Something I have always wondered

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SUMMARY

The discussion centers on the qualitative differences between the Root Mean Square (RMS) and the mean of the absolute values of a dataset. RMS is defined as proportional to the distance from the origin in R^n, and its formula aligns with Pythagoras's theorem when the 1/n term is factored out. In contrast, the mean of the absolute values lacks a geometric interpretation, making RMS a more insightful measure for certain datasets. The differences between these two metrics become particularly pronounced in datasets with varying distributions of values.

PREREQUISITES
  • Understanding of Root Mean Square (RMS) calculation
  • Familiarity with mean and absolute value concepts
  • Basic knowledge of geometric interpretations in mathematics
  • Concept of R^n in data representation
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  • Explore the mathematical derivation of RMS and its applications
  • Investigate the properties of the mean of absolute values in statistical analysis
  • Learn about the implications of using RMS in signal processing
  • Study datasets where RMS and mean of absolute values diverge significantly
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Mathematicians, data analysts, and statisticians interested in understanding the differences between RMS and mean of absolute values, particularly in the context of data representation and 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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