How to calculate a random measurement error?

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Discussion Overview

The discussion revolves around the calculation of random measurement error, specifically comparing two formulas: one for standard deviation and another for mean absolute deviation. The scope includes theoretical considerations and practical applications in fields like experimental physics and machine learning.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant presents two formulas for measuring error: one for standard deviation and one for mean absolute deviation.
  • Another participant asserts that the first formula is the standard deviation, which is the usual measure of error.
  • A third participant agrees that the first formula is standard deviation and notes that the second formula is rarely used.
  • Some participants mention that the mean absolute deviation may be used in machine learning applications due to its computational efficiency, but express a preference for standard deviation in experimental physics.

Areas of Agreement / Disagreement

Participants generally agree that the first formula represents standard deviation, while there is some contention regarding the utility of the second formula (mean absolute deviation) in different contexts.

Contextual Notes

The discussion does not resolve the appropriateness of using mean absolute deviation versus standard deviation in various applications, and it remains unclear under what specific conditions each measure might be preferred.

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TL;DR
How to calculate a random measurement error? If we did a measurement and want to calculate a random error, what formula are we to use?
I have seen this formula

$$\sigma=\sqrt{\frac {\sum_{i=1}^{N}{(X_i- \bar{X})^2}}{N(N-1)}}$$

but also this formula $$\sigma =\frac{\sum_{i=1}^{N}{|X_i- \bar {X}|}}{N}.$$ Which of them is correct?
 
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The first one :smile:

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The first one is the standard deviation. That is the usual one. The second one is the mean absolute deviation and it is rarely used at all.
 
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Dale said:
The second one is the mean absolute deviation and it is rarely used at all.
Some machine learning applications seem to use it - I think because it's cheaper to calculate, so it gives you a time saving if you can live with the less mathematically nice behaviour. But in experimental physics I agree it's a no, you want the standard deviation.
 
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