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WeightedData
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WeightedData

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WeightedData[{x1,x2,…},{w1,w2,…}]

represents observations xi with weights wi.

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WeightedData[{x1,x2,…},fn]

represents observations xi with weighting function fn.

Details

  • WeightedData augments data with weights for each data point.
  • The data {x1,x2,…} and weights {w1,w2,…} should be lists of equal length.
  • The weight function fn is applied to the list {x1,x2,…} and should return an explicit list of weights {w1,w2,…}.
  • WeightedData can be used in statistics functions including:
  • Mean,Variance,…descriptive statistics functions
    EmpiricalDistribution,…nonparametric distribution estimation
    EstimatedDistribution,…parametric distribution estimation
  • WeightedData[{x1,x2,…}] gives data with equal weights.
  • Properties of WeightedData can be obtained by specifying WeightedData[…]["property"].
  • A list of available properties can be obtained using WeightedData[…]["Properties"].
  • WeightedData has the following properties:
  • "EmpiricalPDF"data values and estimated weights
    "InputData"unweighted input data values
    "MetaInformation"a list containing meta-information rules
    "Weights"a list containing the data weights

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Create data with weights:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-f1vfw
In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-i9ljxw
Out[3]=3

Compute a weighted Mean and StandardDeviation:

In[4]:=4
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https://wolfram.com/xid/0h2rom4k7m-qyv0h
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0h2rom4k7m-bpk5jo
Out[5]=5

Scope  (10)Survey of the scope of standard use cases

Create weighted univariate data:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-df2a5v
In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-co21so

Some weighted descriptive statistics:

In[4]:=4
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https://wolfram.com/xid/0h2rom4k7m-ckyp8g
Out[4]=4

Add weights to a set of multivariate values:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-gc3b4e
In[2]:=2
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https://wolfram.com/xid/0h2rom4k7m-cknl3y
Out[2]=2

A set of weighted multivariate descriptive statistics:

In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-e9d6ls
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0h2rom4k7m-i0doty
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0h2rom4k7m-eect9s
Out[5]=5

Use a pure function to create weighted data values:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-e8pspd
In[2]:=2
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https://wolfram.com/xid/0h2rom4k7m-hk9hx2

Weighted means and variances:

In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-c6zxo0
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0h2rom4k7m-b4zlp1
Out[4]=4

Visualize the impact of the various weighting schemes:

In[5]:=5
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https://wolfram.com/xid/0h2rom4k7m-4zm2o
Out[5]=5

Fit nonparametric distributions to weighted data:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-f7es
In[2]:=2
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https://wolfram.com/xid/0h2rom4k7m-iwsv9
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-ihmsy
In[4]:=4
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https://wolfram.com/xid/0h2rom4k7m-fzof5e
Out[4]=4

Fit parametric distributions to weighted data:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-epb0fr
In[2]:=2
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https://wolfram.com/xid/0h2rom4k7m-nnu1d
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-jumbt
Out[3]=3

Compare the estimated and empirical distributions:

In[4]:=4
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https://wolfram.com/xid/0h2rom4k7m-2wxss
In[5]:=5
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https://wolfram.com/xid/0h2rom4k7m-ge33gc
Out[5]=5

Extract the input data from a WeightedData object:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-blw7tf
In[2]:=2
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https://wolfram.com/xid/0h2rom4k7m-bq197b

Compare the distributions of the unweighted and weighted data:

In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-03yqi
In[4]:=4
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https://wolfram.com/xid/0h2rom4k7m-fjjo65
In[5]:=5
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https://wolfram.com/xid/0h2rom4k7m-lsgjf
Out[5]=5

Obtain the weights from a WeightedData object:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-eazgvh
In[2]:=2
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https://wolfram.com/xid/0h2rom4k7m-i0ejbf

Visually inspect the effect of the weights on the data values:

In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-zlfu3
Out[3]=3

Compute a weighted mean from the empirical PDF:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-hqafaf
In[2]:=2
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https://wolfram.com/xid/0h2rom4k7m-p2vfv8
In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-ikmgyz
Out[3]=3

The weighted mean can be computed directly using Mean:

In[4]:=4
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https://wolfram.com/xid/0h2rom4k7m-b5jug9
Out[4]=4

Find the weighted average of an irregularly sampled TimeSeries:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-i8vsw
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0h2rom4k7m-u3uc3
Out[2]=2

Compare with the average of values:

In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-q2rkh
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0h2rom4k7m-qx5fif
Out[4]=4

Create weighted data involving quantities:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-dp8eet
In[2]:=2
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https://wolfram.com/xid/0h2rom4k7m-eqpfu5

