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ScientificErrorAnalysis

 Variance
 return the variance of a quantity-with-error

 Calling Sequence Variance( obj )

Parameters

 obj - quantity-with-error

Description

 • 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

${u\left(y\right)}^{2}=\left(\sum _{i=1}^{N}{\left(\frac{\partial }{\partial {x}_{i}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y\right)}^{2}{u\left({x}_{i}\right)}^{2}\right)+2\left(\sum _{i=1}^{N-1}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\sum _{j=i+1}^{N}\left(\frac{\partial }{\partial {x}_{i}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y\right)\left(\frac{\partial }{\partial {x}_{j}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y\right)u\left({x}_{i},{x}_{j}\right)\right)$

 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.

Examples

 > with(ScientificConstants):
 > with(ScientificErrorAnalysis):
 > a := Quantity( 10., 2. ):
 > Variance( a );
 ${4.}$ (1)
 > GetConstant( h );
 ${\mathrm{Planck_constant}}{,}{\mathrm{symbol}}{=}{h}{,}{\mathrm{value}}{=}{6.626070040}{}{{10}}^{{-34}}{,}{\mathrm{uncertainty}}{=}{8.1}{}{{10}}^{{-42}}{,}{\mathrm{units}}{=}{J}{}{s}$ (2)
 > Variance( Constant( h ) );
 ${6.561}{}{{10}}^{{-83}}$ (3)
 > GetConstant( m[e] );
 ${\mathrm{electron_mass}}{,}{\mathrm{symbol}}{=}{{m}}_{{e}}{,}{\mathrm{derive}}{=}\frac{{2}{}{{R}}_{{\mathrm{\infty }}}{}{h}}{{c}{}{{\mathrm{\alpha }}}^{{2}}}$ (4)
 > Variance( Constant( m[e] ) );
 ${1.241654324}{}{{10}}^{{-76}}$ (5)
 > GetError( Constant( m[e] ) )^2;
 ${1.241654323}{}{{10}}^{{-76}}$ (6)