Chameli Devi Institute of Management and research

Guided by : vandana

mam

Presented by: Lokesh jain

Pankaj kukreja

Presentation on correlation

Sir Francis Galton 1822-1911

Initially developed by Sir Francis Galton (1888) andKarl Pearson (1896)

Introduction Specific exampleNeed of study correlationPositive & negative

correlationLinear CorrelationNon-linear correlationMethods for study of

correlation

“Correlation is the relationship that exists between two or more variables. If variable are related in such a way that change in one creates a corresponding change in the other, then the variable are said to be correlated”

-W.I.KING

“When the relationship is of a quantitative nature, the appropriate statistical tool for discovering and measuring the relationship and expressing it in a brief formula is known as correlation” –CORXTON & COWDEN

1.)relationship between height and weight.2.)relationship between price of a commodity

and demand of commodity.3.)relationship between the age of individuals

and their blood pressure.4.)relationship between the quantum of rain

fall and the yield of wheat.5.)relationship between the advertisement

and sales

Correlation A measure of association between two numerical variables.

Example (positive correlation)Typically, in the summer as the temperature increases people are thirstier.

For seven random summer days, a person recorded the temperature and their water consumption, during a three-hour period spent outside.  

Temperature (F)

Water Consumption

(ounces)

75 16

83 20

85  25

85 27

92 32

97 48

99 48

Correlation help in study economic theory and business studies, it help in establishing relationship between variable like price and quantity demanded, advertising and sales promotion measuresCorrelation analysis helps in deriving precisely the degree and direction of such relation.The effect of correlation is to reduce the range of uncertanity of our prediction. The prediction based on correlation analysis will be more reliable and near to reality.The measure of coefficient of correlation is relative measure of change.

Positive correlation: if both the variables vary in same direction, correlation is said to be positive correlation .if one variable increase, the other also increases or, if one variable decreases, the other variable is said to be a positive correlation.

Negative correlation: if both the variable vary in opposite direction, the correlation is said to be negative. In other word if one variable increases, but other variable decreases or, if one variable decreases but the other variables increases, than correlation between two variables is said to be negative correlation.

Linear correlation: if the amount of changes in one variable bears a constant ratio to the amount of changes in the other variable, then correlation is said to be linear. if such variable plotted on graph paper all the plotted points would fall on straight line.

Non-linear correlation : if change in one variable does not bear a constant ratio to the amount of changes in the other variable, then correlation is said to be non-linear. if such variable plotted on graph, the point will fall on a curve and not a straight line.

Scatter diagram is diagrammatic representation of bivariant data to ascertain the correlation between two variable.

Perfect negative correlation

High degree negative

Low degree negative

X-values

Y-values

Value of r interpretation

(a) If r = +1 Perfect positive correlation

(b) If r =-1 Perfect negative correlation

(c) If r =0 No correlation

(d) If +.75 ≤ r < +1 High positive correlation

(e)If -.75 ≥ r > -1 High negative correlation

(f)If + 0.50 ≤ r < 0.75 Moderate positive correlation

(g)If -0.50 ≥ r > -0.75 Moderate negative correlation

(h)If r < +0.50 Low positive correlation

(i)If r> -0.50 Low negative correlation

Karl Pearson’s

Correlation Coefficient, r

Measures the direction and the strength of the linear association between two numerical paired variables.

1.Linear relationship between variable

2.Cause & affect relationship3.Normality

1.)correlation coefficient gives direction as well as degree of relationship between the variables.

2.)correlation coefficient along with other information help in estimating the values of the dependent variable from the known value of independent variable.

1.)Assumption of linear relationship:-The assumption of linear relationship between the variables may or may not hold always true .

2.)Time consuming :- Its computation is time consuming as compared to other method .

3.)Affected by extreme values: -It is affected by the value of extreme item.

4.)Requires careful interpretation:- The investigation should reach a conclusion based on logical reasoning and investigation on significantly related matter.

r = ∑ XY or ∑ XY

Nσx σy

Where

X is calculation of deviation from the actual mean of X series.i.e. X=X-X

Y is calculation of deviation from the actual mean of Y series i.e. Y=Y-Y

√∑ X2. ∑ Y2

1.)ACTUAL MEAN METHOD

2.)ASSUMED MEAN METHOD (when deviation are taken)

Formula: r= ∑dxdy -

√ √

where as: dx is calculation of deviation from assumed mean of X i.e. dx=X-A

dy is calculation of deviation from assumed mean of Y i.e. dy =Y-A

(∑dx)(∑dy)

N∑dx

2-(∑dx)

2

N

∑dy

2-(∑dy)

2

N

r = N

√ √

Where as:- X is variable of X series; Y is variable of Y series

∑XY -∑X.∑Y

∑X2 -(∑X)2N

∑Y2 - (∑Y)2

N

Standard Error = 1 – r2 √ N

Where, r = Coefficient of correlation, N=Number of pair of observationProbable Error = .6745 Standard error or

=.6745 1 – r2 √ N

h

R = 1-

Where as : R = Rank Correlation Coefficient D = Difference of the ranks

between paired items in two series N = Number of pairs of ranks

6∑D2

N3-N

In case of tied Ranks:

R = 1 - 6(∑D2 + m3 – m…)

12N3 - N

Simple to understand &Easy to apply

Suitable for qualitative Data

Only Method for ranks

Unsuitable For group data

Tedious CalculationApproximation since

actual data are not taken in to account

limitation

Formula:

√ ( )Where, rc= Coefficient of Concurrent deviation C = Number of Positive signs after multiplying the

direction of change of X series and Y series

n = Number of pairs of observations compared

2C - n n

+_+_rc =

Simple to understand and Easy to apply

Suitable for large N

Does not differentiate between Small and Big variable

Result obtained from this method is only an approximate indicator of the presence and absence of correlation

Limitation

GoogleR S Bhardwaj Bharat JhunjhunwalaS.M Shukla

……?Queries

are Invited.

..

Thank You