We consider a distribution of conductance fluctuations in quantum dots with single channel leads and continuous level spectra, and demonstrate that it has a distinctly non-Gaussian shape and strong dependence on time-reversal symmetry, in contrast to an almost Gaussian distribution of conductances in a disordered metallic sample connected to a reservoir by broad multichannel leads. In the absence of time-reversal symmetry, our results obtained within the diagrammatic approach coincide with those derived within non-perturbative techniques. In addition, we show that the distribution has log-normal tails for weak disorder, similar to the case of broad leads, and that it becomes almost log-normal as the amount of disorder is increased towards the Anderson transition.