Covariance is a measure of how much two random variables vary together. It’s similar to variance, but where variance tells you how a single variable varies, co variance tells you how two variables vary together.
The Covariance Formula
The formula is:
Cov(X,Y) = Σ E((X-μ)E(Y-ν)) / n-1 where:
X is a random variable
E(X) = μ is the expected value (the mean) of the random variable X and
E(Y) = ν is the expected value (the mean) of the random variable Y
n = the number of items in the data set
Calculate covariance for the following data set:
x: 2.1, 2.5, 3.6, 4.0 (mean = 3.1)
y: 8, 10, 12, 14 (mean = 11)
Substitute the values into the formula and solve:
Cov(X,Y) = ΣE((X-μ)(Y-ν)) / n-1
= (2.1-3.1)(8-11)+(2.5-3.1)(10-11)+(3.6-3.1)(12-11)+(4.0-3.1)(14-11) /(4-1)
= (-1)(-3) + (-0.6)(-1)+(.5)(1)+(0.9)(3) / 3
= 3 + 0.6 + .5 + 2.7 / 3
The result is positive, meaning that the variables are positively related.
Note on dividing by n or n-1:
When dealing with samples, there are n-1 terms that have the freedom to vary (see: Degrees of Freedom). If you are finding the covariance of just two random variables, just divide by n.
Problems with Interpretation
A large covariance can mean a strong relationship between variables. However, you can’t compare variances over data sets with different scales (like pounds and inches). A weak covariance in one data set may be a strong one in a different data set with different scales.
The main problem with interpretation is that the wide range of results that it takes on makes it hard to interpret. For example, your data set could return a value of 3, or 3,000. This wide range of values is cause by a simple fact; The larger the X and Y values, the larger the covariance. A value of 300 tells us that the variables are correlated, but unlike the correlation coefficient, that number doesn’t tell us exactly how strong that relationship is. The problem can be fixed by dividing the covariance by the standard deviation to get the correlation coefficient.
Corr(X,Y) = Cov(X,Y) / σXσY
Advantages of the Correlation Coefficient
The Correlation Coefficient has several advantages over covariance for determining strengths of relationships:
- Covariance can take on practically any number while a correlation is limited: -1 to +1.
- Because of it’s numerical limitations, correlation is more useful for determining how strong the relationship is between the two variables.
- Correlation does not have units. Covariance always has units
- Correlation isn’t affected by changes in the center (i.e. mean) or scale of the variables
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