Propagation of uncertainty

Let ${\displaystyle \{f_{k}(x_{1},x_{2},\dots ,x_{n})\}}$ be a set of m functions, which are linear combinations of ${\displaystyle n}$ variables ${\displaystyle x_{1},x_{2},\dots ,x_{n}}$ with combination coefficients ${\displaystyle A_{k1},A_{k2},\dots ,A_{kn},(k=1,\dots ,m)}$:

$${\displaystyle f_{k}=\sum _{i=1}^{n}A_{ki}x_{i},}$$

or in matrix notation,

$${\displaystyle \mathbf {f} =\mathbf {A} \mathbf {x} .}$$

Also let the variance–covariance matrix of x = (x₁, ..., x_n) be denoted by ${\displaystyle {\boldsymbol {\Sigma }}^{x}}$ and let the mean value be denoted by ${\displaystyle {\boldsymbol {\mu }}}$:

$${\displaystyle {\boldsymbol {\Sigma }}^{x}=E[(\mathbf {x} -{\boldsymbol {\mu }})\otimes (\mathbf {x} -{\boldsymbol {\mu }})]={\begin{pmatrix}\sigma _{1}^{2}&\sigma _{12}&\sigma _{13}&\cdots \\\sigma _{21}&\sigma _{2}^{2}&\sigma _{23}&\cdots \\\sigma _{31}&\sigma _{32}&\sigma _{3}^{2}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{pmatrix}}={\begin{pmatrix}{\Sigma }_{11}^{x}&{\Sigma }_{12}^{x}&{\Sigma }_{13}^{x}&\cdots \\{\Sigma }_{21}^{x}&{\Sigma }_{22}^{x}&{\Sigma }_{23}^{x}&\cdots \\{\Sigma }_{31}^{x}&{\Sigma }_{32}^{x}&{\Sigma }_{33}^{x}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{pmatrix}}.}$$

${\displaystyle \otimes }$ is the outer product.

Then, the variance–covariance matrix ${\displaystyle {\boldsymbol {\Sigma }}^{f}}$ of f is given by

$${\displaystyle {\boldsymbol {\Sigma }}^{f}=E[(\mathbf {f} -E[\mathbf {f} ])\otimes (\mathbf {f} -E[\mathbf {f} ])]=E[\mathbf {A} (\mathbf {x} -{\boldsymbol {\mu }})\otimes \mathbf {A} (\mathbf {x} -{\boldsymbol {\mu }})]=\mathbf {A} E[(\mathbf {x} -{\boldsymbol {\mu }})\otimes (\mathbf {x} -{\boldsymbol {\mu }})]\mathbf {A} ^{\mathrm {T} }=\mathbf {A} {\boldsymbol {\Sigma }}^{x}\mathbf {A} ^{\mathrm {T} }.}$$

In component notation, the equation

$${\displaystyle {\boldsymbol {\Sigma }}^{f}=\mathbf {A} {\boldsymbol {\Sigma }}^{x}\mathbf {A} ^{\mathrm {T} }}$$

reads

$${\displaystyle \Sigma _{ij}^{f}=\sum _{k}^{n}\sum _{l}^{n}A_{ik}{\Sigma }_{kl}^{x}A_{jl}.}$$

This is the most general expression for the propagation of error from one set of variables onto another. When the errors on x are uncorrelated, the general expression simplifies to

$${\displaystyle \Sigma _{ij}^{f}=\sum _{k}^{n}A_{ik}\Sigma _{k}^{x}A_{jk},}$$

where ${\displaystyle \Sigma _{k}^{x}=\sigma _{x_{k}}^{2}}$ is the variance of k-th element of the x vector. Note that even though the errors on x may be uncorrelated, the errors on f are in general correlated; in other words, even if ${\displaystyle {\boldsymbol {\Sigma }}^{x}}$ is a diagonal matrix, ${\displaystyle {\boldsymbol {\Sigma }}^{f}}$ is in general a full matrix.

