propagates uncertainty in expr by replacing all occurrences of si by Around[xi,δi].


propagates uncertainty in expr using a series expansion to order n.


  • In AroundReplace[expr,{s1->Around[xi,],}], each occurrence of a given si in expr is assumed to represent the same value, but different replacements Around[xi,] are assumed to be uncorrelated.
  • In AroundReplace[expr,rules], rules can contain scalar rules of the form sAround[] for an uncorrelated variable s, or vector rules of the form {s,r,}VectorAround[] for correlated variables s,r,.
  • In AroundReplace[f[s,r],rules], uncertainty propagation is based on the following formula:
  • In AroundReplace[f[s,r],rules,n], uncertainty propagation involves derivatives up to order n.
  • For n=1, the result of AroundReplace[f[s],sAround[x,δ],n] is centered at f[x], but for higher n orders the central value may depend on δ as well.


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Basic Examples  (2)

Perform an operation with Around objects:

Converting x into two uncorrelated variables makes the result have smaller uncertainty:

Specify an intermediate level of correlation with VectorAround:

By default, uncertainty propagation uses first-order expansions:

Use a higher-order expansion to obtain better results:

Properties & Relations  (1)

AroundReplace[expr,rules] is not equivalent to ReplaceAll[expr,rules] in general:

They coincide if each replaced variable appears once in the original expression:

Wolfram Research (2019), AroundReplace, Wolfram Language function,


Wolfram Research (2019), AroundReplace, Wolfram Language function,


Wolfram Language. 2019. "AroundReplace." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2019). AroundReplace. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_aroundreplace, author="Wolfram Research", title="{AroundReplace}", year="2019", howpublished="\url{}", note=[Accessed: 18-July-2024 ]}


@online{reference.wolfram_2024_aroundreplace, organization={Wolfram Research}, title={AroundReplace}, year={2019}, url={}, note=[Accessed: 18-July-2024 ]}