Weighted and geometric means

A weighted mean gives some values more influence than others; a geometric mean averages multiplicative change such as growth rates.

\xbar_w = \frac{\sum_{i=1}^{n} w_i x_i}{\sum_{i=1}^{n} w_i}

Weighted mean.

$w_i$: Non-negative weight of value $i$ (credits, group sizes, portfolio shares).
$x_i$: The $i$ -th value.

G = \left(\prod_{i=1}^{n} x_i\right)^{1/n} = \exp\!\left(\frac{1}{n}\sum_{i=1}^{n} \ln x_i\right)

Geometric mean (all values must be positive).

$\prod$: Product of the terms.
$x_i$: Positive values, e.g. growth factors $1 + r_i$ .

\bar r_{\text{geo}} = \left(\prod_{i=1}^{n} (1 + r_i)\right)^{1/n} - 1

Average growth rate per period.

$r_i$: Growth (return) in period $i$ as a proportion, e.g. 0.10 for +10%.

When to use

Weighted mean: values represent groups of different sizes (mean of group means, average price over items sold in different quantities), or the weights are set by a rule (course weights, index weights).
Geometric mean: quantities that compound — growth rates, returns, ratios, indices. It answers "what constant rate would give the same overall change?"

How to interpret

The weighted mean is pulled towards values with large weights. The geometric average growth rate, applied every period, reproduces the total change exactly. For positive data the geometric mean is never larger than the arithmetic mean (equal only when all values are equal).

Common mistakes

Averaging group means without weighting by group size.
Taking the geometric mean of the rates $r_i$ themselves (which can be negative) instead of the growth factors $1 + r_i$ .
Using the arithmetic mean of returns to describe compounded performance — it overstates it.
Using the geometric mean with zeros or negative values (not defined).

Example

Course marks 80, 70, 90 with weights 0.2, 0.3, 0.5: $\xbar_w = 16 + 21 + 45 = 82$ .

Returns +10%, −20%, +30%: factors 1.1, 0.8, 1.3, product 1.144. $G = 1.144^{1/3} = 1.0459$ , so the average growth is 4.59% per year; the arithmetic mean of the returns, 6.67%, would overstate it.