Some weighted descriptive statistics:

In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-ql964x

Applications  (2)Sample problems that can be solved with this function

Create a weighted histogram:

In[16]:=16
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https://wolfram.com/xid/0h2rom4k7m-hvndff
In[17]:=17
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https://wolfram.com/xid/0h2rom4k7m-blozch
In[18]:=18
✖
https://wolfram.com/xid/0h2rom4k7m-hnis3j
In[19]:=19
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https://wolfram.com/xid/0h2rom4k7m-x67ciz
Out[21]=21

Estimate confidence interval for maximum likelihood estimates of distribution parameters:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-fr0igp
In[2]:=2
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https://wolfram.com/xid/0h2rom4k7m-c0fwx
Out[2]=2

Apply fractional random weight bootstrap to estimate confidence interval, by repeating weighted estimation with weights sampled from a DirichletDistribution with unit parameters:

In[3]:=3
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https://wolfram.com/xid/0h2rom4k7m-dyp41z
In[4]:=4
✖
https://wolfram.com/xid/0h2rom4k7m-lpayg0

Generate one thousand estimates of the distribution parameters:

In[5]:=5
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https://wolfram.com/xid/0h2rom4k7m-kdcgm

Visualize bootstrap estimates:

In[6]:=6
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https://wolfram.com/xid/0h2rom4k7m-fjc8se
Out[6]=6

Fit joint Gaussian distribution to the bootstrapped parameters:

In[7]:=7
✖
https://wolfram.com/xid/0h2rom4k7m-7swc
Out[7]=7

Properties & Relations  (2)Properties of the function, and connections to other functions

Descriptive statistics are based on the underlying EmpiricalDistribution:

In[1]:=1
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https://wolfram.com/xid/0h2rom4k7m-dqtueh
In[2]:=2
✖
https://wolfram.com/xid/0h2rom4k7m-flzoxq
In[3]:=3
✖
https://wolfram.com/xid/0h2rom4k7m-fi9wvz
Out[3]=3

Sample estimates are given when they differ from population estimates:

In[4]:=4
✖
https://wolfram.com/xid/0h2rom4k7m-zbp4w
Out[4]=4

WeightedData works for TimeSeries objects:

In[1]:=1
✖
https://wolfram.com/xid/0h2rom4k7m-c80k57
Out[1]=1

For , the weights are proportional to :

In[2]:=2
✖
https://wolfram.com/xid/0h2rom4k7m-brb9vm
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0h2rom4k7m-k632yu
Out[3]=3

Compare with bin‐sampled time average:

In[4]:=4
✖
https://wolfram.com/xid/0h2rom4k7m-lujmoy
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0h2rom4k7m-ewirn0
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0h2rom4k7m-bz952v

Compute integration limits:

In[7]:=7
✖
https://wolfram.com/xid/0h2rom4k7m-cadij9
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0h2rom4k7m-ea6e5
Out[8]=8
Wolfram Research (2012), WeightedData, Wolfram Language function, https://reference.wolfram.com/language/ref/WeightedData.html.
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Wolfram Research (2012), WeightedData, Wolfram Language function, https://reference.wolfram.com/language/ref/WeightedData.html.

Text

Wolfram Research (2012), WeightedData, Wolfram Language function, https://reference.wolfram.com/language/ref/WeightedData.html.

✖
Wolfram Research (2012), WeightedData, Wolfram Language function, https://reference.wolfram.com/language/ref/WeightedData.html.

CMS

Wolfram Language. 2012. "WeightedData." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WeightedData.html.

✖
Wolfram Language. 2012. "WeightedData." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WeightedData.html.

APA

Wolfram Language. (2012). WeightedData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeightedData.html

✖
Wolfram Language. (2012). WeightedData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeightedData.html

BibTeX

@misc{reference.wolfram_2025_weighteddata, author="Wolfram Research", title="{WeightedData}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/WeightedData.html}", note=[Accessed: 07-March-2025 ]}

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@misc{reference.wolfram_2025_weighteddata, author="Wolfram Research", title="{WeightedData}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/WeightedData.html}", note=[Accessed: 07-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_weighteddata, organization={Wolfram Research}, title={WeightedData}, year={2012}, url={https://reference.wolfram.com/language/ref/WeightedData.html}, note=[Accessed: 07-March-2025 ]}

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@online{reference.wolfram_2025_weighteddata, organization={Wolfram Research}, title={WeightedData}, year={2012}, url={https://reference.wolfram.com/language/ref/WeightedData.html}, note=[Accessed: 07-March-2025 ]}