The document discusses various statistical concepts related to correlation and regression. It defines the coefficient of correlation as a measure of the strength and direction of the relationship between variables ranging from +1 to -1. A value close to 0 indicates no relationship, while values close to +1 or -1 indicate a strong positive or negative linear relationship, respectively. It also discusses the covariance and correlation of random variables, Pearson correlation coefficient, Spearman rank correlation, partial correlation coefficient, and multiple correlation coefficients. Finally, it provides a definition of regression as a technique to determine the mathematical relationship between two variables using a regression line equation.

1. 1
Coefficient of Correlation
Statistical correlation is measured by what is called the coefficient of
correlation (r). Its numerical value ranges from +1.0 to -1.0. It gives us an indication of
both the strength and direction of the relationship between variables.
In general, 0r indicates a positive relationship, 0r indicates a negative
relationship and r= 0 indicates no relationship (or that the variables are independent of
each other and not related). Here r= +1.0 describes a perfect positive correlation and r= -
1.0 describes a perfect negative correlation.
The closer the coefficients are to +1.0 and -1.0, the greater the strength of the relationship
between the variables.
As a rule of thumb, the following guidelines on strength of relationship are often useful
(though many experts would somewhat disagree on the choice of boundaries).
Value of r Strength of relationship
-1.0 to -0.5 or 1.0 to 0.5 Strong
-0.5 to -0.3 or 0.3 to 0.5 Moderate
-0.3 to -0.1 or 0.1 to 0.3 Weak
-0.1 to 0.1 None or very weak
Correlation is only appropriate for examining the relationship between
meaningful quantifiable data (e.g. air pressure, temperature) rather than categorical data
such as gender, color etc.
i.e. if the correlation coefficient is close to 1, it would indicate that the variables are
positively linearly related and the scatter plot falls almost along a straight line with
positive slope. For -1, it indicates that the variables are negatively linearly related and the
scatter plot almost falls along a straight line with negative slope. And for zero, it would
indicate a weak linear relationship between the variables.

2. 2
Covariance Correlation
Covariance and correlation are both describe the degree of similarity between two
random variables. Suppose that X and Y are real-valued random variables for the
experiment with means E(X), E(Y) and variances var(X), var(Y), respectively.
The covariance of X and Y is defined by
Cov (X, Y) = E [(X − E(X)) (Y − E(Y))],
and the correlation of X and Y is defined by
   
   YstdXstd
YX
YXcor
,cov
, 
The correlation coefficient of two variables in a data set equals to their covariance
divided by the product of their individual standard deviations. It is a normalized
measurement of how the two are linearly related.
Formally, the sample correlation coefficient is defined by the following formula, where
xs and ys are the sample standard deviations, and xys is the sample covariance.
yx
yx
ss
s
xyr 
Similarly, the population correlation coefficient is defined as follow, where x and y
are the population standard deviations, and xy is the population covariance.
yx
yx
xy


 
The correlation coefficient is measured on a scale that varies from + 1 through 0
to - 1. Complete correlation between two variables is expressed by either + 1 or -1. When
one variable increases as the other increases the correlation is positive; when one
decreases as the other increases it is negative. Complete absence of correlation is
represented by 0. This figure is Correlation illustrated gives some graphical
representations of correlation.

3. 3
If the points plotted were all on a straight line we would have perfect correlation,
but it could be positive or negative as shown in the diagrams above,
a. Strong positive correlation between x and y. The points lie close to a straight line with
y increasing as x increases.
b. Weak, positive correlation between x and y. The trend shown is that y increases as x
increases but the points are not close to a straight line
c. No correlation between x and y; the points are distributed randomly on the graph.
d. Weak, negative correlation between x and y. The trend shown is that y decreases as x
increases but the points do not lie close to a straight line
e. Strong, negative correlation. The points lie close to a straight line, with y decreasing as
x increases
Correlation can have a value:
1. 1 is a perfect positive correlation
2. 0 is no correlation (the values don't seem linked at all)
3. -1 is a perfect negative correlation
The value shows how good the correlation is (not how steep the line is), and if it
is positive or negative.
Correlation Symbol
Symbol of correlation = r
The Pearson correlation coefficient is given by the following formula
Correlation (r) =
  
     2222

 


YYnXXn
YXXYn

4. 4
or
  
     nYYnXX
nYXXY
r
//
/
2222

 



or
   
    
 



22
YYXX
YYXX
r
ii
ii
Where,
X and Y are the variables.
n= Number of values or elements
X= First score
Y= Second score
∑XY = Sum of the product of the first and second scores.
∑X = Sum of first scores.
∑Y = Sum of second scores.
∑X2 = Sum of square first scores.
∑Y2 = Sum of square second scores.
Correlation Example
Given below is example to calculate the correlation coefficient.
To determine the correlation value for the given set of X and Y values:
X Values 21 23 37 19 24 33
Y Values 2.5 3.1 4.2 5.6 6.4 8.4
Solution:
Let us count the number of values.
N = 6
Determine the values for XY, X2, Y2
X Value Y Value X*Y X*X Y*Y
21 2.5 52.5 441 6.25
23 3.1 71.3 529 9.61
37 4.2 155.4 1369 17.64
19 5.6 106.4 361 31.36
24 6.4 153.6 576 40.96
33 8.4 277.2 1089 70.56

5. 5
Determine the following values ∑X, ∑Y , ∑XY , ∑X2 , ∑y2.
∑X=157
∑Y=30.2
∑XY=816.4
∑X2=4365
∑Y2=176.38
Correlation (r) =
  
     2222

 


YYNXXN
YXXYN
=
.24)(1541)(146
157
(r) =0.33 It means the correlation between X, Y is weak and positive.
Usually, in statistics, there are three types of correlations: Pearson correlation, Kendall
rank correlation and Spearman correlation.
Spearman correlation
Spearman's rank correlation coefficient allows us to identify easily the strength of
correlation within a data set of two variables, and whether the correlation is positive or
negative. The Spearman coefficient is denoted with the Greek letter rho (ρ).
 1
6
1 2
2



nn
d

For example to calculate the Spearman correlation is,
Spearman correlation from the data
Child
number
Rank
height
Rank dead
space
d d2
1 1 3 2 4
2 2 1 -1 1
3 3 2 -1 1
4 4 4 0 0
5 5 5.5 0.5 0.25
6 6 11 5 25
7 7 7 0 0
8 8 5.5 -2.5 6.25
9 9 8 -1 1
10 10 13 3 9
11 11 10 -1 1
12 12 9 -3 3
13 13 12 -1 1
14 14 15 1 1

6. 6
15 15 14 -1 1
Total 60.5
Thus we get that the value is very close to that of the Pearson correlation coefficient.
There are two kinds of coefficient correlation
1- Partial correlation coefficient
  2
23
2
13
231312
3.12
11 rr
rrr
r



or
  2
32
2
12
321213
2.13
11 rr
rrr
r



or
  2
31
2
21
312123
11
1.23
rr
rrr
r



Example 1:-
Assume that X1 is the income family for every month, X2 represent spending
family for every month also and X3 the no. of person in the family and no. of random
sample is 20 family, the following of corr. Coef. are 12r =0.91, 13r =0.39 and 23r =0.62,
and 12r = 21r , 13r = 31r and 23r = 32r
Calculate partial correlation coefficient between income and spending by no. of person in
family.
  2
23
2
13
231312
3.12
11 rr
rrr
r


 =
  22
)62.0(1)39.0(1
62.039.091.0

 x
= H.W.
  2
32
2
12
321213
2.13
11 rr
rrr
r


 =
  22
)62.0(1)91.0(1
62.091.039.0

 x
H.W.
  2
31
2
21
312123
11
1.23
rr
rrr
r


 =
  22
)39.0(1)91.0(1
39.091.062.0

 x
H.W.

7. 7
Theorem1:- If 12r = 13r = 32r = r prove that 3.12r = 2.13r = 1.23r =
r
r
1
Sol:-
  2
23
2
13
231312
3.12
11 rr
rrr
r



 12r = 13r = 32r = r =
  22
11 rr
rrxr


=
 22
2
1 r
rr


=
  rr
rr


11
)1(
=
r
r
1
The same procedure you do for 2.13r and 1.23r
  2
32
2
12
321213
2.13
11 rr
rrr
r



H.W.
1.23r =
  2
31
2
21
312123
11 rr
rrr


H.W.
Multiple correlationcoefficients
If X1, X2, X3 have three random variables then multiple corr. Coef. between
X1and both X2 X3 than calculate according to the following formula.
2
23
231312
2
13
2
12
23.1
1
2
r
rrrrr
r



