Harmonic Mean

For a given set of non-zero observations , H.M is defined as the reciprocal of the A.M of the reciprocals of a given set of observations , with their reciprocals as 1/x₁ , 1/x₂ , ------- 1/x_n , the harmonic mean , denoted as H.M.

H.M. =

n/ 1/x₁ , 1/x₂ , ------- 1/x_n

=

n/ ∑(1/x_i)

H.M. =

∑f/ f₁/x₁ , f₂/x₂ , ------- f_n/x_n

=

∑f_i/ ∑(f_i/x_i)

x_i = Mid - Point

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Weighted Harmonic Mean (HM_w)

HM_w =

w₁/

w₁/x₁

+

w₂/

w₂/x₂

+ ------- +

w_i/

w_i/x_i

=

∑w₁/ ∑

w₁/x₁

x_i = mid - point (In the case of group data).

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Combined Harmonic Mean

If there are two groups with n₁ and n₂ observations and H₁ and H₂ as respective HM's , So combined H.M will be

n₁ + n₂/

n₁/H₁

+

n₂/H₂

Note : (1) If all the observations of the data are equal, say k, then the H.M of the observations will also be k .

(2) H.M is useful in averaging rates and ratios . when units of observations are per day , per unit, per share , per hour , per worker etc , H.M. is most appropriate average .

Note : - (1) A.M ≥ G.M ≥ H.M

when all the observations are equal , then sign of equality exist .

(2) For two positive numbers a and b

AH = G ²