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Mean
✖
Mean

Updated in 14.1

✖
Mean[data]

gives the mean estimate of the elements in data.

✖
Mean[dist]

gives the mean of the distribution dist.

Details

  • Mean is also known as an expectation or average.
  • Mean is a location measure for data or distributions.
  • For VectorQ data , the mean estimate is given by .
  • For MatrixQ data, the mean estimate is computed for each column vector with Mean[{{x1,y1,…},{x2,y2,…},…}] equivalent to {Mean[{x1,x2,…}],Mean[{y1,y2,…}],…}. »
  • For ArrayQ data, the mean estimate is equivalent to ArrayReduce[Mean,data,1]. »
  • For WeightedData[{x1,x2,…},{w1,w2,…}], the mean estimate is given by . »
  • Mean handles both numerical and symbolic data.
  • The data can have the following additional forms and interpretations:
  • Associationthe values (the keys are ignored) »
    WeightedDataweighted mean, based on the underlying EmpiricalDistribution »
    EventDatabased on the underlying SurvivalDistribution »
    TimeSeries, TemporalData, …vector or array of values (the time stamps ignored) »
    Image,Image3DRGB channels values or grayscale intensity value »
    Audioamplitude values of all channels »
    DateObject,TimeObjectlist of dates or list of times »
  • For a list of dates , the mean is given by , which is date plus sum of durations .
  • For a univariate distribution dist, the mean is given by μ=Expectation[x,xdist]. »
  • For multivariate distribution dist, the mean is given by {μx ,μy,…}=Expectation[{x,y,…},{x,y,…}dist]. »
  • For a random process proc, the mean function can be computed for slice distribution at time t, SliceDistribution[proc,t], as μ[t]=Mean[SliceDistribution[proc,t]]. »

Examples

open allclose all

Basic Examples  (5)Summary of the most common use cases

Mean of numeric values:

In[1]:=1
✖
https://wolfram.com/xid/02cai-hrx
Out[1]=1

Mean of symbolic values:

In[1]:=1
✖
https://wolfram.com/xid/02cai-idj
Out[1]=1

Means of elements in each column:

In[1]:=1
✖
https://wolfram.com/xid/02cai-zf6
Out[1]=1

Mean of a list of dates:

In[1]:=1
✖
https://wolfram.com/xid/02cai-t7ech
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-ziof1v
Out[2]=2

Mean of a parametric distribution:

In[1]:=1
✖
https://wolfram.com/xid/02cai-1ikye
Out[1]=1

Scope  (22)Survey of the scope of standard use cases

Basic Uses  (6)

Exact input yields exact output:

In[1]:=1
✖
https://wolfram.com/xid/02cai-ug7y2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-bcry2t
Out[2]=2

Approximate input yields approximate output:

In[1]:=1
✖
https://wolfram.com/xid/02cai-ksx55
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-d02ofx
Out[2]=2

Find the mean of WeightedData:

In[1]:=1
✖
https://wolfram.com/xid/02cai-d0wc9z
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-f1vfw
In[3]:=3
✖
https://wolfram.com/xid/02cai-qyv0h
Out[3]=3

Find the mean of EventData:

In[1]:=1
✖
https://wolfram.com/xid/02cai-e67u14
In[2]:=2
✖
https://wolfram.com/xid/02cai-or2nrz
Out[2]=2

Find the mean of a TimeSeries:

In[1]:=1
✖
https://wolfram.com/xid/02cai-iaswcf
In[2]:=2
✖
https://wolfram.com/xid/02cai-ffhpdi
Out[2]=2

The mean depends only on the values:

In[3]:=3
✖
https://wolfram.com/xid/02cai-fy9fte
Out[3]=3

Compute a weighted mean:

In[4]:=4
✖
https://wolfram.com/xid/02cai-kfbxcw
Out[4]=4

Find the mean of data involving quantities:

In[1]:=1
✖
https://wolfram.com/xid/02cai-jopin9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-e8c21s
Out[2]=2

Array Data  (5)

Mean for a matrix gives columnwise means:

In[1]:=1
✖
https://wolfram.com/xid/02cai-ezu2uz
Out[1]=1

Mean for a arrays gives columnwise means at the first level:

In[2]:=2
✖
https://wolfram.com/xid/02cai-lw96ov
Out[2]=2

Works with large arrays:

In[1]:=1
✖
https://wolfram.com/xid/02cai-nknun
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-ma3v2m
Out[2]=2

When the input is an Association, Mean works on its values:

In[1]:=1
✖
https://wolfram.com/xid/02cai-cs7n5q
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-rvy4yi
Out[2]=2