The general expressions for a scalar-valued function f are a little simpler (here a is a row vector):

$${\displaystyle f=\sum _{i}^{n}a_{i}x_{i}=\mathbf {ax} ,}$$

$${\displaystyle \sigma _{f}^{2}=\sum _{i}^{n}\sum _{j}^{n}a_{i}\Sigma _{ij}^{x}a_{j}=\mathbf {a} {\boldsymbol {\Sigma }}^{x}\mathbf {a} ^{\mathrm {T} }.}$$

Each covariance term ${\displaystyle \sigma _{ij}}$ can be expressed in terms of the correlation coefficient ${\displaystyle \rho _{ij}}$ by ${\displaystyle \sigma _{ij}=\rho _{ij}\sigma _{i}\sigma _{j}}$, so that an alternative expression for the variance of f is

$${\displaystyle \sigma _{f}^{2}=\sum _{i}^{n}a_{i}^{2}\sigma _{i}^{2}+\sum _{i}^{n}\sum _{j(j\neq i)}^{n}a_{i}a_{j}\rho _{ij}\sigma _{i}\sigma _{j}.}$$

In the case that the variables in x are uncorrelated, this simplifies further to

$${\displaystyle \sigma _{f}^{2}=\sum _{i}^{n}a_{i}^{2}\sigma _{i}^{2}.}$$

In the simple case of identical coefficients and variances, we find

$${\displaystyle \sigma _{f}={\sqrt {n}}\,|a|\sigma .}$$

For the arithmetic mean, ${\displaystyle a=1/n}$, the result is the standard error of the mean:

$${\displaystyle \sigma _{f}={\frac {\sigma }{\sqrt {n}}}.}$$

When f is a set of non-linear combination of the variables x, an interval propagation could be performed in order to compute intervals which contain all consistent values for the variables. In a probabilistic approach, the function f must usually be linearised by approximation to a first-order Taylor series expansion, though in some cases, exact formulae can be derived that do not depend on the expansion as is the case for the exact variance of products.[7] The Taylor expansion would be:

$${\displaystyle f_{k}\approx f_{k}^{0}+\sum _{i}^{n}{\frac {\partial f_{k}}{\partial {x_{i}}}}x_{i}}$$

where ${\displaystyle \partial f_{k}/\partial x_{i}}$ denotes the partial derivative of f_k with respect to the i-th variable, evaluated at the mean value of all components of vector x. Or in matrix notation,

$${\displaystyle \mathrm {f} \approx \mathrm {f} ^{0}+\mathrm {J} \mathrm {x} \,}$$

where J is the Jacobian matrix. Since f⁰ is a constant it does not contribute to the error on f. Therefore, the propagation of error follows the linear case, above, but replacing the linear coefficients, A_ki and A_kj by the partial derivatives, ${\displaystyle {\frac {\partial f_{k}}{\partial x_{i}}}}$ and ${\displaystyle {\frac {\partial f_{k}}{\partial x_{j}}}}$. In matrix notation,[8]

$${\displaystyle \mathrm {\Sigma } ^{\mathrm {f} }=\mathrm {J} \mathrm {\Sigma } ^{\mathrm {x} }\mathrm {J} ^{\top }.}$$

That is, the Jacobian of the function is used to transform the rows and columns of the variance-covariance matrix of the argument. Note this is equivalent to the matrix expression for the linear case with ${\displaystyle \mathrm {J=A} }$.

Simplification

Neglecting correlations or assuming independent variables yields a common formula among engineers and experimental scientists to calculate error propagation, the variance formula:[9]

$${\displaystyle s_{f}={\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}s_{x}^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}s_{y}^{2}+\left({\frac {\partial f}{\partial z}}\right)^{2}s_{z}^{2}+\cdots }}}$$

where ${\displaystyle s_{f}}$ represents the standard deviation of the function ${\displaystyle f}$, ${\displaystyle s_{x}}$ represents the standard deviation of ${\displaystyle x}$, ${\displaystyle s_{y}}$ represents the standard deviation of ${\displaystyle y}$, and so forth.

It is important to note that this formula is based on the linear characteristics of the gradient of ${\displaystyle f}$ and therefore it is a good estimation for the standard deviation of ${\displaystyle f}$ as long as ${\displaystyle s_{x},s_{y},s_{z},\ldots }$ are small enough. Specifically, the linear approximation of ${\displaystyle f}$ has to be close to ${\displaystyle f}$ inside a neighbourhood of radius ${\displaystyle s_{x},s_{y},s_{z},\ldots }$.[10]

