A new type of mean?

Question

Usually, when we say mean, we are talking about$$mean=\frac{\sum_{i=1}^na_n}{n}$$But I thought of a similar type of "mean" or measure of center.$$mean\approx\sqrt[n]{\Pi_{i=1}^na_n}$$

Instead of adding and dividing by the amount of numbers you added, you could multiply then all and take the $n^{th}$ root, which yields values that seem to be somewhat of a mean.

I'm going to call this my product mean, or $\mathbb{P}$.

If anybody knows anything about this not being new, please tell me, and some links would be nice.

I noted a few things about this sort of mean.

It tends to be less than the regular mean, in fact, lesser and lesser than the regular mean the larger the variance or standard deviation. Could someone explain that?

I noticed that my mean tends to enjoy larger values rather than smaller values.

In fact:$$\mathbb{P}=\sqrt[n]{\Pi_{i=1}^na_n}\approx\sqrt[n]{\Pi_{i=1}^n(a_n+m)}-m$$

And I believe, if my hunch is correct, that$$\lim_{m\to\infty}\sqrt[n]{\Pi_{i=1}^n(a_n+m)}-m=\frac{\sum_{i=1}^na_n}{n}$$

However, I can't figure out how to evaluate that limit.

Also, $a_i\ne0$, or else the product mean becomes $\mathbb{P}=0$

We also have to assume that $a_i>0$, or else the mean won't evaluate well.

My questions:

1. Is this already a thing? Some links would be nice.

2. Explain the affect of the standard deviation or variance.

3. Evaluate the limit.

4. How should I deal with negative numbers?

5. How do outliers affect $\mathbb{P}$ in comparison with a regular mean?

6. When should this be used instead of the regular mean?

For the comments below, what's the professional word for regular mean? Thank you.

Answer 1

A partial answer to 6: measures of central tendency, such as the arithmetic mean (the one you opened with) or geometric mean, are useful because of theorems concerning them. For example, according to the central limit theorem the arithmetic mean of many independent identically distributed variables has a distribution approximated by the Normal distribution with the same arithmetic mean and variance. Approximately Normal distributions readily occur in nature when many small cumulative effects average in a manner that generalises this result. However, since measures of physical size (such as mass or length) are positive, a Normal approximation is more feasible for their logarithms, in which case we say the original variable's distribution is log-normal. The central limit theorem implies that the geometric mean of many independent identically distributed positive variables has an approximate log-normal distribution.

Answer 2

$\frac{\sum_{i=1}^n a_i}n$ is arithmetic mean. Your product mean is called geometric mean. Take a look at the wiki.