File talk:Mean estimator consistency.gif
Being a little picky here, but the failure here is really in using the (ordinary) sample mean and the sample variance to estimate the underlying location and scale parameters, not that there is no way to use the data to estimate them. This is noted in the section on of parameters. (It's also intuitively clear from the histogram in this animation--the center of symmetry is pretty darn obvious, for example.) Of course if the reason you are interested in the sample mean is that it has independent significance (such as your average winnings if you play a game repeatedly), then the unpredictability of that sample mean is quite relevant.
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