Mean and SD of the inverse of a population

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Discussion Overview

The discussion revolves around whether it is possible to calculate or estimate the mean and standard deviation of the inverse of a population, given the mean and standard deviation of the original population. The scope includes theoretical considerations and mathematical reasoning related to statistical distributions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant suggests using the expectation formula E(g(X))=∫g(x)f(x)dx, where f(x) is the probability density function (p.d.f) of X, but questions whether the mean and standard deviation were derived from known values or just provided.
  • Another participant argues that it is not possible to derive the mean and standard deviation of the inverse population based solely on the original data, although they mention that bounds might be established if additional information, such as a minimum value for X > 0, is known.
  • A further inquiry is made about the definition of "population," asking whether it refers to a statistical distribution from a known family or a collection of data from an unknown probability distribution.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the possibility of calculating the mean and standard deviation of the inverse population. Multiple competing views remain regarding the necessary conditions and definitions involved.

Contextual Notes

There are limitations regarding the assumptions about the population and the nature of the data, particularly concerning whether it is derived from a known distribution or a collection of empirical data.

nokia8650
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If one has the mean and standard deviation of a population, is it possible to calculate (or estimate) the mean and standard deviation of the inverse population (ie. 1/(every value in the original population)?

Thank you!
 
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Can't you use :

E(g(X))=∫g(x)f(x)dx ,

where f(x) is the p.d.f of X?

I don't know if you know just the values E(X) and σ(X), or if you got those using the known
value f(x).
 
No, it's not possible on those data alone. You might be able to derive some bounds if you know a bit more, like a minimum value for X > 0.
 
nokia8650 said:
If one has the mean and standard deviation of a population, is it possible to calculate (or estimate) the mean and standard deviation of the inverse population (ie. 1/(every value in the original population)?

Clarify what you mean by "population". Do you mean a statistical distribution from a known family of distributions, like a "normal distribution"? Or do you mean the mean and standard deviation computed from a collection of data that comes from an unknown probability distribution ( like the "mean height of all emergency personnel in the city") ?
 

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