Around
represents an approximate number or quantity with a value around x and an uncertainty δ.
represents a number or quantity with a value around x and asymmetric uncertainties δ-, δ+.
gives an approximate number or quantity around the mean of the distribution dist, with an uncertainty corresponding to the standard deviation of the distribution.
gives an approximate object around the mean of the elements of list and with an uncertainty corresponding to their standard deviation.
gives an approximate object derived from the number, interval or string specification s.
Details


- Around[x,δ] is typically displayed as x±δ. If δ is very small compared to x, as in Around[1.2345678,0.0000012], it is instead displayed in a form like
.
- Around[x,δ] can be used to represent results of measurements in which there is statistical or other uncertainty.
- Around[x,Scaled[ϕ]] represents a number with relative error ϕ, corresponding to Around[x,ϕ x].
- When Around is used in computations, uncertainties are by default propagated using a first-order series approximation, assuming no correlations.
- Around[…]["prop"] can be used to extract the following properties:
-
"Value" central value x in Around[x,δ] "Uncertainty" uncertainty δ in Around[x,δ] "Number" number with value x and accuracy corresponding to δ "Interval" Interval[{x-δ,x+δ}] - In Around[s], numbers with uncertainty can be specified as follows:
-
x (approximate number) Around[x,(10^-Accuracy[x])/2] Interval[{xmin,xmax}] Around[(xmax+xmin)/2,(xmax-xmin)/2] dist (statistical distribution) Around[Mean[dist],StandardDeviation[dist]] list (list of elements) Around[Mean[list],StandardDeviation[list]] "nn.dddd" (number string) (uncertainty determined by number of significant digits) - For linear computations, Around[x,δ] behaves like a number whose values are distributed according to the normal distribution NormalDistribution[x,δ].
- Relational operators like Less, Equal and Greater on Around objects Around[x1,δ1] and Around[x2,δ2] return True or False depending on whether the distance between centers x1 and x2 is larger or smaller than 2
. Numbers are assigned zero uncertainty when compared to Around objects.
- NumericalOrder[Around[x1,δ1],Around[x2,δ2]] returns 0 if the distance between centers x1, x2 is smaller than 0.5
. Otherwise it returns 1 or
, depending on the ordering of centers. Numbers are assigned zero uncertainty when sorted numerically together with Around objects.
- In Around[x,δ], the value x and the uncertainty δ can be any numeric or symbolic expressions. If δ is a numeric expression, then both x and δ will be made numerical. By default, machine precision will be used, but higher precision may be used if needed to represent the numbers faithfully.
- Around[x,δ] displays with one or two digits of the uncertainty δ shown; x is shown with the same number of digits to the right of the decimal point as is shown in δ.
- In Around[x,δ], x and δ can be quantities with different, though compatible, units.
- Around[{x1,x2,…},δ] threads over the list in its first argument, effectively treating the uncertainties in the xi as being uncorrelated.
Examples
open allclose allBasic Examples (10)Summary of the most common use cases
Uncertain numbers of different sizes and uncertainties:

https://wolfram.com/xid/0j44recy-h1gb62


https://wolfram.com/xid/0j44recy-z3omk


https://wolfram.com/xid/0j44recy-okkv0b

Uncertain Quantity objects with different units:

https://wolfram.com/xid/0j44recy-k90xv2


https://wolfram.com/xid/0j44recy-l4qlw2


https://wolfram.com/xid/0j44recy-go5fvc

An Around object with asymmetric uncertainties:

https://wolfram.com/xid/0j44recy-kpypwp

Specify a 5% relative uncertainty in input:

https://wolfram.com/xid/0j44recy-3utr9t


https://wolfram.com/xid/0j44recy-x8d72b

Perform operations with Around objects:

https://wolfram.com/xid/0j44recy-ob4ddj


https://wolfram.com/xid/0j44recy-ixy2ma

Plot a list of Around objects:

https://wolfram.com/xid/0j44recy-c4upy

Extract the parts of an Around object:

https://wolfram.com/xid/0j44recy-blarkh


https://wolfram.com/xid/0j44recy-cu8ju1


https://wolfram.com/xid/0j44recy-emrdm

Two different instances of the same Around object are assumed to be uncorrelated:

https://wolfram.com/xid/0j44recy-0xmrog

Therefore, the resulting uncertainty is smaller than that obtained by multiplication by 2:

https://wolfram.com/xid/0j44recy-63od8y

Use a symbolic Around object:

https://wolfram.com/xid/0j44recy-cnx1m8


https://wolfram.com/xid/0j44recy-nn0w0a


https://wolfram.com/xid/0j44recy-v5b36x

Add symbolic Around objects, with uncertainties assumed to be uncorrelated:

https://wolfram.com/xid/0j44recy-riy7n5

Scope (20)Survey of the scope of standard use cases
Uncertain Objects (6)

