# Around

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[{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

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## Basic Examples(10)

Uncertain numbers of different sizes and uncertainties:

Uncertain Quantity objects with different units:

An Around object with asymmetric uncertainties:

Specify a 5% relative uncertainty in input:

Perform operations with Around objects:

Plot a list of Around objects:

Extract the parts of an Around object:

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

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

Use a symbolic Around object:

Perform operations on it:

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

## Scope(20)

### Uncertain Objects(6)

Uncertain numbers:

Uncertain numbers with asymmetric uncertainties:

Uncertain Quantity objects:

Use a value and an uncertainty of compatible units:

Use the same units in value and uncertainty:

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

Use asymmetric uncertainties with compatible units:

Values of different sizes for the same uncertainty:

Uncertainties of different sizes for the same value:

An uncertain number with 30% relative uncertainty:

Express the same object using a Quantity percentage:

Use Quantity values of any dimension:

### Accessors and Conversions(4)

Extract the value and uncertainty of an Around object:

Construct a finite precision number from an Around object:

Its accuracy coincides with the uncertainty of a:

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

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

Reconstruct the original Around object from the interval:

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

### Operations with Uncertain Objects(4)

Basic operations with numbers:

Basic operations with Quantity objects:

Construct a QuantityArray object whose elements are Around objects:

Perform operations, preserving the QuantityArray structure:

Normalize the result:

Basic operations with symbolic Around objects:

### Comparisons and Ordering of Uncertain Objects(6)

Compare two Around objects with far apart centers:

Their distance is significantly larger than zero:

Compare two Around objects with close centers:

Their distance is not significantly larger than zero:

Compare an Around object with a number:

The distance is significantly larger than zero:

Compare Around objects with Quantity centers and uncertainties:

The distance between them is significantly larger than zero:

Order numerically a collection of Around objects and numbers:

Order numerically a collection of velocities:

Convert to a common base unit to compare values directly:

## Applications(3)

Plot data with uncertainty:

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

Compute the mean value of the masses and radii:

Compare with the mass and radius of the Earth:

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:

## Properties & Relations(3)

Square an Around object, using a first-order series approximation:

Perform the corresponding exact computation using TransformedDistribution:

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

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

Take a normal distribution and simulate it:

Around[scalars] estimates the mean and standard deviation of the distribution:

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

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

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

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).

#### CMS

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

#### BibTeX

@misc{reference.wolfram_2024_around, author="Wolfram Research", title="{Around}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/Around.html}", note=[Accessed: 04-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_around, organization={Wolfram Research}, title={Around}, year={2023}, url={https://reference.wolfram.com/language/ref/Around.html}, note=[Accessed: 04-August-2024 ]}