# The Mean of the Mean is the Mean

# The Mean of the Mean is the Mean

Join the DZone community and get the full member experience.

Join For FreeHortonworks Sandbox for HDP and HDF is your chance to get started on learning, developing, testing and trying out new features. Each download comes preconfigured with interactive tutorials, sample data and developments from the Apache community.

There’s a theorem in statistics that says

You could read this aloud as “the mean of the mean is the mean.” More explicitly, it says that the expected value of the average of some number of samples from some distribution is equal to the expected value of the distribution itself. The shorter reading is confusing since “mean” refers to three different things in the same sentence. In reverse order, these are:

- The mean of the distribution, defined by an integral.
- The sample mean, calculated by averaging samples from the distribution.
- The mean of the sample mean as a random variable.

The hypothesis of this theorem is that the underlying distribution **has** a mean. Lets see where things break down if the distribution does not have a mean.

It’s tempting to say that the Cauchy distribution has mean 0. Or some might want to say that the mean is infinite. But if we take any value to be the mean of a Cauchy distribution — 0, ∞, 42, etc. — then the theorem above would be false. The mean of *n* samples from a Cauchy has the same distribution as the original Cauchy! The variability does not decrease with *n*, as it would with samples from a normal, for example. The sample mean doesn’t converge to any value as *n* increases. It just keeps wandering around with the same distribution, no matter how large the sample. That’s because the mean of the Cauchy distribution simply doesn’t exist.

Hortonworks Community Connection (HCC) is an online collaboration destination for developers, DevOps, customers and partners to get answers to questions, collaborate on technical articles and share code examples from GitHub. Join the discussion.

Published at DZone with permission of John Cook , DZone MVB. See the original article here.

Opinions expressed by DZone contributors are their own.

## {{ parent.title || parent.header.title}}

## {{ parent.tldr }}

## {{ parent.linkDescription }}

{{ parent.urlSource.name }}