CORRELATION

Meaning of Correlation

 the degree of relationship between two or more variables.

 For example, smoking and lung cancer are correlated:

- if we look at the number of people who smoke and the number of people who die of lung cancer over the course of 80 years, we see a correlation.

Correlation is the relationship that exist between two or more variables. 

Examples

Relationship between the heights and weights.

Relationship between the advertising expenditure and sales.

* Correlation is sometimes termed as "co-variation".

* The measure of correlation is called coefficient of correlation.

AUTHORS DEFINITION 

According to Simph son & Kafka  Correlation analysis deals with the association between two or more variables.

According to Conner     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.

According to Ya-Lun Chou  Correlation analysis attempts to determine the degree of relationship between variables

 Karl Pearson has given the concept of correlation.

 Correlation denotes from“r”

 Correlation lies between -1 to 1

Types of correlation are as follows-

1) Positive or negative correlation

2) Simple,partial,multiple correlation

3) Linear and non-linear correlation

* Positive correlation- If both the variables are varying in the same direction

 i.e. if as one variable is increasing the other on an average is also increasing it is known as positive correlation. 

* Negative correlation :- if the variables are varying in opposite directions i.e. as one variable is increasing the other is decreasing or vice versa, correlation is said to be negative.

 * Simple, partial or multiple correlation- when one, two variables are studied it is a problem of simple correlation, 

when three or more variables are studied it is a problem of either multiple or partial correlation. 

In multiple correlation three or more variables are studied simultaneously.

* Linear or non-linear correlation- if the amount of change in one variable tends to be a constant ratio to the amount of change in the other variable then correlation is said to be linear. 

 * Non-linear or curvilinear if the amount of change in one variable does not bear a constant ratio to the amount of change in the other variable.


METHODS OF  CORRELATION

 Followings are the methods of correlation

1) Scatter diagram method

2) Graphic method

3) Karl Pearson coefficient of correlation.

4) Rank correlation

5) Concurrent deviation method

6) Method of least squares

1) Scatter diagram method- The simplest device for ascertaining whether two variables are related is to prepare dot chart called scatter diagram. The greater the scatter of the plotted points on the chart the lesser is the relationship between the two variables. The more closely the points come to a straight line, the higher the degree of relationships.

 If all the points lie on the straight line falling from the lower left hand corner to the upper right hand, correlation is said to be perfect correlation r= +1

 If all the points are lying on a straight line rising from upper left hand to the corner right hand correlation is said to be perfect negative r= -1

 If the plotted points lie on a straight line parallel to the x-axis or in haphazard manner it shows absence of any relationship between the variables and it is called no correlation r=0

 Perfect positive r = +1, 

perfect negative= -1, positive r>0, negative r<0, 

no correlation= 0.

 As much as relationships come closer to zero it is called weak correlation or low

degree correlation.

 As much as relationships come closer to 1 it is called strong correlation or high degree correlation.

2) Graphic method-when values are plotted on a graph paper we obtain curves,one for x variable and another for y variables. If both the curves drawn on the graph are moving in the same direction (either up or down) correlation is said to be positive.  On the other hands if the curves are moving in the opposite direction, correlation is said to be negative.

3) Karl Pearson coefficient of correlation or product moment coefficient of correlation Karl Pearson method popularly known as Pearsonian co-efficient of correlation.

 The Pearsonian co-efficient of correlation is denoted by the symbol r, r=∑xy/nσx.σy.

 This method is to be applied only where the deviations of items are taken from actual means and not from assumed means.

 Value of co-efficient of correlation lies between -1 to 1.

 The co-efficient of correlation describes not only the magnitude of correlation but also its decision. r= ∑xy /√∑x2.∑y2.

 The coefficient of correlation is said to be a measure of covariance between two series.

The covariance of two series x&y, covariance =∑xy/n

 In order to find out the value of correlation coefficient, first we calculate covariance & then in order to convert it to a relative measure we divide the covariance by the standard deviation of the two series. The ratio so obtained is called Karl Pearson’s coefficient.

 Correlation is independent of change of scale & origin.

 R= ∑xy /√σx.σy.

 R= cov (xy)/σx.σy.

 Probable error= P.Er= 0.6745, 1-r2Ï®/√n

 Standard error= S.E R = 1- r2/√n

Co-efficient of Determination

 Square of co-efficient of correlation is called co-efficient of determination.

 Co-efficient of determination= r2

 R2= explained variance/ total variance.

 

 Co-efficient of determination (r2) means the percentage of variation in the (y) dependent variable which is explained by the independent variable (x).

 Y = ∞+βx where y is dependent variable, ∞ is intercept, β is slope, x is independent variable.

 Co-efficient of determination lies between 0 and 1.

 R2= bxy.byx

 The ratio of unexplained variance to total variance is frequently called the co- efficient of non-determination (k2)

 Square root of non-determination is called co-efficient of alienation or k.

Properties of coefficient of correlation are as follows-

The coefficient of correlation lies between-1 to 1.

The coefficient of correlation is independent of change of scale&origin.

 The coefficient of correlation is the geometric mean of two regression coefficient r= √bxy.byx

4) The degree of relationship between two variable is symmetric rxy=ryx. RANK CORRELATION COEFFICINT

 EDWARD SPEARMAN has developed this.

 Sometimes we are required to examine the extent of association between two ordinary scaled variables such has two rank orderings.

 A measure to ascertain the degree of association between the ranks of the two variables x and y is called rank correlation.

Spearman denotes it by p(rho) P= 1-Ï‚∑d2/n3-n


Features are as follows-

The sum of the differences or ranks between two variables shall be zero,∑d=0 It is distribution free or non-parametric 

If ranks are equal then p=1-6∑d2/n(n2-1)+m(m2-1)/12.

solve 



CONCURRENT DEVIATION METHOD

 It is the simplest method

 The limits of the population correlation are given by r+- P.E

 To find out the direction of change of x variables & y variables.

 Rc=+-√(2c-n)/n

 C= concurrent deviations

 When we observe numerical data in relation to time the set of observations so obtained is known as time series.