✖

Around[x,δ]

represents an approximate number or quantity with a value around x and an uncertainty δ.

✖

Around[x,{δ_-,δ₊}]

represents a number or quantity with a value around x and asymmetric uncertainties δ_-, δ₊.

✖

Around[dist]

gives an approximate number or quantity around the mean of the distribution dist, with an uncertainty corresponding to the standard deviation of the distribution.

✖

Around[list]

gives an approximate object around the mean of the elements of list and with an uncertainty corresponding to their standard deviation.

✖

Around[s]

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[{x_min,x_max}]	Around[(x_max+x_min)/2,(x_max-x_min)/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 x_i as being uncorrelated.

Examples

open allclose all

Basic Examples (10)Summary of the most common use cases

Uncertain numbers of different sizes and uncertainties:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

Uncertain Quantity objects with different units:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

An Around object with asymmetric uncertainties:

In[1]:=1

✖

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

Out[1]=1

Specify a 5% relative uncertainty in input:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

Perform operations with Around objects:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

Plot a list of Around objects:

In[1]:=1

✖

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

Out[1]=1

Extract the parts of an Around object:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

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

In[1]:=1

✖

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

Out[1]=1

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

In[2]:=2

✖

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

Out[2]=2

Use a symbolic Around object:

In[1]:=1

✖

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

Out[1]=1

Perform operations on it:

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

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

In[1]:=1

✖

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

Out[1]=1

Scope (20)Survey of the scope of standard use cases

Uncertain Objects (6)

Uncertain numbers:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

In[4]:=4

✖

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

Out[4]=4

In[5]:=5

✖

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

Out[5]=5

Uncertain numbers with asymmetric uncertainties:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

Uncertain Quantity objects:

In[1]:=1

✖

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

Out[1]=1

Use a value and an uncertainty of compatible units:

In[2]:=2

✖

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

Out[2]=2

Use the same units in value and uncertainty:

In[3]:=3

✖

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

Out[3]=3

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

In[4]:=4

✖

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

Use asymmetric uncertainties with compatible units:

In[5]:=5

✖

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

Out[5]=5

Values of different sizes for the same uncertainty:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

Uncertainties of different sizes for the same value:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

An uncertain number with 30% relative uncertainty:

In[1]:=1

✖

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

Out[1]=1

Express the same object using a Quantity percentage:

In[2]:=2

✖

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

Out[2]=2

Use Quantity values of any dimension:

In[3]:=3

✖

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

Out[3]=3

In[4]:=4

✖

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

Out[4]=4

Accessors and Conversions (4)

Extract the value and uncertainty of an Around object:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

Construct a finite precision number from an Around object:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

Its accuracy coincides with the uncertainty of a:

In[3]:=3

✖

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

Out[3]=3

In[4]:=4

✖

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

Out[4]=4

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

In[5]:=5

✖

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

Out[5]=5

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

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

Reconstruct the original Around object from the interval:

In[3]:=3

✖

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

Out[3]=3

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

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

In[4]:=4

✖

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

Out[4]=4

Operations with Uncertain Objects (4)

Basic operations with numbers:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

In[4]:=4

✖

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

Out[4]=4

In[5]:=5

✖

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

Out[5]=5

In[6]:=6

✖

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

Out[6]=6

Basic operations with Quantity objects:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

Construct a QuantityArray object whose elements are Around objects:

In[1]:=1

✖

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

Out[1]=1

Perform operations, preserving the QuantityArray structure:

In[2]:=2

✖

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

Out[2]=2

Normalize the result:

In[3]:=3

✖

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

Out[3]=3

Basic operations with symbolic Around objects:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

In[4]:=4

✖

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

Out[4]=4

In[5]:=5

✖

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

Out[5]=5

Comparisons and Ordering of Uncertain Objects (6)

Compare two Around objects with far apart centers:

In[1]:=1

✖

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

Out[1]=1

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

Their distance is significantly larger than zero:

In[4]:=4

✖

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

Out[4]=4

Compare two Around objects with close centers:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

Their distance is not significantly larger than zero:

In[3]:=3

✖

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

Out[3]=3

Compare an Around object with a number:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

The distance is significantly larger than zero:

In[4]:=4

✖

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

Out[4]=4

Compare Around objects with Quantity centers and uncertainties:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

The distance between them is significantly larger than zero:

In[4]:=4

✖

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

Out[4]=4

Order numerically a collection of Around objects and numbers:

In[1]:=1

✖

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

Out[1]=1

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

Order numerically a collection of velocities:

In[1]:=1

✖

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

Out[1]=1

Convert to a common base unit to compare values directly:

In[2]:=2

✖

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

Out[2]=2

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

Plot data with uncertainty:

In[1]:=1

✖

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

Out[1]=1

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

In[1]:=1

✖

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

In[2]:=2

✖

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

Out[2]=2

In[3]:=3

✖

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

Out[3]=3

In[4]:=4

✖

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

Out[4]=4

Compute the mean value of the masses and radii:

In[5]:=5

✖

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

Out[5]=5

In[6]:=6

✖

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

Out[6]=6

Compare with the mass and radius of the Earth:

In[7]:=7

✖

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

Out[7]=7

In[8]:=8

✖

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

Out[8]=8

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: