# Uncertainty Wednesday: Sample Mean under Fat Tails (Cont’d)

By [Continuations](https://continuations.com) · 2017-11-15

uncertainty wednesday, cauchy, fat tails, sample mean

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Today’s [Uncertainty Wednesday](http://continuations.com/tagged/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](http://continuations.com/post/167265621000/uncertainty-wednesday-sample-mean-under-fat-tails). Here is a plot (narrowed down to the -25 to +25 range again) for sample size 10:

![](https://storage.googleapis.com/papyrus_images/89a930ffb29eda12894b7096427ddb98.png)

And here is one for sample size 1,000:

![](https://storage.googleapis.com/papyrus_images/0f586f5dec5dd3db3b953067020aa85a.png)

Yup. They look essentially identical. As it turns out this is not an accident. The sample mean of the [Cauchy distribution](http://continuations.com/post/166778060320/uncertainty-wednesday-fat-tails) 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](http://continuations.com/post/165834047085/uncertainty-wednesday-sample-mean). There we saw the sample mean [following a normal distribution](http://continuations.com/post/166037942720/uncertainty-wednesday-sample-mean-contd) that [converged ever tighter](http://continuations.com/post/166305910455/uncertainty-wednesday-sample-mean-part-3) around the [expected value](http://continuations.com/post/162373873320/uncertainty-wednesday-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?

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*Originally published on [Continuations](https://continuations.com/uncertainty-wednesday-sample-mean-under-fat-tails-contd)*