https://wolfram.com/xid/0j44recy-2okixx


https://wolfram.com/xid/0j44recy-pqycks


https://wolfram.com/xid/0j44recy-dld4nq


https://wolfram.com/xid/0j44recy-cmnxv6


https://wolfram.com/xid/0j44recy-78586n

Uncertain numbers with asymmetric uncertainties:

https://wolfram.com/xid/0j44recy-1v197y


https://wolfram.com/xid/0j44recy-3arwrn


https://wolfram.com/xid/0j44recy-jg8ein

Uncertain Quantity objects:

https://wolfram.com/xid/0j44recy-tx9hyz

Use a value and an uncertainty of compatible units:

https://wolfram.com/xid/0j44recy-ncckzu

Use the same units in value and uncertainty:

https://wolfram.com/xid/0j44recy-qhwrh3

The result is a single Quantity object with an Around magnitude:

https://wolfram.com/xid/0j44recy-d7tyur

Use asymmetric uncertainties with compatible units:

https://wolfram.com/xid/0j44recy-eimr9s

Values of different sizes for the same uncertainty:

https://wolfram.com/xid/0j44recy-r1cnsr


https://wolfram.com/xid/0j44recy-dizgmo


https://wolfram.com/xid/0j44recy-0895l1

Uncertainties of different sizes for the same value:

https://wolfram.com/xid/0j44recy-npmi71


https://wolfram.com/xid/0j44recy-hc5jsu


https://wolfram.com/xid/0j44recy-un7rbn

An uncertain number with 30% relative uncertainty:

https://wolfram.com/xid/0j44recy-xa8ge1

Express the same object using a Quantity percentage:

https://wolfram.com/xid/0j44recy-jpox1n

Use Quantity values of any dimension:

https://wolfram.com/xid/0j44recy-ubrzh6


https://wolfram.com/xid/0j44recy-7f1n56

Accessors and Conversions (4)
Extract the value and uncertainty of an Around object:

https://wolfram.com/xid/0j44recy-ly8aw7


https://wolfram.com/xid/0j44recy-ua1qhu


https://wolfram.com/xid/0j44recy-zw73s5

Construct a finite precision number from an Around object:

https://wolfram.com/xid/0j44recy-r15xhq


https://wolfram.com/xid/0j44recy-3q6ehu

Its accuracy coincides with the uncertainty of a:

https://wolfram.com/xid/0j44recy-vzs6bd


https://wolfram.com/xid/0j44recy-kc2va9

Reconstruct the original Around object from the finite-precision number:

https://wolfram.com/xid/0j44recy-n26fov

Construct an interval centered at the value of an Around object and with a semi-width given by its uncertainty:

https://wolfram.com/xid/0j44recy-m5gh42


https://wolfram.com/xid/0j44recy-kx2lss

Reconstruct the original Around object from the interval:

https://wolfram.com/xid/0j44recy-n9vvj2

Use Around to convert strings containing numbers, assuming uncertainty 0.5 in the last significant digit:

https://wolfram.com/xid/0j44recy-ygcagk


https://wolfram.com/xid/0j44recy-rp835i


https://wolfram.com/xid/0j44recy-erxwyw


https://wolfram.com/xid/0j44recy-ne7m5a

Operations with Uncertain Objects (4)
Basic operations with numbers:

https://wolfram.com/xid/0j44recy-bb44vs


https://wolfram.com/xid/0j44recy-jxtter


https://wolfram.com/xid/0j44recy-kr7xug


https://wolfram.com/xid/0j44recy-wfco85


https://wolfram.com/xid/0j44recy-fvvxzr


https://wolfram.com/xid/0j44recy-6cx3p8

Basic operations with Quantity objects:

https://wolfram.com/xid/0j44recy-qjy0k4


https://wolfram.com/xid/0j44recy-dorhiu


https://wolfram.com/xid/0j44recy-oe4kkh

Construct a QuantityArray object whose elements are Around objects:

https://wolfram.com/xid/0j44recy-kpgria

Perform operations, preserving the QuantityArray structure:

https://wolfram.com/xid/0j44recy-0r9lei


https://wolfram.com/xid/0j44recy-nzuigh

Basic operations with symbolic Around objects:

https://wolfram.com/xid/0j44recy-bmsq6v


https://wolfram.com/xid/0j44recy-bfshr6


https://wolfram.com/xid/0j44recy-qplzto


https://wolfram.com/xid/0j44recy-g4tnla


https://wolfram.com/xid/0j44recy-3skaze

Comparisons and Ordering of Uncertain Objects (6)
Compare two Around objects with far apart centers:

https://wolfram.com/xid/0j44recy-daap1c



https://wolfram.com/xid/0j44recy-62ax55

Their distance is significantly larger than zero:

https://wolfram.com/xid/0j44recy-7fj0w

Compare two Around objects with close centers:

https://wolfram.com/xid/0j44recy-ti5ufl



https://wolfram.com/xid/0j44recy-9jdokx

Their distance is not significantly larger than zero:

https://wolfram.com/xid/0j44recy-xfwhxd

Compare an Around object with a number:

https://wolfram.com/xid/0j44recy-dsy1zm


https://wolfram.com/xid/0j44recy-oybtb4


https://wolfram.com/xid/0j44recy-2sb5mt

The distance is significantly larger than zero:

https://wolfram.com/xid/0j44recy-3ho9dy

Compare Around objects with Quantity centers and uncertainties:

https://wolfram.com/xid/0j44recy-4g9ltn


https://wolfram.com/xid/0j44recy-e5fz7s


https://wolfram.com/xid/0j44recy-lws7bn

The distance between them is significantly larger than zero:

https://wolfram.com/xid/0j44recy-222e9b

Order numerically a collection of Around objects and numbers:

https://wolfram.com/xid/0j44recy-3r7r8r


https://wolfram.com/xid/0j44recy-o61yo0


https://wolfram.com/xid/0j44recy-2fm7b2

Order numerically a collection of velocities:

https://wolfram.com/xid/0j44recy-79i4cb

Convert to a common base unit to compare values directly:

https://wolfram.com/xid/0j44recy-hlxzyu

Applications (3)Sample problems that can be solved with this function

https://wolfram.com/xid/0j44recy-e5y0ib

Plot exoplanet radius versus mass, including uncertainties in both variables:

https://wolfram.com/xid/0j44recy-uwjddt

https://wolfram.com/xid/0j44recy-lt6f3g


https://wolfram.com/xid/0j44recy-fltrgv


https://wolfram.com/xid/0j44recy-kp0lsp

Compute the mean value of the masses and radii:

https://wolfram.com/xid/0j44recy-fhg7gy


https://wolfram.com/xid/0j44recy-4ql4z6

Compare with the mass and radius of the Earth:

https://wolfram.com/xid/0j44recy-ou7cwj


https://wolfram.com/xid/0j44recy-gcqwss

Compute the period of oscillation of a pendulum of length, using a value of
for Earth's gravity and assuming an uncertainty of one unit in the last significant digit of those quantities:

https://wolfram.com/xid/0j44recy-t5lk5l


https://wolfram.com/xid/0j44recy-de1aba

Properties & Relations (3)Properties of the function, and connections to other functions
Square an Around object, using a first-order series approximation:

https://wolfram.com/xid/0j44recy-s6apaq

Perform the corresponding exact computation using TransformedDistribution:

https://wolfram.com/xid/0j44recy-6jqekd


https://wolfram.com/xid/0j44recy-o5rzcf

Using a higher-order series expansion gives a better approximation to the exact result:

https://wolfram.com/xid/0j44recy-xi0grc

Using Around directly on the asymmetric distribution returns an object with asymmetric uncertainty:

https://wolfram.com/xid/0j44recy-9e8qgu

Take a normal distribution and simulate it:

https://wolfram.com/xid/0j44recy-9ptwn6

https://wolfram.com/xid/0j44recy-lje035
Around[scalars] estimates the mean and standard deviation of the distribution:

https://wolfram.com/xid/0j44recy-oyrvlf

Around[dist] gives the true parameters in the distribution dist:

https://wolfram.com/xid/0j44recy-bbovpj

MeanAround[scalars] describes the mean of the distribution and the standard error of the mean:

https://wolfram.com/xid/0j44recy-rgw491

Around[x,δ] and CenteredInterval[x,δ] use different propagation rules in numeric operations:

https://wolfram.com/xid/0j44recy-dp7bu6


https://wolfram.com/xid/0j44recy-f0q8fv


https://wolfram.com/xid/0j44recy-6szvtu


https://wolfram.com/xid/0j44recy-rc0nsp

Wolfram Research (2019), Around, Wolfram Language function, https://reference.wolfram.com/language/ref/Around.html (updated 2023).
Text
Wolfram Research (2019), Around, Wolfram Language function, https://reference.wolfram.com/language/ref/Around.html (updated 2023).
Wolfram Research (2019), Around, Wolfram Language function, https://reference.wolfram.com/language/ref/Around.html (updated 2023).
CMS
Wolfram Language. 2019. "Around." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/Around.html.
Wolfram Language. 2019. "Around." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/Around.html.
APA
Wolfram Language. (2019). Around. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Around.html
Wolfram Language. (2019). Around. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Around.html
BibTeX
@misc{reference.wolfram_2025_around, author="Wolfram Research", title="{Around}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/Around.html}", note=[Accessed: 09-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_around, organization={Wolfram Research}, title={Around}, year={2023}, url={https://reference.wolfram.com/language/ref/Around.html}, note=[Accessed: 09-July-2025
]}