Example

Any non-linear differentiable function, ${\displaystyle f(a,b)}$, of two variables, ${\displaystyle a}$ and ${\displaystyle b}$, can be expanded as

$${\displaystyle f\approx f^{0}+{\frac {\partial f}{\partial a}}a+{\frac {\partial f}{\partial b}}b}$$

now, taking variance on both sides, and using the formula[11] for variance of a linear combination of variables:

$${\displaystyle \operatorname {Var} (aX+bY)=a^{2}\operatorname {Var} (X)+b^{2}\operatorname {Var} (Y)+2ab\operatorname {Cov} (X,Y)}$$

hence:

$${\displaystyle \sigma _{f}^{2}\approx \left|{\frac {\partial f}{\partial a}}\right|^{2}\sigma _{a}^{2}+\left|{\frac {\partial f}{\partial b}}\right|^{2}\sigma _{b}^{2}+2{\frac {\partial f}{\partial a}}{\frac {\partial f}{\partial b}}\sigma _{ab}}$$

where ${\displaystyle \sigma _{f}}$ is the standard deviation of the function ${\displaystyle f}$, ${\displaystyle \sigma _{a}}$ is the standard deviation of ${\displaystyle a}$, ${\displaystyle \sigma _{b}}$ is the standard deviation of ${\displaystyle b}$ and ${\displaystyle \sigma _{ab}=\sigma _{a}\sigma _{b}\rho _{ab}}$ is the covariance between ${\displaystyle a}$ and ${\displaystyle b}$.

In the particular case that ${\displaystyle f=ab}$, ${\displaystyle {\frac {\partial f}{\partial a}}=b}$, ${\displaystyle {\frac {\partial f}{\partial b}}=a}$. Then

$${\displaystyle \sigma _{f}^{2}\approx b^{2}\sigma _{a}^{2}+a^{2}\sigma _{b}^{2}+2ab\,\sigma _{ab}}$$

or

$${\displaystyle \left({\frac {\sigma _{f}}{f}}\right)^{2}\approx \left({\frac {\sigma _{a}}{a}}\right)^{2}+\left({\frac {\sigma _{b}}{b}}\right)^{2}+2\left({\frac {\sigma _{a}}{a}}\right)\left({\frac {\sigma _{b}}{b}}\right)\rho _{ab}}$$

where ${\displaystyle \rho _{ab}}$ is the correlation between ${\displaystyle a}$ and ${\displaystyle b}$.

When the variables ${\displaystyle a}$ and ${\displaystyle b}$ are uncorrelated, ${\displaystyle \rho _{ab}=0}$. Then

$${\displaystyle \left({\frac {\sigma _{f}}{f}}\right)^{2}\approx \left({\frac {\sigma _{a}}{a}}\right)^{2}+\left({\frac {\sigma _{b}}{b}}\right)^{2}.}$$

This table shows the variances and standard deviations of simple functions of the real variables ${\displaystyle A,B\!}$, with standard deviations ${\displaystyle \sigma _{A},\sigma _{B},\,}$ covariance ${\displaystyle \sigma _{AB}=\rho _{AB}\sigma _{A}\sigma _{B}\,}$, and correlation ${\displaystyle \rho _{AB}}$. The real-valued coefficients ${\displaystyle a}$ and ${\displaystyle b}$ are assumed exactly known (deterministic), i.e., ${\displaystyle \sigma _{a}=\sigma _{b}=0}$.

In the columns "Variance" and "Standard Deviation", A and B should be understood as expectation values (i.e. values around which we're estimating the uncertainty), and ${\displaystyle f}$ should be understood as the value of the function calculated at the expectation value of ${\displaystyle A,B\!}$.

