ppt

1. Correlation and Regression

2.  Meaning of Correlation
⚫ Co – Means two, therefore correlation is a relation between
two variables (like XandY)
⚫ Correlation isaStatistical method that iscommonly usedto
compare two or more variables
⚫ For example, comparison between income and expenditure,
price and demand etc...

3.  Definition of Correlation
⚫ Correlation is astatistical measure for finding out degree
(strength) of association between two or more than two
variables.

4.  T
ypes of Correlation
⚫ There are three types of correlation asfollows :
1. T
ype – 1 correlation
2. T
ype – 2 correlation
3. T
ype – 3 correlation

5.  T
ype –1 correlation
T
ype – 1
correlation
Positive
correlation
Negative
correlation

6.  Positive Correlation
⚫ The correlation is said to be positive, if the values of two
variables changing with same direction.
⚫ In other words asXincreasing,Yis in increasing similarly as
Xdecreasing ,Yis in decreasing.
⚫ For example :Water consumption andTemperature.

7.  Negative Correlation
⚫ The correlation is said to be negative,if the values of two
variables changing with opposite direction.
⚫ In other words asXincreasing ,Yis in decreasing similarly as
Xdecreasing,Yis in increasing.
⚫ For example :Alcohol consumption and Driving ability.

8.  T
ype –2 correlation
T
ype – 2
correlation
Simple
correlation
Multiple
correlation
Partial
correlation
Total
correlation

9.  Simple Correlation
⚫ Under simple correlation problemthere are only two
variables are studied.
 Multiple Correlation
⚫ Under multiple correlation problemthere are three or
more than three variables are studied.

10.  Partial Correlation
⚫ Under multiple correlation problem there are two
variables considered and other variables keeping as
constant, known as partial correlation.
 T
otal Correlation
⚫ T
otal correlation is based on all the relevant variables, which
is normally not feasible .

11.  T
ype –3 correlation
T
ype –3
correlation
Linear
correlation
Non-Linear
correlation

12.  Linear Correlation
⚫ Acorrelation is said to be linear when the amount of change
in one variable tends to bear aconstant ratio to the amount
of change in the other.
⚫ The graph of the variables havingalinear relationship will
form astraight line.
⚫ For example:
⚫ Y= 3+2X (as per above table)
X 1 2 3 4 5
Y 5 7 9 11 13

13.  Non –Linear Correlation
⚫ The correlation would be non-linear,if the amount of change
in one variable does not bear a constant ratio to the amount
of change in the other variable.

14.  The methods tomeasure of correlation
⚫ There are three methods to measure of correlation :
1. Karl Pearson’scoefficient of correlation method
2. Coefficient of correlation for Bivariate Grouped data
method
3. Spearman’sRank correlation method
4. Scatter diagram method

15.  The methods tomeasure of correlation
Karl Pearson’s
coefficient of
correlation method
Ifmean ofx-seriesand
y-series are must be
integers
Direct method
Ifmid value of x-series
and y-series are not
given in instruction
Short-cut method
Ifeither mean of x-
series and y-series are
not an integer
Ifmid value of x-series
and y-series are given
in instruction
Datagiven in term of
middlevalues ofXand
Y.

16.  Definition : Covariance

17. ⚫Case -1:If cov(X , Y )
x y
are integers then r 
 
n
cov( X ,Y ) 
(xi  X ) (yi  Y )
, n  no 'of obsevations
i
x
y
n
n
 ( x  X ) 2
  s t .d e v i a t i o n o f X 
 ( y i  Y ) 2
  s t . d e v i a t i o n o f Y 
X a n d Y
X  mean of x  series
Y  meanof y  series
 KarlPearson’scoefficient(
r
)ofcorrelationmethod
Direct method (Frequency is not given)

18.  
2
dx
n n
r  n




 
 
 

 





  dy 2
 dx 2
  dy 2

dx  x  A, A is assumed mean of x  series
dy  y  B, B is assumed mean of y  series
 KarlPearson’scoefficient(
r
)ofcorrelationmethod
Short-cut Method (Frequency is not given)
⚫Case -2: If either X o r Y may not be integers then
 dx dy 
 dx  dy 

