Weighted Harmonic Mean

The weighted harmonic mean is a variation of the harmonic mean where each element is multiplied by a specific weight. It is calculated by dividing the sum of the weights by the sum of the weighted reciprocals of the values. The formula is: $$ \mu = \frac{ \sum w_i}{ \sum \frac{1}{x_i} \cdot w_i} $$

The choice of weights for each term is up to you.

In general, you assign a greater weight to the values you consider more important.

These weights can also represent the absolute frequencies of the terms in a distribution.

A Practical Example

Let's say a math exam is divided into three sections: A, B, and C.

A student scores 18 on the first section (A), 22 on the second (B), and 26 on the third (C).

The simple harmonic mean of these three scores is 21.58:

$$ \mu = \frac{3}{\frac{1}{18}+ \frac{1}{22} + \frac{1}{26} } $$

$$ \mu = \frac{3}{0.139} $$

$$ \mu = 21.58 $$

However, the teacher decides that each section should have a different weight:

$$ w_A = 4 \\ w_B = 3 \\ w_C=2 $$

To find the weighted mean, we use the weighted harmonic mean formula:

$$ \mu = \frac{ \sum w_i}{ \sum \frac{1}{x_i} \cdot w_i} $$

$$ \mu = \frac{ 4+3+2}{ \frac{1}{18} \cdot 4 + \frac{1}{22} \cdot 3 + \frac{1}{26} \cdot 2 } $$

$$ \mu = \frac{ 9 }{ \frac{4}{18} + \frac{3}{22} + \frac{2}{26} } $$

$$ \mu = \frac{ 9 }{ \frac{2}{9} + \frac{3}{22} + \frac{1}{13} } $$

$$ \mu = 20.66 $$

The weighted harmonic mean of the scores is μ = 20.66.

In this case, it's lower than the simple harmonic mean (μ = 21.58) because the score in section A carries more weight than the other two sections.

And that's how it's done.