What is covariance?

Intuitively, covariance is the measure of how much two variables vary together (Wikipedia 2006). That is to say, the covariance becomes more positive for each pair of values which differ from their mean in the same direction, and becomes more negative with each pair of values which differ from their mean in opposite directions (Wikipedia 2006). In this way, the more often they differ in the same direction, the more positive the covariance, and the more often they differ in opposite directions, the more negative the covariance (Wikipedia 2006).

The covariance between two real-valued random variables X and Y, with expected values E(X) = μ and E(Y) = ν is defined as:

Figure 1

where E is the expected value (Wikipedia 2006).

The units of measurement of the covariance cov(X, Y) are those of X times those of Y (Wikipedia 2006). By contrast, the correlation, which depends on the covariance, is a dimensionless measure of linear dependence (Wikipedia 2006).

That is, from the definition, the following formula which is commonly used in calculations:

Figure 2

If X and Y are independent, then their covariance is zero (Wikipedia 2006). This is because under independence,

Figure 3

The converse, however, is not true (Wikipedia 2006). It is possible that X and Y are not independent, yet their covariance is zero (Wikipedia 2006). Random variables whose covariance is zero are called uncorrelated.

If X and Y are real-valued random variables and c is a constant ("constant", in this context, means non-random), then the following characteristics of covariance result:

Figure 4

Figure 5

Figure 6

Figure 7

The covariance is sometimes called a measure of "linear dependence" between the two random variables (Wikipedia 2006). The correlationis a closely related concept used to measure the degree of linear dependence between two variables.

References:

Wikipedia. "Covariance", Wikimedia Foundation Inc,  http://en.wikipedia.org/wiki/Covariance, Last accessed 14 February 2006.

Brandon Hodgson