Correlation Analysis: Introduction, definition, utility, types, properties, and degree of correlation, correlation and causation. Measures of coefficient of correlation; Scatter diagram, Karl Pearson’s methods (Deviation method, Product moment method, Variance-Covariance method), Probable error and Karl Pearson’s method of coefficient of correlation. Spearman’s Rank correlation method ( when ranks are given, when ranks are not given, equal or tied ranks), coefficient of determination. Regression Analysis: Meaning, definition, utility of regression. Comparison between correlation and regression. Two lines of regression; Regression equation of line Y on X, Regression equation of line X on Y, Properties of regression lines. Relationship between correlation and regression coefficient. Time Series Analysis: Meaning, definition and utility of time series. Component of time series (Trend, Seasonal variations, Cyclical variations, Irregular variations), Decomposition of time series. Linear trend analysis using freehand method and least square method.

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Quantitative Techniques
Volume-4
(Revised)
1. Correlation and Regression analysis
2. Time series analysis(Measurement of Linear Trend and Seasonal Variations)
E-Book Code : QTVOL4
by
Narender Sharma
“Save Paper, Save Trees, Save Environment”

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E-mail : shakehandwithlife@gmail.com , narender@shakehandwithlife.in
Click on Contents
Correlation Analysis ........................................................................................................................................................................................ 3
Introduction .............................................................................................................................................................................................. 3
Definition of Correlation ...................................................................................................................................................................... 3
Utility of Correlation .............................................................................................................................................................................. 3
Types of Correlation .............................................................................................................................................................................. 3
Properties of Correlation ..................................................................................................................................................................... 5
Degree of Correlation:........................................................................................................................................................................... 6
Correlation and Causation .................................................................................................................................................................. 6
Measures of Coefficient of Correlation .................................................................................................................................................... 7
Scatter Diagram ............................................................................................................................................................................................ 7
Karl Pearson Coefficient of Correlation .............................................................................................................................................. 9
Deviation Method: .................................................................................................................................................................................. 9
Product Moment Method: ................................................................................................................................................................ 10
Variance – Covariance Method ....................................................................................................................................................... 10
Probable Error and Karl Pearson’s Coefficient of Correlation .......................................................................................... 11
Spearman’s Rank Correlation Method ............................................................................................................................................. 12
Rank Correlation coefficient when Ranks are given ............................................................................................................. 12
Rank Correlation coefficient when Ranks are not given ..................................................................................................... 13

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When equal or tied ranks ................................................................................................................................................................. 13
Coefficient of Determination................................................................................................................................................................ 15
Regression Analysis ...................................................................................................................................................................................... 17
Regression (Meaning) ....................................................................................................................................................................... 17
Definition of Regression.................................................................................................................................................................... 17
Utility of Regression ........................................................................................................................................................................... 17
Comparison Between Correlation and Regression ................................................................................................................ 17
Two Lines of Regression ........................................................................................................................................................................ 17
Regression equation of line y on x. ............................................................................................................................................... 17
Regression equation of line x on y ................................................................................................................................................ 17
Properties of regression Lines ....................................................................................................................................................... 18
Relationship between Correlation and Regression Coefficient ............................................................................................. 19
Time series analysis ..................................................................................................................................................................................... 22
Meaning of Time Series ..................................................................................................................................................................... 22
Definitions of Time Series ................................................................................................................................................................ 22
Utility of Time Series .......................................................................................................................................................................... 22
Components of Time Series ............................................................................................................................................................. 22
Analysis or Decompositions of Time Series .............................................................................................................................. 23
Measuring Linear Trends ..................................................................................................................................................................... 24
Free hand curve method ................................................................................................................................................................... 24
Least Square Method .......................................................................................................................................................................... 25
Fitting Straight Line Trend ............................................................................................................................................................. 25
Measurement of Seasonal Variations ............................................................................................................................................... 27
Simple Averages Method .................................................................................................................................................................. 27
References ........................................................................................................................................................................................................ 29
Feedback ........................................................................................................................................................................................................... 29

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Correlation Analysis
Introduction
Consider the following statement
“ Price of a commodity rises as the demand goes up.”
“Temperature rises as the intensity of sunlight increases.”
The above statement are very true. Price with demand and temperature with sunlight have direct or linear relation. Here demand and sunlight are independent variable and price and temperature are dependent variables.
But what degree the price is related to demand and what degree the temperature is related to the intensity of sunlight is the question of discussion.
In the above example one is independent (demand and intensity) and other one is dependent(Price and Temperature) hence,
The correlation gives an indication of how well the two variables move together in a straight line manner. The correlation between X and Y is the same as the correlation between Y and X.
Correlation for a sample is indicated by correlation coefficient(r).
Definition of Correlation
On the bases of above discussion we can define the Correlation as
“The study of relationship between two variables is called correlation analysis”
“The study or measure of degree of linear relationship between an independent variable and dependent variable is called correlation.”
 Correlation analysis deals with the association between two or more variables----------------Simpson and Kafka
 If two or more quantities vary in sympathy, so that movement in one tend to be accompanied by corresponding movements in the other, then they are said to be correlated --------------------Conner
 Correlation analysis attempts to determine the degree of relationship between variables-----------Ya-Lun Chou
Utility of Correlation
 We can measure the degree of relationship between different variables.
 Correlation is the foundation of regression analysis
 Estimation of various factor in economics, business and trade
Types of Correlation
Positive and Negative Correlation:
I) Positive Correlation: If two variables X and Y moves in same direction i.e. if one rises, other rises too and vice versa, then it is called a positive correlation e. g. money and supply
X
Demand
Sunlight
Y
Price
Temperature
Independent Variable
Dpendent Variable

