# MeanDeviation

MeanDeviation[data]

gives the mean absolute deviation from the mean of the elements in data.

# Details

• MeanDeviation is also known as MAD.
• MeanDeviation is a measure of dispersion.
• For VectorQ data , the mean deviation is given by , where is the mean of data.
• For MatrixQ data, the mean deviation is computed for each column vector with MeanDeviation[{{x1,y1,},{x2,y2,},}] equivalent to {MeanDeviation[{x1,x2,}],MeanDeviation[{y1,y2,}],}. »
• For ArrayQ data, the mean deviation is equivalent to ArrayReduce[MeanDeviation,data,1]. »
• MeanDeviation handles both numerical and symbolic data.
• The data can have the following additional forms and interpretations:
•  Association the values (the keys are ignored) » SparseArray as an array, equivalent to Normal[data] » QuantityArray quantities as an array » WeightedData based on the underlying EmpiricalDistribution » EventData based on the underlying SurvivalDistribution » TimeSeries, TemporalData, … vector or array of values (the time stamps ignored) » Image,Image3D RGB channel's values or grayscale intensity value » Audio amplitude values of all channels »

# Examples

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## Basic Examples(3)

MeanDeviation of a list of numbers:

MeanDeviation of symbolic data:

MeanDeviation of the columns of a matrix:

## Scope(13)

### Basic Uses(6)

Exact input yields exact output:

Approximate input yields approximate output:

Find the mean deviation of WeightedData:

Find the mean deviation of EventData:

Find the mean deviation for TimeSeries:

The mean deviation depends only on the values:

Find the mean deviation of data involving quantities:

### Array Data(5)

MeanDeviation for a matrix works columnwise:

MeanDeviation for a tensor works across the first index: »

Works with large arrays:

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

SparseArray data can be used just like dense arrays:

Find mean deviation of a QuantityArray:

### Image and Audio Data(2)

Channelwise mean deviation value of an RGB image:

Mean deviation value of a grayscale image:

On audio objects, MeanDeviation works channelwise:

## Applications(3)

Identify periods of high volatility in stock data using a five-year moving mean deviation:

Compute mean deviations for slices of a collection of paths of a random process:

Choose a few slice times:

Plot mean deviations over these paths:

Find the mean deviation of the heights for the children in a class:

Plot the mean deviation respective of the mean:

## Properties & Relations(4)

MeanDeviation is the Mean of absolute deviations from the Mean:

MeanDeviation is equivalent to the 1norm of the deviations divided by the Length:

For large uniform datasets, MeanDeviation and MedianDeviation are nearly the same:

MeanDeviation as a scaled ManhattanDistance from the Mean:

## Neat Examples(1)

Ratio of MeanDeviation to MedianDeviation for increasing sample size:

Wolfram Research (2007), MeanDeviation, Wolfram Language function, https://reference.wolfram.com/language/ref/MeanDeviation.html (updated 2023).

#### Text

Wolfram Research (2007), MeanDeviation, Wolfram Language function, https://reference.wolfram.com/language/ref/MeanDeviation.html (updated 2023).

#### CMS

Wolfram Language. 2007. "MeanDeviation." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/MeanDeviation.html.

#### APA

Wolfram Language. (2007). MeanDeviation. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MeanDeviation.html

#### BibTeX

@misc{reference.wolfram_2024_meandeviation, author="Wolfram Research", title="{MeanDeviation}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/MeanDeviation.html}", note=[Accessed: 24-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_meandeviation, organization={Wolfram Research}, title={MeanDeviation}, year={2023}, url={https://reference.wolfram.com/language/ref/MeanDeviation.html}, note=[Accessed: 24-June-2024 ]}