Function	Variance	Standard Deviation
${\displaystyle f=aA\,}$	${\displaystyle \sigma _{f}^{2}=a^{2}\sigma _{A}^{2}}$	${\displaystyle \sigma _{f}=|a|\sigma _{A}}$
${\displaystyle f=aA+bB\,}$	${\displaystyle \sigma _{f}^{2}=a^{2}\sigma _{A}^{2}+b^{2}\sigma _{B}^{2}+2ab\,\sigma _{AB}}$	${\displaystyle \sigma _{f}={\sqrt {a^{2}\sigma _{A}^{2}+b^{2}\sigma _{B}^{2}+2ab\,\sigma _{AB}}}}$
${\displaystyle f=aA-bB\,}$	${\displaystyle \sigma _{f}^{2}=a^{2}\sigma _{A}^{2}+b^{2}\sigma _{B}^{2}-2ab\,\sigma _{AB}}$	${\displaystyle \sigma _{f}={\sqrt {a^{2}\sigma _{A}^{2}+b^{2}\sigma _{B}^{2}-2ab\,\sigma _{AB}}}}$
${\displaystyle f=A-B,}$	${\displaystyle \sigma _{f}^{2}=\sigma _{A}^{2}+\sigma _{B}^{2}-2\sigma _{AB}}$	${\displaystyle \sigma _{f}={\sqrt {\sigma _{A}^{2}+\sigma _{B}^{2}-2\sigma _{AB}}}}$
${\displaystyle f=AB\,}$	${\displaystyle \sigma _{f}^{2}\approx f^{2}\left[\left({\frac {\sigma _{A}}{A}}\right)^{2}+\left({\frac {\sigma _{B}}{B}}\right)^{2}+2{\frac {\sigma _{AB}}{AB}}\right]}$[15][16]	${\displaystyle \sigma _{f}\approx \left|f\right|{\sqrt {\left({\frac {\sigma _{A}}{A}}\right)^{2}+\left({\frac {\sigma _{B}}{B}}\right)^{2}+2{\frac {\sigma _{AB}}{AB}}}}}$
${\displaystyle f={\frac {A}{B}}\,}$	${\displaystyle \sigma _{f}^{2}\approx f^{2}\left[\left({\frac {\sigma _{A}}{A}}\right)^{2}+\left({\frac {\sigma _{B}}{B}}\right)^{2}-2{\frac {\sigma _{AB}}{AB}}\right]}$[17]	${\displaystyle \sigma _{f}\approx \left|f\right|{\sqrt {\left({\frac {\sigma _{A}}{A}}\right)^{2}+\left({\frac {\sigma _{B}}{B}}\right)^{2}-2{\frac {\sigma _{AB}}{AB}}}}}$
${\displaystyle f={\frac {A}{A+B}}\,}$	${\displaystyle \sigma _{f}^{2}\approx {\frac {f^{2}}{\left(A+B\right)^{2}}}\left({\frac {B^{2}}{A^{2}}}\sigma _{A}^{2}+\sigma _{B}^{2}-2{\frac {B}{A}}\sigma _{AB}\right)}$	${\displaystyle \sigma _{f}\approx \left|{\frac {f}{A+B}}\right|{\sqrt {{\frac {B^{2}}{A^{2}}}\sigma _{A}^{2}+\sigma _{B}^{2}-2{\frac {B}{A}}\sigma _{AB}}}}$
${\displaystyle f=aA^{b}\,}$	${\displaystyle \sigma _{f}^{2}\approx \left({a}{b}{A}^{b-1}{\sigma _{A}}\right)^{2}=\left({\frac {{f}{b}{\sigma _{A}}}{A}}\right)^{2}}$	${\displaystyle \sigma _{f}\approx \left|{a}{b}{A}^{b-1}{\sigma _{A}}\right|=\left|{\frac {{f}{b}{\sigma _{A}}}{A}}\right|}$
${\displaystyle f=a\ln(bA)\,}$	${\displaystyle \sigma _{f}^{2}\approx \left(a{\frac {\sigma _{A}}{A}}\right)^{2}}$[18]	${\displaystyle \sigma _{f}\approx \left|a{\frac {\sigma _{A}}{A}}\right|}$
${\displaystyle f=a\log _{10}(bA)\,}$	${\displaystyle \sigma _{f}^{2}\approx \left(a{\frac {\sigma _{A}}{A\ln(10)}}\right)^{2}}$[18]	${\displaystyle \sigma _{f}\approx \left|a{\frac {\sigma _{A}}{A\ln(10)}}\right|}$
${\displaystyle f=ae^{bA}\,}$	${\displaystyle \sigma _{f}^{2}\approx f^{2}\left(b\sigma _{A}\right)^{2}}$[19]	${\displaystyle \sigma _{f}\approx \left|f\right|\left|\left(b\sigma _{A}\right)\right|}$
${\displaystyle f=a^{bA}\,}$	${\displaystyle \sigma _{f}^{2}\approx f^{2}(b\ln(a)\sigma _{A})^{2}}$	${\displaystyle \sigma _{f}\approx \left|f\right|\left|b\ln(a)\sigma _{A}\right|}$
${\displaystyle f=a\sin(bA)\,}$	${\displaystyle \sigma _{f}^{2}\approx \left[ab\cos(bA)\sigma _{A}\right]^{2}}$	${\displaystyle \sigma _{f}\approx \left|ab\cos(bA)\sigma _{A}\right|}$
${\displaystyle f=a\cos \left(bA\right)\,}$	${\displaystyle \sigma _{f}^{2}\approx \left[ab\sin(bA)\sigma _{A}\right]^{2}}$	${\displaystyle \sigma _{f}\approx \left|ab\sin(bA)\sigma _{A}\right|}$
${\displaystyle f=a\tan \left(bA\right)\,}$	${\displaystyle \sigma _{f}^{2}\approx \left[ab\sec ^{2}(bA)\sigma _{A}\right]^{2}}$	${\displaystyle \sigma _{f}\approx \left|ab\sec ^{2}(bA)\sigma _{A}\right|}$
${\displaystyle f=A^{B}\,}$	${\displaystyle \sigma _{f}^{2}\approx f^{2}\left[\left({\frac {B}{A}}\sigma _{A}\right)^{2}+\left(\ln(A)\sigma _{B}\right)^{2}+2{\frac {B\ln(A)}{A}}\sigma _{AB}\right]}$	${\displaystyle \sigma _{f}\approx \left|f\right|{\sqrt {\left({\frac {B}{A}}\sigma _{A}\right)^{2}+\left(\ln(A)\sigma _{B}\right)^{2}+2{\frac {B\ln(A)}{A}}\sigma _{AB}}}}$
${\displaystyle f={\sqrt {aA^{2}\pm bB^{2}}}\,}$	${\displaystyle \sigma _{f}^{2}\approx \left({\frac {A}{f}}\right)^{2}a^{2}\sigma _{A}^{2}+\left({\frac {B}{f}}\right)^{2}b^{2}\sigma _{B}^{2}\pm 2ab{\frac {AB}{f^{2}}}\,\sigma _{AB}}$	${\displaystyle \sigma _{f}\approx {\sqrt {\left({\frac {A}{f}}\right)^{2}a^{2}\sigma _{A}^{2}+\left({\frac {B}{f}}\right)^{2}b^{2}\sigma _{B}^{2}\pm 2ab{\frac {AB}{f^{2}}}\,\sigma _{AB}}}}$

