Ernest Rutherford reportedly said something like, if your experiment needs statistics, you should have done a better experiment. Sometimes this point is driven home by an experimental technique that gives results that are strikingly clear. To the right is an example of this, from Zhu et al., Nano Lett. (in press), doi: 10.1021/acs.nanolett.5b03075. The technique here is called "density gradient ultracentrifugation".
You know that the earth's atmosphere is denser at ground level, with density decreasing as you go up in altitude. If you ignore temperature variation in the atmosphere, you get a standard undergraduate statistical physics problem ("the exponential atmosphere") - the gravitational attraction to the earth pulls the air molecules down, but the air has a non-zero temperature (and therefore kinetic energy). A density gradient develops so that the average gravitational "drift" downward is balanced on average by "diffusion" upward (from high density to low density).
The idea of density gradient ultracentrifugation is to work with a solution instead of the atmosphere, and generate a vastly larger effective gravitational force (to produce a much sharper density gradient within the fluid) by using an extremely fast centrifuge. If there are particles suspended within the solution, they end up sitting at a level in the test tube that corresponds to their average density. In this particular paper, the particles in question are little suspended bits of hexagonal boron nitride, a quasi-2d material similar in structure to graphite. The little hBN flakes have been treated with a surfactant to suspend them, and depending on how many layers are in each flake, they each have a different effective density in the fluid. After appropriate dilution and repeated spinning (41000 RPM for 14 hours for the last step!), you can see clearly separated bands, corresponding to layers of suspension containing particular thickness hBN flakes. This paper is from the Hersam group, and they have a long history with this general technique, especially involving nanotubes. The results are eye-popping and seem nearly miraculous. Very cool.