Something I have always wondered

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The discussion explores the qualitative differences between the root mean square (RMS) and the mean of the absolute value, particularly in the context of datasets with values ranging from 0% to 100%. RMS is linked to the geometric concept of distance from the origin in R^n, resembling Pythagorean theorem when the 1/n term is factored out. In contrast, the mean of the absolute value lacks a similar geometric interpretation. The differences between these two metrics become more pronounced in datasets with varying distributions, especially those with outliers or skewed values. Understanding these distinctions is crucial for selecting the appropriate metric for data analysis.
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What is the qualitative difference between rms and the mean of the absolute value? For what datasets do they differ most?
 
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0xDEADBEEF said:
What is the qualitative difference between rms and the mean of the absolute value? For what datasets do they differ most?

Say you take a sample of n test results. Each test result ranges between 0% and 100%. You can then imagine the space of the possible test results as a subset of R^n. The particular sample you drew is then a single point in R^n.

The RMS is proportional to the distance between this point and the origin.

You can see in the formula for RMS, if you move the 1/n term outside of the square root radical, you get pythagoras's theorem.

As far as I know (which isn't much), the mean of the absolute value has no such neat geometric interpretation.
 

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