Where r12 is the corr. between x1, x2
Where r13 is the corr. between x1, x3
Where r23 is the corr. between x2, x3
Example 2:-
Assume that X1 represent of producing wheat, X2 represent the type of seeds and
X3 represent the type used of manufactures, calculated corr. coef. between these
variables with the following results
Calculate the multiple corr. coef. between X1 and both X2 and X3.
12r =0.70, 13r =0.80 and 23r =0.55

8. 8
2
23
231312
2
13
2
12
23.1
1
2
r
rrrrr
r



2
22
)55.0(1
)55.0)(80.0)(70.0(280.070.0


 = H.W.
Theorem2:-
If 12r = 13r = 23r = r prove that 12.313.223.1 rrr  =
r
r
1
2 2
Sol:-
2
23
231312
2
13
2
12
23.1
1
2
r
rrrrr
r



 12r = 13r = 32r = r = 2
22
1
...2
r
rrrrr


= 2
32
1
.22
r
rr


=
)1)(1(
)1(2 2
rr
rr


=
r
r
1
2 2
The same procedure you do for 13.2r and 12.3r .
Regression
In mathematics, regression is one of the important topics in statistics. The process
of determining the relationship between two variables is called as regression. It is also
one of the statistical analysis methods that can be used to assessing the association
between the two different variables. Here, we will study about the regression and also
some example problems in regression.
Regression Definition
A regression is a statistical analysis assessing the association between two
variables. It is used to find the relationship between two variables. A technique used to
discover a mathematical relationship between two variables. The equation for the
regression line be:
bxay 

where a is a constant i.e. Intercept   NXbNYa //
, b is the regression coefficient i.e. Slope     22
/    XXNYXXYNb
, x is the value of the independent variable, and y

is the predicted value of the dependent
variable. It means x and y are the variables.
b= the slope of the regression line is also called as regression coefficient

9. 9
a= intercept point of the regression line which is in the y-axis.
N= Number of values or elements.
X= First score
Y= Second score
∑XY = Sum of the product of the first and second scores
∑X = Sum of first scores
∑Y = Sum of second scores
∑X2 = Sum of square first scores.
There are two kinds of regression:-
1- Simple Regression
Simple regression analysis involves a single independent, or predictor variable
and a single dependent variable. It is analysis whereas the correlation does not distinguish
between independent and dependent variables. In simple regression analysis, there is no
partialling out of other variables because no other variables are included in the regression.
The equation of the probabilistic simple regression is
y = β0+β1x1+ε
where, y is the value of the dependent variable
β0 is the population y intercept
β0 is the population slope
ε the error of prediction.
2- Multiple Regression
Regression analysis with two or more independent variables or with at least one
nonlinear predictor is called multiple regression analysis. Multiple regression analysis is
similar to simple regression analysis. However it is more complex conceptually and
computationally.
The equation of the probabilistic multiple regression is
y = β0+β1 x1+β2 x2+..............+βi xi +ε
where, y is the value of the dependent variable, β0 is the regression constant and β1, β2,
..., βi is the partial regression coefficient for the independent variables, 1, 2,...., i
respectively. 'i' is the number of independent variables.
Example of Regression
Determine the regression equation by using the regression slope coefficient and intercept
value as shown in the regression table given below.

10. 10
X Values 55 60 65 70 80
Y Values 52 54 56 58 62
For the given data set of data, solve the regression slope and intercept values.
Solution:
Let us count the number of values.
N = 5
Determine the values for XY, X2
X Value Y Value X*Y X*X
55 52 2860 3025
60 54 3240 3600
65 56 3640 4225
70 58 4060 4900
80 62 4960 6400
Determine the following values ∑X, ∑Y, ∑XY, ∑X2.
∑X=330
∑Y=282
∑XY=18760
∑X2=22150
Substitute values in the slope formula
Slope     22
/    XXNYXXYNb =(5)×(18760)−(330)×(282)/
(5)×(22150)−(330)2  Then b = 0.4
Substitute the values in the intercept formula given.
Intercept   NXbNYa // = 282/5−0.4 (330)/ 5  Then a=30
Substitute the regression coefficient value and intercept value in the regression equation
 bxay 

= 30 + 0.4x