What is the significance of covariance? What does it mean when the covariance of two variables is positive?
Covariance is a linear measure of "connectivity." It is positive when the two variables you have at hand are positively connected.
Covariance is a linear measure of "connectivity." It is positive when the two variables you have at hand are positively connected. it means that whenever one variable increases, the other increases, e.g. sunlight and ice cream consumption, sugar consumption and tooth cavity incidents.
Covariance is a linear statistical measure of dependence. It is applied when you have two variables that must be interpreted in terms of dependence. If you have more than one, you must use matrix of covariance. On the picture below, it is shown the drawback of covariance, it cannot detect non-linearity, all the cases possesses the same covariance.
Thus, covariance is significant because it is a measure of "variable connectivity", or even randomness, it is close to zero in random variables. See that in some case, we can have fake covariances, such as the number sells of ice cream increases as a result of a certain singer becoming famous.
Some mathematical results
The last result mathematical connect covariance and correlation.