Does the magnitude of covariance have any real meaning?

[dhiraj]

The title of the question may not be clear. But that is the real difficulty I am facing. I am not able to understand logic behind coming up with this formula for covariance.

We know that the sample covariance formula is:-

\(\displaystyle cov(x,y)=\frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{n-1}\)

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 compare e.g. Having covariance of 50 for example is more of proof of x and y moving together than having it as 30. So their order 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.

 

[I like Serena]

The title of the question may not be clear. But that is the real difficulty I am facing. I am not able to understand logic behind coming up with this formula for covariance.

We know that the sample covariance formula is:-

\(\displaystyle cov(x,y)=\frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{n-1}\)

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 compare e.g. Having covariance of 50 for example is more of proof of x and y moving together than having it as 30. So their order 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.

Hi dhiraj! Welcome to MHB! (Smile)

It seems you already know.

If both values in the numerator are positive they provide a positive product.
And if both values are negative they also provide a positive product.
So if the values co-vary in a positive sense, we'll get a strong positive value, as we should.

A single value does indeed not make much sense.
We cannot derive a covariance, and neither for that matter, can we derive a standard deviation (for the same reason).