SparseArray data can be used just like dense arrays:

In[1]:=1
✖
https://wolfram.com/xid/02cai-n691tv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-drrysl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-l4ct3
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02cai-d6csj0
Out[4]=4

Find mean of a QuantityArray:

In[1]:=1
✖
https://wolfram.com/xid/02cai-lgwnaj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-k03qc6
Out[2]=2

Image and Audio Data  (2)

Channel-wise mean value of an RGB image:

In[1]:=1
✖
https://wolfram.com/xid/02cai-hfby9q
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-phlz4o
Out[2]=2

Mean intensity value of a grayscale image:

In[3]:=3
✖
https://wolfram.com/xid/02cai-ue2gq5
Out[3]=3

On audio objects, Mean works channel-wise:

In[1]:=1
✖
https://wolfram.com/xid/02cai-nq1jnz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-mjmudf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-bs38vd
Out[3]=3

Date and Time  (4)

Compute mean of dates:

In[3]:=3
✖
https://wolfram.com/xid/02cai-b1smxx
In[5]:=5
✖
https://wolfram.com/xid/02cai-pa4nmn
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/02cai-uok1il
Out[6]=6

Compute the weighted mean of dates:

In[1]:=1
✖
https://wolfram.com/xid/02cai-c98kbd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-8c1had
In[3]:=3
✖
https://wolfram.com/xid/02cai-t71b2h
Out[3]=3

Compute the mean of dates given in different calendars:

In[1]:=1
✖
https://wolfram.com/xid/02cai-wbzcuv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-lyrjbs
Out[2]=2

The mean is given in one of the input calendars:

In[3]:=3
✖
https://wolfram.com/xid/02cai-9ius88
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02cai-qe5gbw
Out[4]=4

Compute the mean of times:

In[1]:=1
✖
https://wolfram.com/xid/02cai-et9bla
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-ztsexm
Out[2]=2

List of times with different time zone specifications:

In[3]:=3
✖
https://wolfram.com/xid/02cai-mrqghz
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02cai-1d7sk5
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/02cai-ldn4s4
Out[5]=5

Distributions and Processes  (5)

Find the mean for univariate distributions:

In[1]:=1
✖
https://wolfram.com/xid/02cai-rxz55
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-hbq28j
Out[2]=2

Multivariate distributions:

In[3]:=3
✖
https://wolfram.com/xid/02cai-ek075b
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02cai-lzwoz3
Out[4]=4

Mean for derived distributions:

In[1]:=1
✖
https://wolfram.com/xid/02cai-rgc72x
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-byqvvz
Out[2]=2

Data distribution:

In[3]:=3
✖
https://wolfram.com/xid/02cai-215ry
In[4]:=4
✖
https://wolfram.com/xid/02cai-fq5ptk
Out[4]=4

Mean for distributions with quantities:

In[1]:=1
✖
https://wolfram.com/xid/02cai-b43qn8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-fxupr5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-j6hwjw
Out[3]=3

Mean function for a continuous-time random and discrete-state process:

In[1]:=1
✖
https://wolfram.com/xid/02cai-c4ojmv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-g9pmgp
Out[2]=2

Find the mean of TemporalData at some time t=0.5:

In[1]:=1
✖
https://wolfram.com/xid/02cai-jfiydh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-cnazd
Out[2]=2

Find the mean function together with all the simulations:

In[3]:=3
✖
https://wolfram.com/xid/02cai-bdty7n
Out[3]=3

Applications  (11)Sample problems that can be solved with this function

Basic Applications  (5)

The mean represents the center of mass for a distribution:

In[1]:=1
✖
https://wolfram.com/xid/02cai-cdcslf
In[2]:=2
✖
https://wolfram.com/xid/02cai-qt88jt
Out[2]=2

The mean for distributions without a single mode:

In[1]:=1
✖
https://wolfram.com/xid/02cai-ce64dk
In[2]:=2
✖
https://wolfram.com/xid/02cai-foavzj
Out[2]=2

The mean for multivariate distributions:

In[1]:=1
✖
https://wolfram.com/xid/02cai-nh16ep
In[2]:=2
✖
https://wolfram.com/xid/02cai-db82mz
Out[2]=2

Mean values of cells in a sequence of steps of 2D cellular automaton evolution:

In[1]:=1
✖
https://wolfram.com/xid/02cai-f6i
Out[1]=1

Compute means for slices of a collection of paths of a random process:

In[1]:=1
✖
https://wolfram.com/xid/02cai-8se1zg

Choose a few slice times:

