Population Variation with two variables

Question

I have a dataset with two variables. I want to treat my dataset as a population not a sample. I am wondering if I can just use the formula for population variance as below:

$$\frac{\sum(X-\mu)^2}{N}$$

Here, do I need to think about the degree of freedom? Does the degree of freedom have anything with the number of variables?

If you have expertise in this, please let me have an idea. Thank you.

Answer 1

If your $\mu$ is known. Then you can consider $\dfrac{\sum(X-\mu)^2}{N}$. Note here your d.f is $N$. If your $\mu$ is unknown. Then you estimate is as $\bar X$ and you can consider $\dfrac{\sum(X-\bar X)^2}{N-1}$. Note here your d.f is $N-1$.

Answer 2

If your population has $N$ equally probable outcomes (so each has probability $1/N$) and a value of the variable $X$ is assigned to each outcome, and $\mu$ is the average of those $N$ values of $X$, then the expression you wrote is the variance of $X$.