Weighted Geometric Mean

The weighted geometric mean is a variation of the geometric mean, where each element is raised to a power determined by an integer weight (w). The product of the weighted values is then taken under a root, with the root's index being the sum of the weights. $$ \mu = \sqrt[ \sum{w_i} \ ]{\prod_{i=1}^n x_i^{w_i} } $$

Weights can be assigned as needed to give more importance to certain elements over others.

For example, more weight might be given to the most recent values in a series.

Alternatively, the absolute frequency of each term could be used as its weight.

Note. In general, the weighted geometric mean is much more sensitive to smaller values compared to the weighted arithmetic mean. However, one of its disadvantages is that the calculations become significantly more complex, especially when there are many terms or when the weights are large. Additionally, if even a single term is zero, the entire result becomes zero.

A Practical Example

A student completes three assignments (A, B, C) as part of a math exam.

They score 18 on the first assignment (A), 22 on the second (B), and 26 on the third (C).

The simple geometric mean of the three scores is 21.75.

$$ \mu = \sqrt[3]{18 \cdot 22 \cdot 26}$$

$$ \mu = \sqrt[3]{10296}$$

$$ \mu = 21.75 $$

However, the professor decides to assign different weights to the assignments.

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

To calculate the mean, I must use the weighted geometric mean.

$$ \mu = \sqrt[4+3+2]{18^4 \cdot 22^3 \cdot 26^2} $$

$$ \mu = \sqrt[9]{104976 \cdot 10648 \cdot 676} $$

$$ \mu = 20.51 $$

In this case, the weighted geometric mean is lower (μ=20.51) than the simple geometric mean (μ=21.75) because the first assignment (A) carries a higher weight than the others.

And so on.