Covariance Formula

Covariance formula

Covariance states the relationship between two random variables. The formula is applied to get the variance between two variables. It states that till what range with the change in one variable changes together. If one variable is of a higher range other will also have a higher range and lower the one variable other would be of a lower range. Covariance is measured in units.

Positive covariance-

When both the variables move in the same direction it is called positive covariance.

Negative covariance-

When both the variables move in opposite directions it is said as negative covariance.

In the formula of covariance, the covariance between two variables X and Y can be denoted as Cov(x,y). The formula of covariance varies in some circumstances. Formulas for the covariance of population and sample are different.

Covariance formula for population

 Cov(x,y) = ∑(x_i – x ) × (y_i – y)/ (N)

Covariance formula for sample

 Cov(x,y) = ∑(x_i – x ) × (y_i – y)/ (N – 1)

Where, X_i –values of the X-variable

             Y_i – values of the Y-variable

             X̄ – mean of the X-variable

             Ȳ – mean of the Y-variable

             n – number of data points

The covariance formula can be also written as

ρ(X,Y)=Cov(X,Y)/σ_X * σ_Y

ρ(X,Y)      – correlation between the variables X and Y

Cov(X,Y) –  covariance between the variables X and Y

σ_X           –  standard deviation of the X-variable

σ_Y            – standard deviation of the Y-variable

Example

1. Find covariance for following data  x = {6,5,3,4,2}, y = {12,10,8,6,4}

Given data, x = {6,5,3,4,2}, y = {12,10,8,6,4} & N = 5

                    X̄ = (6 + 5 + 3 + 4 + 2) / 5

                       = 20 / 5

                       = 4

                   Ȳ  = (12 + 10 +8 + 6 + 4) / 5

                       = 40 / 5

                       = 8

Using the sample covariance formula

  Cov(x,y) = ∑(x_i – x ) × (y_i – y)/ (N – 1)

    = [(6 – 4)(12 – 8) + (5 – 4)(10 – 8) + (3 – 4)(8 – 8) + (4 – 4)(6 – 8) + (2 – 4)(4 – 8)] / 5 – 1

     = 8 + 2 + 0 + 0 + 6 / 4

      = 16 / 4

       = 4

Using the Population covariance formula

 Cov(x,y) = ∑(x_i – x ) × (y_i – y)/ (N)

= [(6 – 4)(12 – 8) + (5 – 4)(10 – 8) + (3 – 4)(8 – 8) + (4 – 4)(6 – 8) + (2 – 4)(4 – 8)] / 5

= 8 + 2 + 0 + 0 + 6 / 5

= 16 / 5

= 3.2

2. The standard deviation of X is 15 and the standard deviation of Y is 20. The correlation between X and Y is 5. Find covariance of X and Y.

Given data, σ_X =15, σ_Y =20 and  ρ(X,Y) = 5

ρ(X,Y) = Cov(X,Y)/σ_X * σ_Y

5= Cov(X,Y)/ 15*20

Cov(X,Y)= 1500