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Harmonic  Mean 

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☼ Harmonic  Mean  :   If the reciprocals of  the terms is an arithmetic series is taken and then summed up, then the number of  observations divided by the sum is defined as Harmonic mean.

Harmonic  mean   of  a_{1},a_{2},a_{3},...,a_{n}    is   =   H. M. = \frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+ \frac{1}{a_{n}}}  = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_{i}}}

 e.g. Find the harmonic mean of  1, 0.5  and 10.

= \frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+ \frac{1}{a_{n}}}

=\frac{3}{\frac{1}{1}+\frac{1}{0.5}+ \frac{1}{10}}

= \frac{3}{\frac{1}{1}+\frac{10}{5}+ \frac{1}{10}}

=  \frac{3}{\frac{10+20+1}{10}}

= \frac{3}{\frac{31}{10}}  =    3  \times \frac{10}{31}      =  \frac{30}{31}

The  Harmonic Mean of  two numbers  a  and  b  =  H.M.  = \frac{2ab}{a+b }  .

Relation  between  AM,  GM  and  HM  :   For two numbers  a  and  b,

AM  = \frac{a+b}{2}  ,     GM   =  \sqrt{ab}  and   HM  =  \frac{2ab}{a+b }

(GM)2  =  AM    HM   (i.e.  G 2  =  HA)            &            AM  >  GM  >  HM

 

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