In an electrical context, power is the mean-square voltage across a load divided by the load resistance. The mean-square voltage corresponds to the expected value of the voltage squared E[V2], estimates of which are found by using the expression sumNi V2i / N, for N time samples. These are definitions of variance, if the V random process is zero mean, i.e. mean(V) = 0.

Dear Talha,

briefly: only if the signal is ergodic and the mean value is null.

I hope that this can help you:

https://en.wikipedia.org/wiki/User:Orcioni/sandbox#Power_and_variance

I'd answer this more intuitively. First, we know that whether this signal is a voltage or a current, in the time domain, the power of the signal is a function of the square of that signal. Either I2R or V2/R. Power is directly related to the square of the signal.

Then we know that in statistics, variance is the square of the difference between the samples and the mean (variance usually called σ2). So that gives us a hint.

variance = σ2 = 1/N * sum from i = 1 to i = N of (xi - µ)2

where xi are the individual samples and µ is the mean. In this case, the signal can go on for an indefinitely long time, analogous to taking a number of samples that approaches ∞.

Take a signal y(t). It has a DC component, its mean value, and it goes above and below that mean.(Possibly, the mean could be 0, but it doesn't have to be.)

So, the variance of the y(t) function is a function of the square of the difference between the mean (DC component) and the y values, over time. Whether that signal y(t) is current or voltage, the power will be directly related to its variance σ2.

For example, what is the power of a voltage represented by a sine function? It's one of those things we learned and take for granted. The power is the root mean square of that sine (RMS value) squared, divided by the resistance. Well, doesn't RMS look exactly like the square root of variance? (RMS is the square Root of the difference between the Mean value of that sine and the instantaneous value, Squared, for all t.)

Dear Albert,

in your formulation there is ambiguity about the need of the mean to be null to satisfy the equality.

Moreover, nothing is true if the signal it is not ergodic. Since a single realization, also if it go on for an indefinitely long time, would not represent the statistical information enclosed in the variance.

These parameters are statistical significance. As the first order moment is average, second as variance, third as skewness and so forth.

As the second moment is variance and calculate as square value so it is represented as signal power. This is also called hjorth activity in EEG signal quantification.

Simone, vediamo,

in your formulation there is ambiguity about the need of the mean to be null to satisfy the equality.

The statistical mean doesn't need to be zero, but the electrical signal created by those statistical samples will have a zero mean. The variance is always the average distance between the mean and the individual samples, squared, over the time period. So yes, the signal that creates is only a function of the square of the distance above and below the mean, not the mean of the samples itself. That should work fine.

Moreover, nothing is true if the signal it is not ergodic. Since a single realization, also if it go on for an indefinitely long time, would not represent the statistical information enclosed in the variance.

I think I agree here, but it's what we should expect in this application, no? We are analyzing an EEG signal, after all. We could make that same ergodicity argument when applying FFT to a signal.