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CORRELATION

A Statistical technique that is used to analyse the

strength and direction of the relationship betweentwo quantitative variable is called Correlational

analysis.

Two variables are said to be in correlation if the

change in one of the variable results in a change in

other variable.

E g :- 1) Frequency of smoking and lungs damage ,

2) Sales revenue and expenses incurred on

advertising.


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Importance of correlation

If variables are linearly related to each otherthen it helps in estimation of one from theother.

Advertisement and sales

Prices and Demand

We use Regression Analysis to find the valueof one variable from the other


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TYPES OF CORRELATION

POSITIVE AND NEGATIVE

LINEAR AND NON-LINEAR

SIMPLE ,PARTIAL AND MULTIPLE


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POSITIVE CORRELATION AND

NEGATIVE CORRELATION

POSITIVECORRELATION NEGATIVE CORRELATION

If the variables vary in

same direction,

correlation is said to bePOSITIVE.

If one variable increases,

the other also increases on

the other hand, if one

variable decreases, the

other also decreases.

If both variables vary in

the opposite direction,

correlation is said to beNEGATIVE.

If one variable increases

and the other decreases, or

one decreases the other

increases.


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LINEAR CORRELATION NON-

LINEAR CORRELATION

LINEAR CORRELATION

If the extent of changein one variable tends tohave a constant ratio inthe extent of change inanother variable, thenthe correlation is said tobe LINEAR.

NON-LINEAR CORRELATION

If the extent of changein one variable tends tohave no consistent ratioin the extent of changein another variable,then the correlation issaid to be NON-LINEAR.


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SIMPLE,PARTIAL AND MULTIPLE

CORRELATION

When only two variables are involved,

it is simple correlation When three or more than three

variables are involved, we can computeeither partial or multiple correlation


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Methods of

correlation

graphic

Scatter diagram

algebraic

1. Karl pearson

2. Rank method


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Scatter Diagram

Scatter diagram is a graph or chart which helps todetermine whether there is a relationship between twovariables by examining the graph of the observed data.

A scattered diagram can give us two types of information:

Pattern that indicate that the variables are related.

If the variables are related,what kind of line orestimating equation,describes this relationship.


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KARLS PEARSONS

COEFFIENT OF CORRELATION

Karl Pearsons Coefficient of Correlationdenoted by- r The coefficient of

correlation r measure the degree oflinear relationship between twovariables say x & y.

r = N dxdy - dxdyN dx-(dx)N dy-(dy)


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The value of correlation coefficient rranges from -1 to +1If r = +1, then the correlation between the

two variables is said to be perfect andpositiveIf r = -1, then the correlation between the

two variables is said to be perfect andnegativeIf r = 0, then there exists no correlation

between the variables

Interpretation of Correlation

Coefficient (r)


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REGRESSION

The statistical technique that express the

relationship between two or more variables in the

form of an equation to estimate the value of a

variable, based on the given value of another

variable is called regression analysis.

eg :- Profit after Sales of a firm.


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Difference between dependent variable

and independent variable

Independent Variable

1. The known variable is called

the independent variable.

2. What we typically call X.

3. Variable that is controlled or

manipulated.

4. It is plotted on horizontal axis.

5. An input variable.

Dependent Variable

1. The variable we are trying to

predict is the dependent

variable.

2. What we typically call Y.

3. Variable that cannot be

controlled or manipulated.

4. It is plotted on vertical axis.

5. An output variable.


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Difference between Regression

and Correlation

Regression

A statistical method used

to describe the nature ofrelationship.

In linear regression analysisone variable is considered

as dependent variable and

other as independent

variable

Correlation

A statistical method used

to determine whether arelationship between two

or more variables exist.

In correlation analysis weexamine the degree of

association between two

variables


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Advantages of Regression Analysis

It helps in developing a regression equation

by which the value of a dependent variable

can be estimated given a value of an

independent variable.

