# Median Median[data]

gives the median estimate of the elements in data.

Median[dist]

gives the median of the distribution dist.

# Details    • Median is a robust location estimator, which means it not very sensitive to outliers.
• For a vector data , the median can be thought of as the "middle value".
• Formally, when data is sorted as , the median is given by center element if is odd and the mean of the two center elements if is even.
• • Median[data] is equivalent to Quantile[data,1/2,{{1/2,0},{0,1}}].
• For matrix data, the median is computed for each column vector with Median[{{x1,y1,},{x2,y2,},}] equivalent to {Median[{x1,x2,}],Median[{y1,y2,}],}. »
• • For array data, median is equivalent to ArrayReduce[Median,data,1]. »
•                         ↓ ↓ ↓ ↓ ↓     • 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 channels values or grayscale intensity value » Audio amplitude values of all channels »
• Median[dist] is the minimum of the set of number(s) m such that Probability[xm,xdist]1/2 and Probability[xm,xdist]1/2. »
• For a continuous distribution dist, the median can be defined using the cumulative distribution function: .
• • Median[dist] is equivalent to Quantile[dist,1/2].
• For a random process proc, the median function can be computed for slice distribution at time t, SliceDistribution[proc,t], as Median[SliceDistribution[proc,t]]. »
• # Examples

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## Basic Examples(3)

Find the middle value in the list:

Average the two middle values:

Median of a parametric distribution:

## Scope(19)

### Basic Uses(8)

Exact input yields exact output:

Approximate input yields approximate output:

Find the median of WeightedData:

Find the median of EventData:

Find the median of TemporalData:

Find the median of a TimeSeries:

The median depends only on the values:

Find a three-element moving median:

Find the median of data involving quantities:

### Array Data(5)

Median for a matrix gives columnwise medians:

Median for a tensor gives columnwise medians at the first level:

Works with large arrays:

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

SparseArray data can be used just like dense arrays:

Find median of a QuantityArray:

### Image and Audio Data(2)

Channel-wise median value of an RGB image:

Median intensity value of a grayscale image:

Median amplitude of all amplitude values of all channels:

### Distributions and Processes(4)

Find the median for a parametric distribution:

Median for a derived distribution:

Data distribution:

Median for distributions with quantities:

Median function for a time slice of a random process:

## Applications(7)

The median represents the center of a distribution:

The median for a distribution without a single mode:

Find the median length, in miles, for 141 major rivers in North America:

Plot a Histogram for the data:

Probability that the length exceeds 90% of the median:

Smooth an irregularly spaced time series using a moving median:

A 90-day moving median:

Obtain a robust estimate of location when outliers are present:

Extreme values have a large influence on the Mean:

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

Choose a few slice times:

Plot medians over these paths:

Find the median height for the children in a class:

## Properties & Relations(7)

Median is equivalent to a parametrized Quantile:

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

For univariate data, Median coincides with SpatialMedian:

The Median of absolute deviations from the Median is MedianDeviation:

MovingMedian is a sequence of medians:

For any distribution, there is InverseCDF[dist,1/2]=Median[dist]:

Similarly for InverseSurvivalFunction:

For a continuous distribution, there is CDF[dist,Median[dist]]=1/2:

Similarly for SurvivalFunction:

For discrete distributions, the identity does not hold:

## Possible Issues(2)

Median requires numeric values: Median of data computed via Quantile does not always agree with Median:

Calculate median directly:

Specify linear interpolation parameters in Quantile:

## Neat Examples(1)

The distribution of Median estimates for 20, 100, and 300 samples: