# Mean 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 a vector data , the mean estimate is given by .
• • For matrix data, mean estimate is computed for each column vector with Mean[{{x1,y1,},{x2,y2,},}] equivalent to {Mean[{x1,x2,}],Mean[{y1,y2,}],}. »
• • For array data, mean estimate is equivalent to ArrayReduce[Mean,data,1]. »
•                         ↓ ↓ ↓ ↓ ↓     • For WeightedData[{x1,x2,},{w1,w2,}], mean estimate is given by . »
• Mean 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 weighted mean, 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 »
• 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]. »
•  symmetric distribution skewed distribution  • 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

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## Basic Examples(4)

Mean of numeric values:

Mean of symbolic values:

Means of elements in each column:

Mean of a parametric distribution:

## Scope(18)

### Basic Uses(6)

Exact input yields exact output:

Approximate input yields approximate output:

Find the mean of WeightedData:

Find the mean of EventData:

Find the mean of a TimeSeries:

The mean depends only on the values:

Compute a weighted mean:

Find the mean of data involving quantities:

### Array Data(5)

Mean for a matrix gives columnwise means:

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

Works with large arrays:

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

SparseArray data can be used just like dense arrays:

Find mean of a QuantityArray:

### Image and Audio Data(2)

Channel-wise mean value of an RGB image:

Mean intensity value of a grayscale image:

On audio objects, Mean works channel-wise:

### Distributions and Processes(5)

Find the mean for univariate distributions:

Multivariate distributions:

Mean for derived distributions:

Data distribution:

Mean for distributions with quantities:

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

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

Find the mean function together with all the simulations:

## Applications(11)

### Basic Applications(5)

The mean represents the center of mass for a distribution:

The mean for distributions without a single mode:

The mean for multivariate distributions:

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

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

Choose a few slice times:

Plot means over these paths:

### Applications(6)

Find the mean height for the children in a class:

Find the mean height for the children in a class:

Find the mean strength for 480 samples of ceramic material:

Plot a Histogram for the data with mean position highlighted:

Compute the probability that the strength exceeds the mean:

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

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

A 90-day moving mean:

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:

Plot the survival function:

Compute the mean time to failure:

## Properties & Relations(17)

Mean is Total divided by Length:

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

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

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

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

The Mean of absolute deviations from the Mean is MeanDeviation:

Mean is logarithmically related to GeometricMean for positive values:

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

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

The n CentralMoment is the Mean of deviations raised to the n power:

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

Expectation for a list is a Mean:

MovingAverage is a sequence of means:

A 0% TrimmedMean is the same as Mean:

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

LocationTest tests whether the mean is close to 0:

The probability ( ) value:

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

The probability ( ) value:

## Possible Issues(1)

Outliers can have a disproportionate effect on Mean:

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

Use Median as something much less sensitive to outliers:

## Neat Examples(1)

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