It helps to determine standard error of

estimate to measure the variability or spread

of values of a dependent variable with

respect to the regression line.


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Estimation using the Regression Line

The equation for a straight line where thedependent variable Y is determined by the

independent variable X is:

Y = a + bxWhere,

a = y-intercept

b = slope of the line

Y = value of dependent variable

X = value of independent variable


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THE METHOD OF LEAST SQUARE

It is a method of having a good fit of a line

which minimizes the error between the

estimated points on the line and actual

points that were used to draw it.

In this method Y represents the individual

value of the observed points measured along

the Y-axis and Y(y-hat) symbolize the

individual values of the estimated points. The Estimated Line is:

= a + bx


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COEFFICIENT OF

DETERMINATION

The convenient way of interpreting thevalue of correlation coefficient is to use ofsquare of coefficient of correlation whichis called Coefficient of Determination.

The Coefficient of Determination is r2.

r2

= 1- (Y- )2

(Y-Y)2


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STANDARD ERROR OF ESTIMATE

Standard error of estimate measures thevariability of the scatter of the observed

values around the regression line.

It is given by:

Se= (Y- )2

n-2If Se=0, the estimating equation is expected to

be a perfect estimator of the dependent

variable.


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WHAT DOES TIME-SERIES MEAN?

A time series is a sequence of data points,

measured typically at successive points in time

spaced at uniform time intervals.

Time series is a set of measurements of avariable that are ordered through time

Time series analysis comprises methods for

analyzing time series data in order to extractmeaningful statistics and other characteristics

of the data

DIFFERENCE WITH REGRESSION


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DIFFERENCE WITH REGRESSIONANALYSIS

Timeseries Analysis

Regression Analysis

Time series forecasting is the

use of a model to predict

future values based on

previously observed values.

Regression analysis is often

employed in such a way as

to test theories that the

current value of one time

series affects the current

value of another time

series.

Regression analysis cannot

explain seasonal and cyclical

effects.

It shows or suggestsperiodicity of a data like

seasonal and cyclical

effects.


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COMPONENTS OF TIME SERIES

SECULAR TREND

CYCLICAL VARIATIONS

SEASONAL VARIATIONS

IRREGULAR VARIATIONS


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Units

years

Upward trend of sales of Laptops in Ranchi

2000 2001 2002 2003 2004 2005 2006 2007

2000

4000

6000

8000

10000


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units

(in

000

)

years

Declining trend of using Landline Phones in India

2000 01 02 03 05 06 07 08 09 10 11

30

60

90

120

150

180

04


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CYCLICAL VARIATION

Cyclical variations are long-term movements that representconsistently recurring rises and declines in activity.

Timing is the most important factor which affect

the Cyclical Variations.

for example- Business Cycle, it consists of the recurrence ofthe up and down movements of business activity


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depression

prosperity

Prosperity orboom

Economicactivities

time

Cyclical Variation(Business cycle)


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SEASONAL VARIATION

Seasonal variations are those periodic movements in business

activity which occur regularly every year.

Since these variations repeat during a period of twelve months

so, they can be predicted fairly accurately.

Seasonal Variations are caused by climate and weather

conditions, customs, festivals and habits.

for example-Sales of Cold-drinks goes up in summer season

than any other season


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U

nits

years2000 2001 2002 20032004 2005 2006

Sales of Cold-drinks

10000

12000

14000

1600018000

20000

IRREGULAR VARIATION


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IRREGULAR VARIATIONIrregular variations refer to such variations in business

activity which do not repeat in a definite pattern.

In these type of variations the pattern of the variable isunpredictable.Irregular Variations are caused by unpredictable

factors like natural disasters (earthquakes, floods,

wars etc.).These are unpredictable and no onehas control over it.

For example-Production of cars tremendously wentdown after earthquake came in Japan in Nov 2011.


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2005 2006 2007 2008 2009 2010 2011

100000

150000

200000

250000

300000

350000

unit

s

Production of cars in Japan

years


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