
(years)
(x)
Serial
n.
144
49
84
12
7
1
64
36
48
8
6
2
144
64
96
12
8
3
100
25
50
10
5
4
121
36
66
11
6
5
169
81
117
13
9
6
∑y2=
742
∑x2=
291
∑xy=
461
∑y=
66
∑x=
41
Total

r = 0.759
strong direct correlation

















6
(66)
742
.
6
(41)
291
6
66
41
461
r
2
2

Regression Analyses
Regression: technique concerned with predicting
some variables by knowing others
The process of predicting variable Y using
variable X

Regression
 Uses a variable (x) to predict some outcome
variable (y)
 Tells you how values in y change as a function
of changes in values of x

Correlation and Regression
 Correlation describes the strength of a linear
relationship between two variables
 Linear means “straight line”
 Regression tells us how to draw the straight line
described by the correlation

Linear Equations
Y
Y = bX + a
a = Y-intercept
X
Change
in Y
Change in X
b = Slope
bX
a
ŷ 


Exercise
A sample of 6 persons was selected the
value of their age ( x variable) and their
weight is demonstrated in the following
table. Find the regression equation and
what is the predicted weight when age is
8.5 years.

Weight (y)
Age (x)
Serial no.
12
8
12
10
11
13
7
6
8
5
6
9
1
2
3
4
5
6

Answer
Y2
X2
xy
Weight (y)
Age (x)
Serial no.
144
64
144
100
121
169
49
36
64
25
36
81
84
48
96
50
66
117
12
8
12
10
11
13
7
6
8
5
6
9
1
2
3
4
5
6
742
291
461
66
41
Total

6.83
6
41
x 
 11
6
66


a
92
.
0
6
)
41
(
291
6
66
41
461
2





b
Regression equation
0.92X
11
ŷ(x) 


0.92x
11
ŷ(x) 

18.82Kg
8.5
*
0.92
11
ŷ(8.5) 


Kg
9
.
17
7.5
*
0.92
11
ŷ(7.5) 



11.4
11.6
11.8
12
12.2
12.4
12.6
7 7.5 8 8.5 9
Age (in years)
Weight
(in
Kg)
we create a regression line by plotting two
estimated values for y against their X component,
then extending the line right and left.

Multiple Regression
Multiple regression analysis is a
straightforward extension of simple
regression analysis which allows more
than one independent variable.

correlation_regression.ppt

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