Algebra theory weighted means - weighted A.M. G.M. H.M.

• Weighted Means

Given values a1,a2,…,ana_1, a_2, \ldots, a_n with corresponding weights ω1,ω2,…,ωn\omega_1, \omega_2, \ldots, \omega_n

Weighted Arithmetic Mean (WAM):

WAM=a1ω1+a2ω2+⋯+anωnω1+ω2+⋯+ωn\text{WAM} = \frac{a_1 \omega_1 + a_2 \omega_2 + \cdots + a_n \omega_n}{\omega_1 + \omega_2 + \cdots + \omega_n}

Weighted Geometric Mean (WGM):

WGM=(a1ω1a2ω2⋯anωn)1ω1+ω2+⋯+ωn\text{WGM} = \left( a_1^{\omega_1} a_2^{\omega_2} \cdots a_n^{\omega_n} \right)^{\frac{1}{\omega_1 + \omega_2 + \cdots + \omega_n}}

Weighted Harmonic Mean (WHM):

WHM=ω1+ω2+⋯+ωnω1a1+ω2a2+⋯+ωnan\text{WHM} = \frac{\omega_1 + \omega_2 + \cdots + \omega_n}{\frac{\omega_1}{a_1} + \frac{\omega_2}{a_2} + \cdots + \frac{\omega_n}{a_n}}

Example:

AM of values:
2,2,2,3,3,3,4,4,4,42, 2, 2, 3, 3, 3, 4, 4, 4, 4

i.e., frequencies:

  • 2→3 times2 \rightarrow 3 \text{ times}

  • 3→3 times3 \rightarrow 3 \text{ times}

  • 4→4 times4 \rightarrow 4 \text{ times}

WAM=2(3)+3(3)+4(4)3+3+4=6+9+1610=3710\text{WAM} = \frac{2(3) + 3(3) + 4(4)}{3 + 3 + 4} = \frac{6 + 9 + 16}{10} = \frac{37}{10}

Inequality Relation:

WAM≥WGM≥WHM\boxed{\text{WAM} \geq \text{WGM} \geq \text{WHM}}


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