19.  Examples
⚫ Ex-1 : Find the correlation coefficient from the following tabular data :
Ans : 0.845
⚫ Ex-2 : Calculate Karl Pearson’scoefficient of correlation between
advertisement cost and sales asper the data given below:
Ans : 0.7807
X 1 2 3 4 5 6 7
Y 6 8 11 9 12 10 14
Add. Cost 39 65 62 90 82 75 25 98 36 78
Sales 47 53 58 86 62 68 60 91 51 84

20.  Examples
⚫Ex-3: Find the correlation coefficient from the following
tabular data :
Ans: -0.99(approx)
• Ex-4:Calculate Pearson’scoefficient of correlation from the
following taking 100 and 50 asthe assumed average of x-
series and y-series respectively:
X 1 2 3 4 5 6 7 8 9 10
Y 46 42 38 34 30 26 22 18 14 10
X 104 111 104 114 118 117 105 108 106 100 104 105
Y 57 55 47 45 45 50 64 63 66 62 69 61
Ans : -0.67

21. n n
 
 
 

 





 Coefficient(r) of correlation for Bivariate Groupeddata method
⚫ In case of bivariate grouped frequency distribution
,coefficient of correlation is given by
 f u v 
 f u  f v 
r  n
 f u 2
 f v 2
 f u 2
  f v 2

c
c i s l e n g t h o f a n i n t e r v a l
d
d i s l e n g t h o f a n i n t e r v a l
X  A




u  , A i s a s s u m e d m e a n o f x  s e r i e s ,
Y  B
v  , B i s a s s u m e d m e a n o f y  s e r i e s ,

22.  Examples
⚫Ex-5: Find the correlation coefficient between the grouped
frequency distribution of two variables (Profit and Sales)
given in the form of atwo wayfrequencytable :
Ans : 0.0946
(
)
Sales (in rupees ) 
P R
r S
o

f
i
t
80-90 90-100 100-110 110-120 120-130 Total
50-55 1 3 7 5 2 18
55-60 2 4 10 7 4 27
60-65 1 5 12 10 7 35
65-70 - 3 8 6 3 20
Total 4 15 37 28 16 100

23.  Examples
⚫Ex-6 : Find the correlation coefficient between the ages of
husbands and the ages of wives given in the form of a two
wayfrequencytable :
Ans : 0.61
(
)
Ages of Husbands (in years )
a
W g
i e
v s
e y
s r
20-25 25-30 30-35 35-40 Total
15-20 20 10 3 2 35
20-25 4 28 6 4 42
25-30 - 5 11 - 16
30-35 - - 2 - 2
35-40 - - - - 0
Total 24 43 22 6 95

24.  Spearman’sRankCorrelationMethod
⚫ The methods,we discussedin previous section are depends on the
magnitude of the variables.
⚫ but there are situations,where magnitude of the variable is not
possible then we will use“Spearman’sRankcorrelation method”.
⚫ For example we can not measure beauty and intelligence
quantitatively
. It possible to rank individualin order.
⚫ Edward Spearman’sformula for Rank Correlation coefficient R,
asfollows:
R 1 
n3
 n
n  no 'of individualsineach series
d Thedifferencebetweentheranksof thetwo series
6 d 2

25.  Examples
⚫Ex-7: Calculate the rank correlation coefficient if two judges
in abeauty contest ranked the entries follows:
Ans : -1
• Ex-8:Ten students got the following percentage of marks in
mathematics and statistics.Evaluate the rank correlation
between them.
Judge X 1 2 3 4 5
JudgeY 5 4 3 2 1
Roll. No. 1 2 3 4 5 6 7 8 9 10
Marks
in
Maths
78 36 98 25 75 82 90 62 65 39
Marks in Stat.
84 51 91 60 68 62 86 58 53 47
Ans : 0.8181

26.  Scatter DiagramMethod
⚫ In this method first we plot the observations in XY– plane .
⚫ X- Independent variable along with horizontal axis.
⚫ Y- Dependent variable along with vertical axis.

40.  Interpretationof correlation coefficient
⚫ The closer the value of the correlation coefficient is to 1 or -1, the
stronger the relationship between the two variables and the more
the impact their fluctuations will have on each other.
⚫ Ifthe valueofr is1, this denotes aperfect positiverelationship
betweenthe two andcanbe plotted on agraphasaline that goes
upwards, with ahighslope.
⚫ Ifthe valueofr is0.5, this will denote apositiverelationship
betweenthe two variablesandit canbe plotted on agraphasaline
that goes upward, with amoderate slope.
⚫ Ifthe value of r is 0, there is no relationship at all between the
two variables.
⚫ Ifthe value of r is -0.5, this will denote anegative relationship
betweenthe two variablesandit canbe plotted on agraphasaline
that goes downwards with amoderate slope.