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I) Negative Correlation: If two variables X and Y move in opposite direction i.e. if one rises , other falls , and if one falls other rises , then it is called as negative correlation e.g. demand and price
Linear and Curvi-Linear Correlation:
I) Linear Correlation: If the ratio of change of two variables X and Y remains constant throughout , then they are said to be linearly correlated their relationship is best described by straight line.
II) Curvi-Linear Correlation: The amount of change in one variable does not bear a constant ratio to the amount of change in the other variable e.g. when every time X rises by 10%, then Y rises by 20%,sometimes by 10% and sometimes by 40% then non linear or Curvi-linear correlation exists between them.
Simple Partial and Multiple Correlation:
I) Simple Correlation: Relationship between Two variables
II) Partial Correlation: Among three or more variables, relationship of two variables are studied assuming other as constant
III) Multiple Correlation: Study the relationship among three or more variable.
0
10
20
30
40
0
2
4
6
8
10
12
Positive Correlation
0
5
10
15
20
25
30
0
2
4
6
8
10
12
Negative Correlation
0
10
20
30
40
50
60
0
2
4
6
8
10
12
Curvi-Linear Correlation

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Properties of Correlation
1. Limits of coefficient of Correlation: value of ‘r’ lies between –1 and +1 i.e.
This implies that ‘r’ can never be greater than 1 and also it can’t take the value less than – 1.
2. Not Affected by Change of Origin and Scale. If the origin is shifted or scale is changed i.e.
If X changed to X – 3 and Y changed to Y – 2.
Then the value of correlation coefficient ‘r’ will be the same as for X and Y.
3. Geometric Mean of Regression Coefficients:
√ .
4. It is a Pure Number: ‘r’ is pure number and is independent of the units of measurements. This means that even if two variables are expressed in two different units of measurements e.g. Production in Tons, and power consume is in KW, the value of correlation comes out with a pure number. Thus it does not require that the units of both the variables should be the same.
5. Coefficient of correlation is Symmetric i.e.
:
It means that either we compute the value of correlation coefficient between x and y or between y and x , the coefficient of correlation remains the same.
6. If X and Y are independent variables, then coefficient of correlation is zero but the converse is not necessarily true.
If X and Y are two independent variables then ( , ) , ( , ) ,
Thus if X and Y are independent they are not correlated.
On the contrary : if r=0, the X and Y may not necessarily be independent.
Let the two variables X and Y has a relation then values of Y for different values of X according to the relation, and so the data will given in below table :
X
-3
-2
-1
0
1
2
3
Y
9
4
1
0
1
4
9
XY
-27
-8
-1
0
1
8
27
Hence , ( , ) . . .
Thus , ( , ) .
From the above calculation it is found that although coefficient of correlation is zero, but X and Y are not independent. In fact the variables are related by a quadratic equation i.e. there is a quadratic relation (i.e. non linear relationship) between the variables. This property implies that ‘r’ is only a measure of the linear relationship between X and Y. If the relationship is non-linear, the computed value of ‘r’ is no longer a measure of the degree of relationship between the two variables.

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Degree of Correlation:
The degree of Correlation can be Known by Coefficient of Correlation (r). Various types of correlation are mentioned in below table
S.No.
Degree of Correlation
Coefficient of Correlation (r)
Positive
Negative
1
Perfect Correlation
+1
-1
2
High Degree of Correlation
+0.75 to +1
-0.75 to -1
3
Moderate Degree of Correlation
+0.25 to + 0.75
-0.25 to - 0.75
4
Low Degree of Correlation
0 to +0.25
0 to -0.25
5
Absence of Correlation
0
0
Correlation and Causation
In a small sample it is possible that two variables are highly correlated but in universe or in population, these variables are unlikely to be correlated. Such type of meaningless correlation called causation and it may be either the fluctuations of pure random sampling or due to the bias of investigator in selecting the sample. The below example makes the point clear
Income (in Rs.)
5000
6000
7000
8000
9000
Weight (in Kg)
100
120
140
160
180
The above data stated, there is perfect positive correlation between monthly income and weight. Weight increases with rise in income. Even this kind of correlation can not be meaningful. Such relation is said to be spurious or non – sense correlation.
Another example of such type of correlation
No. of death cases and No. of manglik and non manglik couple
Manglik and Non Manglik Couple
200
300
400
500
600
700
Death Cases
2
3
4
5
6
7
The above data shows a perfect positive correlation between the no. of Manglik and Non – Manglik Couple and death cases in such type of couples. But this type of data has not any ground reality, this type of correlation in the data is called stupid correlation.