I have read that variance of a deterministic signal is 0 because it is deterministic and does not vary over time.
In this explanation, I am considering a $\sin(t)$ wave where value at each t is a variable(it will be a random variable when we are doing measurements but I am not sure what to call it here, so just variables). We can have sampled values of $\sin(t)$ for limited number of variables but it is not related to my question.
I understand very well for a single variable that it's variance is zero. For eg: value of sine wave at $t=1$ which is given by $x=\sin(1)$ . Since $\sin(1)$ is constant every time you try to measure it, variance given by the following formula is zero as well: $$ var(x)= E[(x-\bar x)(x-\bar x)^T] $$ I understand this is zero because $x=\bar x$ every time you calculate x. Now, suppose you have a deterministic signal($\sin(x)$ from $0$ to $2\pi$). Now, mean of the signal is 0 and variance is just the squared sum of signal values from $0$ to $2\pi$ which is not zero. Even more, this is providing the energy of the signal. Does that mean variance of a deterministic signal is defined and does not have to be zero. Is variance even defined for a whole signal or is this concept valid only for a single random variable.