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IEVref:103-02-04ID:
Language:enStatus: Standard
Term: geometric mean value
Synonym1: geometric average
[Preferred]
Synonym2:
Synonym3:
Symbol:
Definition: quantity representing the quantities in a finite set or in an interval,

  1. for n positive quantities x 1 , x 2 , x n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadIhadaWgaaWcba qcLboacaaIXaaaleqaaOGaaiilaiaaysW7caWG4bWaaSbaaSqaaKqz GdGaaGOmaaWcbeaakiaacYcacaaMe8UaaGPaVlablAciljaaykW7ca aMc8UaaGjbVlaadIhadaWgaaWcbaGaamOBaaqabaaaaa@49C5@ , by the positive nth root of their product:

    X g = ( x 1 x 2 ... x n ) 1/n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadIfadaWgaaWcba qcLboacaqGNbaaleqaaOGaeyypa0JaaiikaiaadIhadaWgaaWcbaqc LboacaaIXaaaleqaaOGaeyyXICTaamiEamaaBaaaleaajug4aiaaik daaSqabaGccqGHflY1caGGUaGaaiOlaiaac6cacqGHflY1caWG4bWa aSbaaSqaaiaad6gaaeqaaOGaaiykamaaCaaaleqabaqcLboacaaIXa WccaGGVaGaamOBaaaaaaa@5095@

  2. for a quantity x depending on a variable t, by the quantity X g MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadIfadaWgaaWcba qcLboacaqGNbaaleqaaaaa@389C@ calculated from the values of the quantity x(t) by the expression

    log X g x ref = 1 T 0 T log x(t) x ref dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaKqzafGaciiBaiaac+ gacaGGNbGcdaWcaaqaaiaadIfadaWgaaWcbaqcLboacaqGNbaaleqa aaGcbaGaamiEamaaBaaaleaajug4aiaabkhacaqGLbGaaeOzaaWcbe aaaaGccqGH9aqpdaWcaaqaaKqzafGaaGymaaGcbaGaamivaaaadaWd XaqaaKqzafGaciiBaiaac+gacaGGNbGcdaWcaaqaaiaadIhacaGGOa GaamiDaiaacMcaaeaacaWG4bWaaSbaaSqaaKqzGdGaaeOCaiaabwga caqGMbaaleqaaaaaaeaacaaMi8EcLboacaaIWaaaleaacaaMi8Uaam ivaaqdcqGHRiI8aKqzafGaaiizaOGaamiDaaaa@5BEA@

    where x ref MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeGaciWaamGadaGadeaabaGaaqaaaOqaaiaadIhadaWgaaWcba qcLboacaqGYbGaaeyzaiaabAgaaSqabaaaaa@3A98@ is a reference value

Note 1 to entry: The geometric mean value of a periodic quantity is usually taken over an integration interval the range of which is the period multiplied by a natural number.

Note 2 to entry: The geometric mean value of a quantity is denoted by adding the subscript g to the symbol of the quantity.


Publication date:2017-07
Source:
Replaces:103-02-04:2009:12
Internal notes:2017-08-25: Added <p> tag before list. LMO
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