For uncorrelated variables (${\displaystyle \rho _{AB}=0}$, ${\displaystyle \sigma _{AB}=0}$) expressions for more complicated functions can be derived by combining simpler functions. For example, repeated multiplication, assuming no correlation, gives

$${\displaystyle f=ABC;\qquad \left({\frac {\sigma _{f}}{f}}\right)^{2}\approx \left({\frac {\sigma _{A}}{A}}\right)^{2}+\left({\frac {\sigma _{B}}{B}}\right)^{2}+\left({\frac {\sigma _{C}}{C}}\right)^{2}.}$$

For the case ${\displaystyle f=AB}$ we also have Goodman's expression[7] for the exact variance: for the uncorrelated case it is

$${\displaystyle V(XY)=E(X)^{2}V(Y)+E(Y)^{2}V(X)+E((X-E(X))^{2}(Y-E(Y))^{2})}$$

and therefore we have:

$${\displaystyle \sigma _{f}^{2}=A^{2}\sigma _{B}^{2}+B^{2}\sigma _{A}^{2}+\sigma _{A}^{2}\sigma _{B}^{2}}$$

• Bevington, Philip R.; Robinson, D. Keith (2002), Data Reduction and Error Analysis for the Physical Sciences (3rd ed.), McGraw-Hill, ISBN 978-0-07-119926-1
• Fornasini, Paolo (2008), The uncertainty in physical measurements: an introduction to data analysis in the physics laboratory, Springer, p. 161, ISBN 978-0-387-78649-0
• Meyer, Stuart L. (1975), Data Analysis for Scientists and Engineers, Wiley, ISBN 978-0-471-59995-1
• Peralta, M. (2012), Propagation Of Errors: How To Mathematically Predict Measurement Errors, CreateSpace
• Rouaud, M. (2013), Probability, Statistics and Estimation: Propagation of Uncertainties in Experimental Measurement (PDF) (short ed.)
• Taylor, J. R. (1997), An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements (2nd ed.), University Science Books
• Wang, C M; Iyer, Hari K (2005-09-07). "On higher-order corrections for propagating uncertainties". Metrologia. 42 (5): 406–410. doi:10.1088/0026-1394/42/5/011. ISSN 0026-1394. S2CID 122841691.

• A detailed discussion of measurements and the propagation of uncertainty explaining the benefits of using error propagation formulas and Monte Carlo simulations instead of simple significance arithmetic
• GUM, Guide to the Expression of Uncertainty in Measurement
• EPFL An Introduction to Error Propagation, Derivation, Meaning and Examples of Cy = Fx Cx Fx'
• uncertainties package, a program/library for transparently performing calculations with uncertainties (and error correlations).
• soerp package, a Python program/library for transparently performing *second-order* calculations with uncertainties (and error correlations).
• Joint Committee for Guides in Metrology (2011). JCGM 102: Evaluation of Measurement Data - Supplement 2 to the "Guide to the Expression of Uncertainty in Measurement" - Extension to Any Number of Output Quantities (PDF) (Technical report). JCGM. Retrieved 13 February 2013.
• Uncertainty Calculator Propagate uncertainty for any expression