= Inverse-variance weighting
\b[Inverse-variance weighting] combines estimates of the same quantity by giving a more precise estimate a larger weight. Suppose $X_i$ are independent, <unbiased estimators> of $\theta$, with known <variances> $0<\sigma_i^2<\infty$. For normalized weights,
$$
w_i=\frac{\sigma_i^{-2}}{\sum_j\sigma_j^{-2}},\qquad
\widehat\theta=\sum_iw_iX_i,\qquad
\operatorname{Var}(\widehat\theta)=\frac1{\sum_i\sigma_i^{-2}}.
$$
The inverse is taken after squaring the <standard uncertainty>: twice the standard uncertainty gives one quarter as much weight. The corresponding <weighted least squares> criterion is $\sum_i(X_i-\theta)^2/\sigma_i^2$; differentiating it with respect to $\theta$ gives the same <weighted arithmetic mean>.<Weighted Least Squares by NIST>[§4.4.3.2, General WLS Criterion, intercept-only specialization]{ref}
The optimality statement is precise. Any linear <unbiased estimator> $\sum_i a_iX_i$ requires $\sum_i a_i=1$. By the <Cauchy-Schwarz inequality>,
$$
1=\left(\sum_i(a_i\sigma_i)\sigma_i^{-1}\right)^2
\le\left(\sum_i a_i^2\sigma_i^2\right)\left(\sum_i\sigma_i^{-2}\right).
$$
Equality holds for the displayed weights. Thus they minimize <variance> among linear unbiased combinations under these assumptions; a <normal distribution> is not needed for that proof.
For an original example, combine observed values 10 and 14 with standard uncertainties 1 and 2 in the same units. The normalized weights are $0.8$ and $0.2$, giving estimate $10.8$, variance $0.8$ and standard uncertainty approximately $0.894$. Equal weights give 12 and standard uncertainty $\sqrt{1.25}\approx1.118$.
Correlated errors change the problem. A contribution common to every reading survives every weighting whose coefficients sum to one; <precision limits of weighting correlated repeats> derives the resulting floor. More generally, use the full <covariance matrix> and <generalized least squares> rather than treating correlated readings as independent.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equation13, and §5.2.4, common inputs]{ref} <Uncertainty in estimated inverse-variance weights> is another limitation: variances estimated from the same data are not fixed known quantities, and an accidentally tiny estimated variance does not justify arbitrary confidence.
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