Standard Deviation: Formula (8) Approximation Explained

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Discussion Overview

The discussion revolves around the approximation of standard deviation for a function V(x1,x2,x3,x4) when the variables are standard normal with small variances. Participants seek clarification on the derivation and application of a specific formula (formula 8) related to this approximation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant, Anders, inquires about the proof or further reading regarding the approximation of standard deviation using formula (8).
  • Another participant suggests that expressing the deviation in Vo as a linear combination of partial derivatives and individual variable deviations is key to understanding the approximation.
  • Anders expresses confusion about the "clever" part of the derivation and questions the relationship between "difference" and "variance," requesting additional formulas.
  • A participant explains that the deviation can be represented as a sum of partial derivatives multiplied by the deviations of the variables, providing a specific example with two variables.
  • Anders seeks clarification on why the term dx is equated to standard deviation.
  • A participant responds that dx can be chosen as a multiple of the standard deviation for each variable, referencing relevant Wikipedia articles for further context.
  • Anders shares that a link from Wikipedia led to helpful information regarding the topic.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the understanding of the derivation and relationship between the terms involved. Multiple viewpoints and levels of understanding are present, indicating ongoing exploration of the topic.

Contextual Notes

Some assumptions regarding the independence of variables and the conditions under which the approximation holds are not fully articulated, leading to potential limitations in the discussion.

daudaudaudau
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Hi all.

If I have a function V(x1,x2,x3,x4) and I want to calculate it's standard deviation when x1,x2,x3,x4 are standard normal and their variances are small, then formula (8) on this page
http://www.devicelink.com/mem/archive/99/09/003.html" is an approximation to the standard deviation. Can anyone offer me a proof or tell me where I can read more about this formula?

-Anders
 
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Hi.

Hmm, I think it's the clever part I don't understand then. Must I taylor expand V and then compute the standard deviation? And also, I can't see the connection between "difference" and "variance". I'd appreciate a few formulas. Thanks.

-Anders
 
I thought it would follow from dV = Sum[(partial V/partial xi) dxi, i=1,2,3,4]. The "d" operator is similar to "deviation" (e.g., from the mean). Let's say you have only 2 x's. Then dv = (Dv/Dx1) dx1 + (Dv/Dx2) dx2 ==> dv^2 = (Dv/Dx1)^2 dx1^2 + (Dv/Dx2)^2 dx2^2 + ignored term* approx. equal to (Dv/Dx1)^2 dx1^2 + (Dv/Dx2)^2 dx2^2. (I've used capital D for "partial.")

In fact, you don't even need the link I posted.

*This is the interaction term 2(Dv/Dx1)(Dv/Dx2) dx1 dx2. If you think that for a given "random draw" either of dx1 or dx2 (but not necessarily both, and you don't know which) is likely to be "very small" then you can assume 2(Dv/Dx1)(Dv/Dx2) dx1 dx2 = 0.
 
Last edited:
Okay. The only part I don't understand now is why dx is the same as a standard deviation ?
 

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