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Karl Pearson’s (Product Moment) Correlation Coefficient

Karl Pearson’s correlation coefficient quantitatively measures degree of relationship between two variables x and y.

 

Definition: A ratio between the co-variance between two variables to the product of their standard deviations is called Karl Pearson’s correlation coefficient.

 

Covariance

When n pairs of observations (x1y1), (x2y2), (x3y3) … (xnyn) are given between two variables x and y, then the covariance of x and y, cov(xy) and is defined as:

 

We can also express the covariance between the variables x and y as

 

 

Note: Variance is always positive, whereas covariance may be positive, negative or zero.

  • Using the above values of cov(xy), Sx and Sy, we can express the correlation coefficient as
     
Example
If r = 0.25, Sx = 6, Sy = 8, then find cov(xy).
Solution

⇒ cov(x, y) =12

 

Example
Compute the correlation co-efficient between x and y for the given data.
 
X 1 2 3 4 5
y 10 20 30 40 50
Solution
 
 
 
Substituting the relevant values, we get
 
 
  • For grouped bivariate data, Karl Pearson’s correlation co-efficient can be written as
     
     

Note:

  • The ratio of explained variance to total variance is known as coefficient of determination.
     
    i.e., Coefficient of determination = 
     
    It can be calculated by taking the square of the coefficient of correlation
     
    i.e., Coefficient of determination = r2
  • The ratio of unexplained variance to total variance is known as coefficient of non-determination.
     
    i.e., Coefficient of determination = 
     
    Coefficient of non-determination = 1 - r2
  • The square root of the coefficient of determination is known as the coefficient of alienation.
     
    i.e., Coefficient of determination =

Properties of Coefficient of Correlation

  1. The value of the coefficient of correlation is independent of the origin i.e., increasing or decreasing the values of xi and yi by some non-zero constant will not affect the value of the coefficient of correlation
  2. The value of the coefficient of correlation is independent of the scale i.e., multiplying or dividing the values of xi and yi by some non-zero constant will not affect the value of the coefficient of correlation
     
    If we apply a change of origin of ‘a’ on x and ‘b’ on y and a change of scale of ‘c’ on x and ‘d’ on y, then we have
     
     
    Using ui and vi, we have
     
     
     
  3. The value of the correlation coefficient lies between –1 and +1
  4. Correlation coefficient is unit free
  5. rXY = {(b.d )/|b||d|} r UV

Merits

  1. It gives us the direction as well as the degree of correlation between the variables.
  2. It helps in estimating the value of a dependent variable from the value of the independent variables.

Demerits

  1. Computation of Karl Pearson’s correlation coefficient consumes a lot of time.
  2. It is affected by the values of extreme observations.





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