41.  Interpretationof correlation coefficient
⚫ If the value of r is -1, it will denote a negative relationship
betweenthe two variablesandit canbe plotted on agraphasaline
that goes downwards with asteep slope.
⚫ Ifthe value of the correlation coefficient is between 0.1 to 0.5 or -
0.1 and -0.5, the two variables in the relationshipare said to be
weakly related.If the value of the correlation coefficient is
between 0.9 and 1 or -0.9 and -1, the two variables are extremely
stronglyrelated.
⚫ Aswe discussed earlier,apositive coefficient will showvariables
that rise at the same time.
⚫ Anegative coefficient,on the other hand,will showvariables that
move in opposite directions.It’s easy to tell the relationship
between bychecking the positive or negative value of the
coefficient.

42. Regression

44.  T
ypes of Regression
⚫ SIMPLEREGRESSION
Studyonlytwo variables at atime.
• MUL
TIPLEREGRESSION
 Studyof more than two variables at atime.

45.  Lines of Regression
(a) Regression EquationYon X
w h e re
Y  Y  b y x ( X  X )
2
2
yx
x
1. b
n
n n
( )

cov(X ,Y)
,
3. x     
 
4. n Total no.of observations
5. byx  regressioncoefficient of regressionlineY on X
2. cov(X ,Y )   XY
 X Y ,
 X 2
  X 
2

46.  Lines of Regression
(b) Regression Equation XonY
w h e re
X  X  b x y ( Y  Y )
2
2
xy
y
1. b
n
n n
( )

cov(X ,Y)
,
3. y     
 
4. n Total no.of observations
5. bxy  regressioncoefficient of regressionline X onY
2. cov(X ,Y )   XY
 X Y ,
Y 2
 Y 
2

47.  Regression Equations
 The algebraic expressions of the regression lines are called
regression equations.
 Sincethere are two regression lines therefore there are two
regression equations.
 Using previous method we haveobtained the regression
equationYon Xas Y= a+ b X and that of XonYas X=a+ bY
 The values of“a”and“b”are dependson the means, the standard
deviation and coefficient of correlation between the two
variables.

48.  Regression equation Yon X
 x
Y  Y  r
 y
( X  X ) w h e re
2
2
n n
n n
1. x      ,
 
2. r  Correlation coefficient between X and Y
3.y     
 
 X 2
  X 
2
Y 2
 Y 
2
4. n Total no.of observation or  f

49.  Regression equation Xon Y
w h e re
 y
X  X  r
 x
(Y  Y )
2
2
n n
n n
1. x      ,
 
2. r  Correlation coefficient between X and Y
3. y     
 
 X 2
  X 
2
Y 2
 Y 
2
4. n Total no.of observation or  f

51. Ex-3 From the following data calculate two equations of
lines regression.
Where correlation coefficient r = 0.5.
Y=0.
4
5X+4
0.5
X=0.
556Y
+22.4
7
Ex-4 From the following data calculate two equations of
lines regression.
Where correlation coefficient r = 0.52.
Y=4.
1
6X+4
09.81
X=0.
065Y
– 9.35
X Y
Mean 60 67.5
Standard
Deviation
15 13.5
X Y
Mean 508.4 23.7
Standard
Deviation
36.8 4.6

52.  Difference between correlation and Regression
1. Describing Relationships
⚫ Correlation describes the degree to which two variables are related.
⚫ Regression gives a method for finding the relationship between two
variables.
2. Making Predictions
⚫ Correlation merely describes how well two variables are related. Analysing
the correlation between two variables does not improve the accuracy with
which the value of the dependent variable could be predicted for a given
value of the independent variable.
⚫ Regression allows us to predict values of the dependent variable for a given
value of the independent variable more accurately
.
3. Dependence BetweenV
ariables
⚫ In analysing correlation, it does not matter which variable is independent
and which isindependent.
⚫ In analysing regression, it is necessary to identify between the dependent
and the independent variable.

