Assume a context with $N$ approximately normal distributions where a lower mean implies a 'better' distribution and a high variance or high standard deviation implies a 'better' distribution as well. Unfortunately, it is unknown which of the two weighs heavier and by how much. Does there exist a sound way to go from here to compare the distributions?
Simply calling the distribution with the minimal coefficient of variation $c_v = \sigma/\mu$ the 'best' assumes equal weights, right?
I have seen something on $\mu-\gamma\sigma$ optimization, but I am not sure when one would use this outside of finance.
Without any knowledge about the weighting, the only thing you can say is the following:
Distribution $F$ is better than distribution $F'$ if and only $F$ has a weakly lower mean and a weakly higher variance (one of them strictly) than $F'$.