Residence time is essentially the time spent by the particles(either the reactants or products) in a reactor. A plot of the time spent by each particle is called the residence time distribution curve. We may plot this curve experimentally by using a non reactive tracer(sometimes even a reactive one).
The spreading of this tracer in the reactor is assumed to be a diffusion like process, and is called "dispersion" or longitudinal diffusion. The measure of spreading of this RTD curve is indicated by the term "variance".
In ideal plug flow, there is no dispersion. As a result, the tracer does not spread, and the variance is zero.
"Variance" and "RTD" are not one and the same. How can Residence time turn to zero? If a particle spends "zero" time in a reactor, can you get what that translates to? The particles do not enter or leave the reactor at all. The reactor has nothing in it. You do not even carry out the reaction.