The simple definition of a mean is that of a numeric quantity which represents the center of a collection of numbers. Here the trick lies in defining the exact type of numeric collection, as beyond the arithmetic mean (AM for short, the sum of all values divided by their number) there are many more, with the other two classical Pythagorean means being the geometric mean (GM) and harmonic mean (HM).
The question that many start off with, is what the GM and AM are and why you’d want to use them, which is why [W.D.] wrote a blog post on that topic that they figure should be somewhat intuitive relative to digging through search results, or consulting the Wikipedia entries.
Compared to the AM, the GM uses the product of the values rather than the sum, which makes it a good fit for e.g. changes in a percentage data set. One thing that [W.D] argues for is to use logarithms to grasp the GM, as this makes it more obvious and closer to taking the AM. Finally, the HM is useful for something like the average speed across multiple trips, and is perhaps the easiest to grasp.
Ultimately, the Pythagorean means and their non-Pythagorean brethren are useful for things like data analysis and statistics, where using the right mean can reveal interesting data, much like how other types using something like the median can make a lot more sense. The latter obviously mostly in the hazy field of statistics.
No matter what approach works for you to make these concepts ‘click’, they’re all very useful things to comprehend, as much of every day life revolves around them, including concepts like ‘mean time to failure’ for parts.
Top image: Cycles of sunspots for the last 400 years as an example data set to apply statistical interpretations to. (Credit: Robert A. Rohde, CC BY-SA 3.0)
