This is e+e- interactions versus energy (sqrt(s)) from the particle data book, fig 49.5 :
Many of the interesting experimental measurements that led to the quark model are in this plot.
Particles like protons, electrons muons have a fixed mass, no width, measured by the "length" of the four vector. Measurement errors will introduce a statistical indeterminacy which can be fitted with the statistically defined gaussian. In the plot though, we see practically a delta function for the J/psi of the plot in the question, because the scale is different. Omega and rho have a large width which is not gaussian, Y is very narrow and the Z is also non gaussian.
What defines the intrinsic width is the type of interaction entering in the possible decays of the resonances and whether strong interaction decays are suppressed or not. The J/ψ is a good example:
It has a rest mass of 3.0969 GeV/c2, just above that of the ηc (2.9836 GeV/c2), and a mean lifetime of 7.2×10^−21 s. This lifetime was about a thousand times longer than expected.
Hadronic decay modes of J/ψ are strongly suppressed because of the OZI Rule. This effect strongly increases the lifetime of the particle and thereby gives it its very narrow decay width of just 93.2±2.1 keV. Because of this strong suppression, electromagnetic decays begin to compete with hadronic decays. This is why the J/ψ has a significant branching fraction to leptons.
The width of the Z also is interesting,and is composed out of all the partial widths in the channels it can decay,page 2 in this link.
In conclusion, the intrinsic interaction width ,( calculable by the standard model), of resonances has to be folded with the experimental error widths ,as explained in the comments to the question. When the intrinsic width is very small , the experimental error dominates and is a gaussian.