# Does the magnitude of covariance have any real meaning? [duplicate]

I am not able to understand logic behind coming up with this formula for covariance. We know that the sample covariance formula is:

$${\rm Cov}(x,y)=\frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{n-1}$$

I am not able to understand the logic behind the numerator. Why are we multiplying the terms? I mean, I know we need to find the change in x is appearing along with change in y (if they co-vary), so this formula will give direction of that change in terms of sign of the number that you get after substituting the values in the formula. And also we will be able to make some comparisons, e.g., having a covariance of 50 is more evidence of x and y moving together than having it as 30. So their ordering will make sense. But if you just think of a single value of covariance without having to compare it with anything else, does it have any real intuitive meaning (just the number in itself)? For example, in case of variance or standard deviation one can easily see that it is average number of 'deviation from the mean' per data point in the sample. But here in case of covariance, I am not able to come up with any such intuitive understanding.

As I was adding tags for this question just now—I saw this as the tag description for the covariance the tag:

Covariance is a quantity used to measure the strength and direction of the linear relationship between two variables. The covariance is unscaled, & thus often difficult to interpret

So probably my difficult to understand covariance can be attributed to this.