This is primarily a question about how to name my filter. Suppose that I have a photon detector, and that I want to measure the rate at which photons arrive, with a simple gated counter (because this is the only hardware I have). The photon rate is just the counter value divided by the gate interval.
It seems to me that the counter is an integrator: so by opening the gate for, say, 1 second, I have made a first-order low pass filter (LPF).
Now, suppose that I take the mean of, say, 10 such measurements. Have I now built a second-order LPF? Or merely a first-order LPF with a longer time-constant? Does the answer change if the measurements are not consecutive?