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MeanAround[{x1,x2,x3,…}]

gives an Around object describing the mean of the xi and its uncertainty.

MeanAround[{{x11,x12,…},{x21,…},…}]

gives a VectorAround object describing the means of the vectors xi and their covariance.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Scalar Mean with Uncertainty  
Vector Mean with Uncertainty  
Applications  
Properties & Relations  
See Also
Related Guides
History
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MeanAround

MeanAround[{x1,x2,x3,…}]

gives an Around object describing the mean of the xi and its uncertainty.

MeanAround[{{x11,x12,…},{x21,…},…}]

gives a VectorAround object describing the means of the vectors xi and their covariance.

Details

  • The objects xi can be numbers, quantities or Around objects.
  • MeanAround[{Around[x1,δ1],…}] computes the weighted mean of the xi, with weights proportional to 1/δi2.
  • For a list of n numbers or quantities, MeanAround[list] gives Around[Mean[list],].
  • For a list of n vectors, MeanAround[list] gives VectorAround[Mean[list],].
  • In MeanAround[{{x11,x12,…},{x21,x22,…},…}], the xij can be quantities, so long as the units of all x1j, all x2j, etc. are compatible.
  • The singular case MeanAround[{x}] is defined to return x with zero uncertainty.

Examples

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Basic Examples  (5)

Find the mean of a list of numbers, tagged with its uncertainty:

Wolfram Language code: MeanAround[{1, 2, 3, 4, 3, 2, 1}]

Find the mean of lists of vectors, with uncertainty expressed as a covariance matrix:

Wolfram Language code: MeanAround[{{1, 1}, {2, 2}, {3, 3}}]
Wolfram Language code: MeanAround[{{1, 1}, {0, 2}, {3, 1}}]

Find the mean of a list of Quantity objects, tagged with its uncertainty:

Wolfram Language code: list = {Quantity[0.49114, "Meters"], Quantity[1.59556, "Yards"], Quantity[58.864, "Centimeters"], Quantity[0.59244, "Meters"], Quantity[14.6046, "Inches"]};
Wolfram Language code: UnitConvert[list]
Wolfram Language code: MeanAround[list]

Find the mean of a simulation of a normal distribution centered at 3:

Wolfram Language code: SeedRandom[123]; data = RandomVariate[NormalDistribution[3, 0.5], 10]
Wolfram Language code: MeanAround[data]

If the simulation contains many more points, the uncertainty of the mean is lower:

Wolfram Language code: SeedRandom[123]; data = RandomVariate[NormalDistribution[3, 0.5], 10 ^ 4];
Wolfram Language code: MeanAround[data]

MeanAround on a list of vectors returns a VectorAround object:

Wolfram Language code: RandomVariate[MultinormalDistribution[{2, 3}, {{1, 0.5}, {0.5, 2}}], 10]
Wolfram Language code: MeanAround[%]

Scope  (4)

Scalar Mean with Uncertainty  (3)

Compute the mean Around object for a list of numbers:

Wolfram Language code: data = {2.745, 2.964, 2.203, 3.768, 4.339, 2.341, 2.452, 3.071, 3.052, 2.163};
Wolfram Language code: MeanAround[data]

Compute the mean Around object for a list of energies:

Wolfram Language code: data = {Quantity[3.62, "Joules"], Quantity[2.91, "Joules"], Quantity[3.16, "Joules"], Quantity[2.95, "Joules"], Quantity[2.24, "Joules"], Quantity[3.160591821875137, "Joules"], Quantity[2.11, "Joules"], Quantity[3.77, "Joules"], Quantity[3.11, "Joules"], Quantity[2.36, "Joules"]}
Wolfram Language code: MeanAround[data]

Find the weighted mean of a list of Around objects:

Wolfram Language code: list = {Around[0.49114, 0.04205430639900287], Around[0.59556, 0.009586202141021988], Around[0.58864, 0.039668541726954415], Around[0.59244, 0.07593245740260027], Around[0.46046, 0.029440780862296734], Around[0.49334, 0.019166637500261305], Around[0.41232, 0.029567347205657818], Around[0.47712, 0.029548139370204562], Around[0.48596, 0.08931941281826966], Around[0.5557, 0.08274315688817477]};
Wolfram Language code: MeanAround[list]

Compare to the result of MeanAround on the bare values, ignoring the original uncertainties:

Wolfram Language code: MeanAround[Through[list["Value"]]]

Compare with a direct mean on the original list:

Wolfram Language code: Mean[list]

Vector Mean with Uncertainty  (1)

Compute the mean of a list of {length,time} pairs:

Wolfram Language code: QuantityArray[RandomVariate[MultinormalDistribution[{2, 3}, {{1, 0.5}, {0.5, 2}}], 100], {"Meters", "Seconds"}]
Wolfram Language code: MeanAround[%]

Applications  (1)

Download 100 random movie entities and find their runtimes:

Wolfram Language code: movies = RandomEntity["Movie", 100];
Wolfram Language code: runtimes = DeleteMissing[EntityValue[movies, "Runtime"]]

There is a wide distribution of runtime values:

Wolfram Language code: MinMax[runtimes]

These are the mean value and the standard deviation of runtimes:

Wolfram Language code: N@{Mean[runtimes], StandardDeviation[runtimes]}

This is the mean with its uncertainty:

Wolfram Language code: MeanAround[runtimes]
Wolfram Language code: Histogram[runtimes, 20]

Properties & Relations  (2)

Take a normal distribution and simulate it:

Wolfram Language code: dist = NormalDistribution[5, 0.3];
Wolfram Language code: scalars = RandomVariate[dist, 100];

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

Wolfram Language code: Around[scalars]

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

Wolfram Language code: Around[dist]

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

Wolfram Language code: MeanAround[scalars]

Take a multinormal distribution for 2D vectors and simulate it:

Wolfram Language code: vdist = MultinormalDistribution[{5, -4}, {{0.3, -0.1}, {-0.1, 0.2}}];
Wolfram Language code: Dimensions[vectors = RandomVariate[vdist, 100]]

VectorAround[vectors] estimates the mean and covariance matrices of the distribution:

Wolfram Language code: VectorAround[vectors]

MeanAround[scalars] describes the mean of the distribution and the covariance matrix associated with that mean:

Wolfram Language code: MeanAround[vectors]
Wolfram Research (2019), MeanAround, Wolfram Language function, https://reference.wolfram.com/language/ref/MeanAround.html.

Text

Wolfram Research (2019), MeanAround, Wolfram Language function, https://reference.wolfram.com/language/ref/MeanAround.html.

CMS

Wolfram Language. 2019. "MeanAround." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MeanAround.html.

APA

Wolfram Language. (2019). MeanAround. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MeanAround.html

BibTeX

@misc{reference.wolfram_2026_meanaround, author="Wolfram Research", title="{MeanAround}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/MeanAround.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_meanaround, organization={Wolfram Research}, title={MeanAround}, year={2019}, url={https://reference.wolfram.com/language/ref/MeanAround.html}, note=[Accessed: 12-June-2026]}