In[2]:=2
✖
https://wolfram.com/xid/02cai-52xxug
In[3]:=3
✖
https://wolfram.com/xid/02cai-iakfqb

Plot means over these paths:

In[4]:=4
✖
https://wolfram.com/xid/02cai-tvmkqe
Out[4]=4

Applications  (6)

Find the mean height for the children in a class:

In[1]:=1
✖
https://wolfram.com/xid/02cai-doufxt
In[2]:=2
✖
https://wolfram.com/xid/02cai-k8lwe
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-bbuq7u
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02cai-wtptgb
Out[4]=4

Find the mean height for the children in a class:

In[1]:=1
✖
https://wolfram.com/xid/02cai-cevfij
In[2]:=2
✖
https://wolfram.com/xid/02cai-fllmtw
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-celepo
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02cai-cny2bx
Out[4]=4

Find the mean strength for 480 samples of ceramic material:

In[1]:=1
✖
https://wolfram.com/xid/02cai-t28mn
In[2]:=2
✖
https://wolfram.com/xid/02cai-kqueul
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-d0uzg2
Out[3]=3

Plot a Histogram for the data with mean position highlighted:

In[4]:=4
✖
https://wolfram.com/xid/02cai-udd21q
In[5]:=5
✖
https://wolfram.com/xid/02cai-vomdk1
Out[5]=5

Compute the probability that the strength exceeds the mean:

In[6]:=6
✖
https://wolfram.com/xid/02cai-bs6d63
Out[6]=6

Compute the mean lifetime for a quantity subject to exponential decay with rate :

In[1]:=1
✖
https://wolfram.com/xid/02cai-ju4zmq
Out[1]=1

Smooth an irregularly spaced time series by computing a moving mean:

In[1]:=1
✖
https://wolfram.com/xid/02cai-tuglf3

A 90-day moving mean:

In[2]:=2
✖
https://wolfram.com/xid/02cai-d4ls8q
In[3]:=3
✖
https://wolfram.com/xid/02cai-fjo2gi
Out[3]=3

A vacuum system in a small electron accelerator contains 20 vacuum bulbs arranged in a circle. The vacuum system fails if at least 3 adjacent vacuum bulbs fail:

In[1]:=1
✖
https://wolfram.com/xid/02cai-ll1p6f
In[2]:=2
✖
https://wolfram.com/xid/02cai-yzvkim
In[3]:=3
✖
https://wolfram.com/xid/02cai-ig5yz3
In[4]:=4
✖
https://wolfram.com/xid/02cai-m9nudo

Plot the survival function:

In[5]:=5
✖
https://wolfram.com/xid/02cai-fmqdbm
Out[5]=5

Compute the mean time to failure:

In[6]:=6
✖
https://wolfram.com/xid/02cai-xext1o
Out[6]=6

Properties & Relations  (17)Properties of the function, and connections to other functions

Mean is Total divided by Length:

In[1]:=1
✖
https://wolfram.com/xid/02cai-kxw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-lqd
Out[2]=2

Mean is equivalent to a 1‐norm divided by Length for positive values:

In[1]:=1
✖
https://wolfram.com/xid/02cai-i96jjd
In[2]:=2
✖
https://wolfram.com/xid/02cai-1z6z5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-jsdb4x
Out[3]=3

Mean of WeightedData is equivalent to the mean of the EmpiricalDistribution of the data:

In[1]:=1
✖
https://wolfram.com/xid/02cai-h4u0sw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-7uxvba
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-ic85bc
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02cai-s3rbyq
Out[4]=4

Mean of EventData is equivalent to the mean of the SurvivalDistribution of the data:

In[1]:=1
✖
https://wolfram.com/xid/02cai-1hdnl4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-q4w4mm
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-gbz2mn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02cai-7uq9qi
Out[4]=4

For nearly symmetric samples, Mean and Median are nearly the same:

In[1]:=1
✖
https://wolfram.com/xid/02cai-t9n
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-cfc
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-erqvhl
In[4]:=4
✖
https://wolfram.com/xid/02cai-eyllwj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/02cai-h1qxa
Out[5]=5

The Mean of absolute deviations from the Mean is MeanDeviation:

In[1]:=1
✖
https://wolfram.com/xid/02cai-793v1
In[2]:=2
✖
https://wolfram.com/xid/02cai-bhsxlb
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-b9x4up
Out[3]=3

Mean is logarithmically related to GeometricMean for positive values:

In[1]:=1
✖
https://wolfram.com/xid/02cai-ebg6dq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-24ovr
Out[2]=2

Mean is the inverse of HarmonicMean of the inverse of the data:

