Variants of Mean Deviation

My ninth mathematics related post is the following:

You definitely have heard of mean deviation in statistics classes. I mean, who has not? The steps of mean deviation follow if you are not used to it yet.

• 1. Find the mean of all values
• 2. Find the distance of each value from that mean (subtract the mean from each value, ignore minus signs)
• 3. Then find the mean of those distances
• ^[1]

The problem is that while those instructions seem simple, they are not, as there are many means available. There are 57^[2] named means available to peruse. That means there are 57², or 2889 named variants available. Turns out that if one is not interested in using the default 57 means, they can create their own mean by defining a multi-variate function that follows certain properties, which brings the number of variants to infinity. For notability purposes, we are going to list only four of those variants, which are derived from the two most common means in existence:

• Geometric Mean
• Arithmetic Mean

One of those variants will not be explored due to the fact that arithmetic mean deviation exists. Let's get started. We are going to define ${\displaystyle A=\left[600,470,170,430,300\right]}$; that will be our list this blog post.

Arithmetic Mean-Geometric Mean Deviation

• 1. Find the arithmetic mean of all values. ${\displaystyle 394}$
• 2. Find the distance of each value from that mean (subtract the mean from each value, ignore minus signs) ${\displaystyle B=\left[206,76,224,36,94\right]}$
• 3. Then find the geometric mean of those distances ${\displaystyle \prod_{n=1}^{5}B\left[n\right]^{\frac{1}{5}}\approx103.5}$

Geometric Mean-Arithmetic Mean Deviation

• 1. Find the geometric mean of all values. ${\displaystyle \mu=\prod_{n=1}^{5}A\left[n\right]^{\frac{1}{5}}\approx361.6}$
• 2. Find the distance of each value from that mean (subtract the mean from each value, ignore minus signs) ${\displaystyle B\approx\left[238.4,108.4,191.6,68.4,61.6\right]}$
• 3. Then find the arithmetic mean of those distances ${\displaystyle \approx133.7}$

Geometric Mean-Geometric Mean Deviation

I will leave it up to you to show that it is 115.8.

The MAD is 127.2, FYI.

That's all I wanted to show you.