Today’s Uncertainty Wednesday will be quite short as I am super swamped. Last week I showed some code and an initial graph for sample means of size 100 from a Cauchy distribution. Here is a plot (narrowed down to the -25 to +25 range again) for sample size 10:
And here is one for sample size 1,000:
Yup. They look essentially identical. As it turns out this is not an accident. The sample mean of the Cauchy distribution has itself a Cauchy distribution. And it has the same shape, independent of how big we make the sample!
There is no convergence here. This is radically different from what we encountered with the sample mean for dice rolling. There we saw the sample mean following a normal distribution that converged ever tighter around the expected value as we increased the sample size.
Next week will look at what the takeaway from all of this. Why does the sample mean for some distributions (e.g. uniform) follow a normal distribution and converge but not so for others? And, most importantly, what does that imply for what we can learn from data that we observe?
