For a different view of how useful statistical practice (when applied mindlessly) is, read "The Black Swan". It's all about how people use statistical models with Normal distributions in places where it's patently unjustified, and the price we pay for it.
I played this game myself with a friend. I sent him ten samples of a (for him) unknown distribution and asked him to estimate the mean. Then 100, then 1000. His estimate of the mean kept changing to higher and higher values, because the samples were drawn from a Pareto (power-law) distribution with a mean of 1000. Such a distribution is almost indistinguishable from one with a mean of infinity, because all the signal is in the very rare, large outliers. If you try to analyze samples from such a process assuming it's Gaussian, nothing will make sense, and the standard deviation will give you an estimated uncertainty of the mean that is far, far below the actual uncertainty.
I played this game myself with a friend. I sent him ten samples of a (for him) unknown distribution and asked him to estimate the mean. Then 100, then 1000. His estimate of the mean kept changing to higher and higher values, because the samples were drawn from a Pareto (power-law) distribution with a mean of 1000. Such a distribution is almost indistinguishable from one with a mean of infinity, because all the signal is in the very rare, large outliers. If you try to analyze samples from such a process assuming it's Gaussian, nothing will make sense, and the standard deviation will give you an estimated uncertainty of the mean that is far, far below the actual uncertainty.