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Covariance and Correlation


Covariance:

Covariance calculates that how much two random variables change together. If small value of one variable corresponds with smaller value of other variable and the same thing for greater values, it means covariance is positive. On the other hand, when smaller value of one variable corresponds with greater value of other variable and vice versa, then in such case covariance is negative. These mathematical signs show the linear relationship between these variables. The magnitude of covariance is not simple to interpret. Magnitude of strength of the linear correlation determines the covariance coefficient.

Properties of Covariance

A more general identity for covariance matrices: Let A be a random vector with covariance matrix sigma(A).Let X be a matrix that can act on The covariance matrix of vector XA is:

Sigma (XA) = X sigma (A) X^T.

It is a direct result of linearity of expectation and it is useful when applying linear transformation to vector.

Uncorrelatedness and Independence: IfA and B are independent, then their covariance is zero.

E(AB) = E(A)*E(B).


Correlation:

Correlation means relationship between two 2 or more variables.

Correlation analysis deals with association of two or more variables-Simpson&Kafka

Steps:

  • To identify the variables that is correlated.
  • Establish degree of relationship.
  • Estimation(regression)

For ex:

  • Rainfall and yield of tea and rice
  • Age of husband and wife.

Significance of Correlation:

  • Most variable show some kind of relationship between them. For ex. relationship between income and expenditure, demand and price and etc. Correlation analysis helps to measure the degree of relationship between two or more variables.
  • We will the estimate the value of one variable given the value of other, when we know that two variables are closely related .This is being done with the help of regression analysis.
  • Correlation analysis helps to understand the economic behaviour. Inbusiness, itenable the manager to estimate cost,price,and other variables on the basis of which various variables like costs, salesetc. functionally related. And also there is no need of any guesswork in decision making because of available relationship between various variables.
  • Development in the methods of philosophy & science has been characterized with an increase in knowledge of correlationanalysis. In nature also one can find multiplicity in their interrelated forces.
  • The purpose of correlation is to reduce the range of uncertainity. Thedata based on correlation analysis is being more reliable and near to reality.

Types of Correlation

1. Positive and Negative: Correlation is said to be positive when both the variables moves in the same direction. For ex.supply and price, age of husband and wife.

Correlation is said to be negative when both variables moves in the opposite direction. For ex. demand and price, expenses and income.

2. Linear and Non-Linear: When the rate of change in both the variables is same, then it is said to be linear correlation. When the rate of change between both the variables is different, then it is said to be non-linear correlation.

3. Simple, Partial &Multiple: When we studies correlation between only two variables, not more than two is called simple correlation.

When we studies 3 or more variables, then it is called multiple correlation.When we studies 3 or more variable, but consider only two at a time, then it is called partial correlation.

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