In[1]:=1
✖
https://wolfram.com/xid/02cai-bvbvtz
In[2]:=2
✖
https://wolfram.com/xid/02cai-2fgcb
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-f73eok
Out[3]=3

The square root of Mean of the data squared is RootMeanSquare:

In[1]:=1
✖
https://wolfram.com/xid/02cai-hqtrtd
In[2]:=2
✖
https://wolfram.com/xid/02cai-cyyyvo
Out[2]=2

The n^(th) CentralMoment is the Mean of deviations raised to the n^(th) power:

In[1]:=1
✖
https://wolfram.com/xid/02cai-g8e1st
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-6s2s3
Out[2]=2

Variance is a scaled Mean of squared deviations from the Mean:

In[1]:=1
✖
https://wolfram.com/xid/02cai-brtwin
In[2]:=2
✖
https://wolfram.com/xid/02cai-hmk853
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/02cai-cibwnc
Out[3]=3

Expectation for a list is a Mean:

In[1]:=1
✖
https://wolfram.com/xid/02cai-ba2oj3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-hemld8
Out[2]=2

MovingAverage is a sequence of means:

In[1]:=1
✖
https://wolfram.com/xid/02cai-bxc2b1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-i0fg4m
Out[2]=2

A 0% TrimmedMean is the same as Mean:

In[1]:=1
✖
https://wolfram.com/xid/02cai-k3hcsh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-fc96q6
Out[2]=2

The Expectation of a random variable in a distribution is the Mean:

In[1]:=1
✖
https://wolfram.com/xid/02cai-cl5rpb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cai-ekl2fe
Out[2]=2

LocationTest tests whether the mean is close to 0:

In[1]:=1
✖
https://wolfram.com/xid/02cai-w9gyw
In[2]:=2
✖
https://wolfram.com/xid/02cai-eafvmq
Out[2]=2

The probability () value:

In[3]:=3
✖
https://wolfram.com/xid/02cai-iosml1
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02cai-ca4gle
Out[4]=4

LocationEquivalenceTest tests for equivalence of means in two or more datasets:

In[1]:=1
✖
https://wolfram.com/xid/02cai-ch52p0
In[2]:=2
✖
https://wolfram.com/xid/02cai-cytau2
Out[2]=2

The probability () value:

In[3]:=3
✖
https://wolfram.com/xid/02cai-h6zobm
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/02cai-c79z3d
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

Outliers can have a disproportionate effect on Mean:

In[1]:=1
✖
https://wolfram.com/xid/02cai-vlx43
Out[1]=1

Use TrimmedMean to ignore a fraction of the smallest and largest elements:

In[2]:=2
✖
https://wolfram.com/xid/02cai-blvoit
Out[2]=2

Use Median as something much less sensitive to outliers:

In[3]:=3
✖
https://wolfram.com/xid/02cai-fueg3r
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

The distribution of Mean estimates for 10, 100, and 300 samples:

(2) In[1]:=2
✖
https://wolfram.com/xid/02cai-8raem
(2) Out[1]=2
Wolfram Research (2003), Mean, Wolfram Language function, https://reference.wolfram.com/language/ref/Mean.html (updated 2024).
✖
Wolfram Research (2003), Mean, Wolfram Language function, https://reference.wolfram.com/language/ref/Mean.html (updated 2024).

Text

Wolfram Research (2003), Mean, Wolfram Language function, https://reference.wolfram.com/language/ref/Mean.html (updated 2024).

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Wolfram Research (2003), Mean, Wolfram Language function, https://reference.wolfram.com/language/ref/Mean.html (updated 2024).

CMS

Wolfram Language. 2003. "Mean." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Mean.html.

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Wolfram Language. 2003. "Mean." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Mean.html.

APA

Wolfram Language. (2003). Mean. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Mean.html

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Wolfram Language. (2003). Mean. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Mean.html

BibTeX

@misc{reference.wolfram_2025_mean, author="Wolfram Research", title="{Mean}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Mean.html}", note=[Accessed: 29-May-2025 ]}

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@misc{reference.wolfram_2025_mean, author="Wolfram Research", title="{Mean}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Mean.html}", note=[Accessed: 29-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_mean, organization={Wolfram Research}, title={Mean}, year={2024}, url={https://reference.wolfram.com/language/ref/Mean.html}, note=[Accessed: 29-May-2025 ]}

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@online{reference.wolfram_2025_mean, organization={Wolfram Research}, title={Mean}, year={2024}, url={https://reference.wolfram.com/language/ref/Mean.html}, note=[Accessed: 29-May-2025 ]}