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The Mean

The popular but humble Mean, a Measure of Central Tendency1.

Types of means​

Arithmetic​

For a given set of observations, it’s the sum of the product of the observation and its Expected Value.

μ=∑i=1nxiE[xi]\mu = \sum_{i=1}^{n}{x_i}E[x_i]

The Average is a special case where the Expected Values are all the same (i.e., it’s a boring-ass distribution and all observations are equally likely). So if th Expected Value of nthn^{th} observation xix_i is just 1n\frac{1}{n}

μAverage=1n∑i=1nxi\mu_\text{Average} = \frac{1}{n}\sum_{i=1}^{n}{x_i}

Geometric​

If you have nn observations of some variable XX (or just a set of numbers), the Geometric Mean is the nthn^{th} root of their product.

μGeometric=∏i=1nxin\mu_\text{Geometric} = \sqrt[n]{\prod_{i=1}^n{x_i}}

Harmonic​

This one’s really fun and a total inversion of the Arithmetic Mean.

μHarmonic=n∑i=1n1xi\mu_\text{Harmonic} = \frac{n}{\sum_{i=1}^n{\frac{1}{x_i}}}

Others​

There are a lot of means. Consider the Stolarsky Mean which you can use to derive min, max, Arithmetic, Geometric, Harmonic, Power, Identric, and Quadratic means. You will almost never use them (and can try and pass that Real Analysis class you dropped in college in another life).

Here’s another general formula:

μ=f−1(∑i=1nf(xn)n)\mu = f^{-1}\left( \frac{\sum_{i=1}^nf(x_n)}{n} \right)

Where ff is some function.

  • If you choose f(x)=xf(x) = x and f−1(y)=yf^{-1}(y) = y, you will get the Arithmetic Mean.
  • Pick f(x)=ln(x)f(x) = ln(x) and f−1(y)=eyf^{-1}(y) = e^y and you get the Geometric Mean.
  • Pick f(x)=1xf(x) = \frac{1}{x} and f−1(y)=1yf^{-1}(y) =\frac{1}{y} and you get the Harmonic Mean.

Neato.

When do we use each?​

What ‘center’ would you like in your Measure of Central Tendency? Generally speaking,

  • Additive? Use Arithmetic.
  • Multiplicative? Use Geometric.
  • Rate-based? Use Harmonic.

Arithmetic​

Simple counts and the like: costs, heights, salaries, durations. Note that it is notorious for being very sensitive to large outliers (think the average net worth of nine people and some fuckhead billionaire).

Footnotes​

  1. Take some Distribution and try balancing it on your finger 👉🔔. Other fingers you could use are Median and Mode. ↩