In statistics, inverse-variance weighting is a method of aggregating two or more random variables to minimize the variance of the weighted average. Each random variable is weighted in inverse proportion to its variance (i.e., proportional to its precision).
Formulation
Given a sequence of independent observations yi with variances σi2, the inverse-variance weighted average is given by
y ^ = ∑ i y i / σ i 2 ∑ i 1 / σ i 2 . {\displaystyle {\hat {y}}={\frac {\sum _{i}y_{i}/\sigma _{i}^{2}}{\sum _{i}1/\sigma _{i}^{2}}}.}
The inverse-variance weighted average has the least variance among all weighted averages, which can be calculated as
V a r ( y ^ ) = 1 ∑ i 1 / σ i 2 . {\displaystyle Var({\hat {y}})={\frac {1}{\sum _{i}1/\sigma _{i}^{2}}}.}
This variance can be used to parametrize a confidence interval. If the variances of the measurements are all equal, then the inverse-variance weighted average becomes the simple average. Inverse-variance weighting is typically used in statistical meta-analysis or sensor fusion to combine the results from independent measurements.
Context Suppose an experimenter wishes to measure the value of a quantity, say the acceleration due to gravity of Earth, whose true value happens to be μ {\displaystyle \mu } . A careful experimenter makes multiple measurements, which we denote with n {\displaystyle n} random variables X 1 , X 2 , . . . , X n {\displaystyle X_{1},X_{2},...,X_{n}} . If they are all noisy but unbiased, i.e., the measuring device does not systematically overestimate or underestimate the true value and the errors are scattered symmetrically, then the expectation value E [ X i ] = μ {\displaystyle E[X_{i}]=\mu } ∀ i {\displaystyle \forall i} . The scatter in the measurement is then characterised by the variance of the random variables V a r ( X i ) := σ i 2 {\displaystyle Var(X_{i}):=\sigma _{i}^{2}} , and if the measurements are performed under identical scenarios, then all the σ i {\displaystyle \sigma _{i}} are the same, which we shall refer to by σ {\displaystyle \sigma } . Given the n {\displaystyle n} measurements, a typical estimator for μ {\displaystyle \mu } , denoted as μ ^ {\displaystyle {\hat {\mu }}} , is given by the simple average X ¯ = 1 n ∑ i X i {\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i}X_{i}} . Note that this empirical average is also a random variable, whose expectation value E [ X ¯ ] {\displaystyle E[{\overline {X}}]} is μ {\displaystyle \mu } but also has a scatter. If the individual measurements are uncorrelated, the square of the error in the estimate is given by
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