Covariance is a measure of how two variables change together. It is a statistical concept that measures the degree to which two variables are related. Specifically, covariance measures the joint variability of two variables. A positive covariance means that the two variables tend to increase or decrease together, while a negative covariance means that they tend to change in opposite directions. A covariance of zero means that the two variables are independent of each other and have no linear relationship.
Covariance is calculated using the following formula:
Cov(X, Y) = Σ[(Xi – μx)(Yi – μy)] / (n – 1)
where Cov(X, Y) is the covariance between variables X and Y, Σ is the summation symbol, Xi and Yi are the individual values of X and Y, μx and μy are the means of X and Y, and n is the sample size.
Covariance is often used in finance to measure the relationship between two financial assets. For example, the covariance between the returns of two stocks can be used to assess the risk of a portfolio containing those stocks. If the covariance between the two stocks is high, it indicates that they tend to move in the same direction, which increases the risk of the portfolio. Conversely, if the covariance is low, it indicates that the two stocks are relatively independent and diversification can help to reduce the portfolio risk.
Covariance step by step
Sure, here are the step-by-step instructions for calculating covariance between two variables:
- Collect the data for the two variables you want to analyze. You should have a set of data for each variable, with each set containing the same number of observations. For example, if you are analyzing the relationship between the age and income of a group of people, you would have one set of data for age and another set of data for income.
- Calculate the mean for each variable. To do this, add up all of the values for each variable and divide by the number of observations. For example, if you have 10 observations for age and their values are 25, 30, 35, 40, 45, 50, 55, 60, 65, and 70, the mean age would be (25+30+35+40+45+50+55+60+65+70)/10 = 47.5.
- Calculate the deviation from the mean for each observation of each variable. To do this, subtract the mean from each observation. For example, if you have an observation for age of 25 and the mean age is 47.5, the deviation from the mean for this observation would be 25 – 47.5 = -22.5.
- Multiply the deviations for each observation of both variables. Take each observation’s deviation from the mean for variable 1, multiply it by the corresponding observation’s deviation from the mean for variable 2, and add up these products. For example, if the deviation for age is -22.5 and the deviation for income is 1500, the product of these two deviations would be (-22.5) x (1500) = -33,750.
- Add up the products from step 4. This will give you the sum of the joint deviations. For example, if you have 10 observations for both age and income, you will have 10 products. Add up these 10 products to get the sum of the joint deviations.
- Divide the sum of the joint deviations by the sample size minus one. The sample size is the number of observations you have for each variable. Subtract one from the sample size before dividing to get the denominator for the covariance formula. For example, if you have 10 observations for both age and income, the sample size is 10. Divide the sum of the joint deviations by 10-1 = 9 to get the covariance.
The resulting value is the covariance between the two variables.