53. Assignment
Q-1 Do as directed (Ex-1 to Ex-5 _ solve using Karl pearson’s method)
Ex-1 Find the correlation coefficient between the serum and diastolic
blood pressure and serum cholesterol levels of 10 randomly selected
data of 10 persons.
Ans.
=
0.809
Person 1 2 3 4 5 6 7 8 9 10
Chole
s
terol
(X)
307 259 341 317 274 416 267 320 274 336
Diast
ol ic
B.P(Y)
80 75 90 74 75 110 70 85 88 78
Ex-2 Find the correlation coefficient between Intelligence Ratio (I.R) and
Emotional Ration(E.R) from the following data
Ans. =
0.596
3
Student 1 2 3 4 5 6 7 8 9 10
I.R(X) 105 104 102 101 100 99 98 96 93 92
E.R(Y) 101 103 100 98 95 96 104 92 97 94

54. Assignment
Ex-3 Find the correlation coefficient from the following data Ans. =
-0.79
X 1100 1200 1300 1400 1500 1600 1700 1800 1900 200
Y 0.30 0.29 0.29 0.25 0.24 0.24 0.24 0.29 0.18 0.15
Ex-4 Find the correlation coefficient from the following data Ans. =
0.958
2
X 1 2 3 4 5 6 7 8 9 10
Y 10 12 16 28 25 36 41 49 40 50
Ex-5 Find the correlation coefficient from the following data Ans. =
0.949
5
X 78 89 97 69 59 79 68 61
Y 125 137 156 112 107 138 123 110

55. ⚫Ex-6 : Find the correlation coefficient between the marks of
classtest for the subjectsmaths and science given in the
form of atwo wayfrequencytable :
Assignment
(
)
Ages of Husbands (in years )
a
W g
i e
v s
e y
s r
10-15 15-20 20-25 25-30 Total
40-50 0 1 1 1 3
50-60 3 3 0 1 7
60-70 3 3 3 1 10
70-80 1 0 1 1 3
80-90 0 3 3 1 7
Total 7 10 8 5 30
Ans :0.1413

56. ⚫Ex-7 : Find the correlation coefficient between the marks of
annualexam for the subjectsAccount and statistics given in
the form of atwo wayfrequencytable :
Assignment
Marks in account 
S m
t a
a r
t k
s
60-65 65-70 70-75 75-80 Total
50-60 5 5 5 5 20
60-70 0 5 5 10 20
70-80 8 10 0 22 40
80-90 3 3 3 3 12
90-100 3 3 0 2 8
Total 19 26 13 42 100
Ans :0.45

57. Assignment
Q-2 Two judges in a beauty contest rank
the 12 contestants as follows :
X 1 2 3 4 5 6 7 8 9 10 11 12
Y 12 19 6 10 3 5 4 7 8 2 11 1
What degree of agreement is there between the
judges?
-0.454
Q-3 Nine Students secured the following
percentage of marks in mathematics and
chemistry
Roll.No 1 2 3 4 5 6 7 8 9
Marks in Maths 78 36 98 25 75 82 90 62 65
Marks in Chem. 84 51 91 60 68 62 86 58 53
Find the rank correlation coefficient and
comment on its value.
0.84

58. Assignment
Q-4 What is correlation ?How will you measure it?
Q-5 Define coefficient of correlation. Explain how you will interpret the
value of coefficient of correlation .
Q-6 What is Scatter diagram? Towhat extent does it help in finding
correlation between two variables ?Or Explain Scatter diagram
method.
Q-7 What is Rank correlation?
Q-8 Explain the following terms with an example .
(i) Positive and negative correlation
(ii) Scatter diagram
(iii) correlation coefficient
(iv) total correlation
(v) partial correlation
Q-9 Explain the term regression and state the difference between
correlation and regression.
Q-10 What are the regression coefficient? Stat their properties.
Q-11 Explain the terms Lines of regression and Regression equations.

59. Assignment
Q-12 Two judges in a beauty contest rank
the 12 contestants as follows :
-0.454
X 1 2 3 4 5 6 7 8 9 10 11 12
Y 12 19 6 10 3 5 4 7 8 2 11 1
What degree of agreement is there between the
judges?
Q-13 Nine Students secured the following
percentage of marks in mathematics and
chemistry
Roll.No 1 2 3 4 5 6 7 8 9
Marks in Maths 78 36 98 25 75 82 90 62 65
Marks in Chem. 84 51 91 60 68 62 86 58 53
Find the rank correlation coefficient and
comment on its value.
0.84

60. Mathematicians are born not made