The Variance( obj ) command returns the variance of the quantity-with-error obj.

The quantity-with-error obj can have functional dependence on other quantities-with-error.

If the quantity-with-error obj does not have functional dependence on other quantities-with-error, the uncertainty of obj is accessed and converted to the variance (by squaring).

If the quantity-with-error obj has functional dependence on other quantities-with-error, the variance is calculated using the usual formula of error analysis involving a first-order expansion with the dependent form and covariances between the other quantities-with-error. This process can be recursive.

The variance $u^{2\left(y\right)}$ in $y$, where $y$ depends on the $x_i$, is

where $u\left(x_i\right)$ is the error in $x_i$, $u\left(x_i\,x_j\right)$ is the covariance between $x_i$ and $x_j$, and the partials are evaluated at the central values of the $x_i$.

Variances involving physical constants are calculated naturally and correctly in the implied system of units because central values and errors are obtained from the interface to ScientificConstants.

$\mathrm{with}\left(\mathrm{ScientificConstants}\right)\:$

$\mathrm{with}\left(\mathrm{ScientificErrorAnalysis}\right)\:$

$a≔\mathrm{Quantity}\left(10.\,2.\right)\:$

$\mathrm{Variance}\left(a\right)$

$4.$

$\mathrm{GetConstant}\left(h\right)$

$\mathrm{Planck\_constant},\mathrm{symbol}=h,\mathrm{value}=6.626070040\times {10}^{\mathrm{-34}},\mathrm{uncertainty}=8.1\times {10}^{\mathrm{-42}},\mathrm{units}=Js$

$\mathrm{Variance}\left(\mathrm{Constant}\left(h\right)\right)$

$6.561\times {10}^{\mathrm{-83}}$

$\mathrm{GetConstant}\left(m\left[e\right]\right)$

$\mathrm{electron\_mass},\mathrm{symbol}=m_e,\mathrm{derive}=\frac{2R_{\mathrm{\infty }}h}{c{\mathrm{\alpha }}^2}$

$\mathrm{Variance}\left(\mathrm{Constant}\left(m\left[e\right]\right)\right)$

$1.241654324\times {10}^{\mathrm{-76}}$

${\mathrm{GetError}\left(\mathrm{Constant}\left(m\left[e\right]\right)\right)}^2$

$1.241654323\times {10}^{\mathrm{-76}}$