How can I find the minimum value for the sum of absolute values?

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Discussion Overview

The discussion revolves around finding the minimum value for the sum of absolute values, specifically the expression |Xi-V| for i=1,...,n. Participants explore different approaches to minimize this sum and clarify the notation used.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about minimizing the sum of absolute values and suggests that V should be equal to the mean value of X.
  • Another participant questions the notation, asking if Xi represents X times i.
  • A clarification is provided that Xi refers to a subscripted variable, indicating a sequence of values from X1 to Xn.
  • One participant argues that the expression is independent of the value of X and suggests that V should be chosen based on the order statistics of the Xi values, specifically between X(n/2) and X(n/2+1) if n is even, or X((n+1)/2) if n is odd.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the approach to minimize the sum of absolute values, with differing views on the role of V and its relationship to the Xi values.

Contextual Notes

There is ambiguity regarding the assumptions about the distribution of Xi values and the implications of choosing V based on the order statistics.

Jorge
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Hello,


I have been having problems finding the way to minimize the sum of absolute values. Specificaly I am looking for the value of X that will minimize the sum|Xi-V|<-- i=1,...n . I know that V should be equal to the mean value of X. But I do not know the correct approach to finding this minimum.

Can I square the Xi-V and differentiate? or is there another approach?

Thanks...
 
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So is Xi X times i?
 
Sorry about that,

i---> is the sub index. Meaning X1...Xn.

Then it is Sum from i={1 to n }of |Xi-V|.
 
Well then that expression is completely independent of the value of X. Do you mean you are looking for a V to minimize that expression? If you are then you can let V be any value between X(n/2) and X(n/2+1) if n is even, and you can let V be X((n+1)/2) if n is odd.
 
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