= Weighted geometric mean
{title2=$G_w=\prod_i x_i^{w_i}$}
For positive <real numbers> $x_i$ and nonnegative weights with $\sum_iw_i=1$, the weighted geometric mean is
$$
G_w=\prod_i x_i^{w_i}=\exp\left(\sum_iw_i\log x_i\right).
$$
It takes a <weighted mean> of <logarithms>. A zero weight excludes that input; equal weights give the ordinary <geometric mean>. If $x_i$ occurs $m_i$ times in a list of total length $M$, its correct weight is $w_i=m_i/M$. For the list $(12,3,3,3)$,
$$
G_w=12^{1/4}3^{3/4}=3\sqrt2\approx4.24264,
$$
and replacing all four entries by $G_w$ preserves the product $324$.
There is also an optimization interpretation. A <logarithmic transformation> turns multiplicative discrepancies into additive ones. For a positive representative $c$, the <weighted mean minimizes squared deviations> identity gives
$$
\sum_iw_i(\log x_i-\log c)^2=\sum_iw_i(\log x_i-\log G_w)^2+(\log c-\log G_w)^2.
$$
Thus $c=G_w$ uniquely minimizes weighted <quadratic loss> in logarithmic coordinates. The <weighted arithmetic mean> minimizes squared discrepancies in the original coordinates. These two objectives explain why different choices of what constitutes an error select different summaries.
The <strictly concave function> $\log x$ and the reversed <Jensen's inequality> give $G_w\leq A_w$, where $A_w=\sum_iw_ix_i$ is the <weighted arithmetic mean>. Apply the same reasoning to the <multiplicative inverses> to get $H_w\leq G_w$ for the <weighted harmonic mean>. Thus $H_w\leq G_w\leq A_w$; equality holds exactly when all positive-weight inputs agree. For this list, $H_w=48/13$, $G_w=3\sqrt2$, and $A_w=21/4$. The weights specify which inputs are represented and must stay the same across the comparison.
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