# Mean

Mean[list]

gives the statistical mean of the elements in list.

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.
• Mean[list] is equivalent to Total[list]/Length[list].
• Mean handles both numerical and symbolic data.
• Mean[{{x1,y1,},{x2,y2,},}] gives {Mean[{x1,x2,}],Mean[{y1,y2,}],}.
• Mean[dist] is equivalent to Expectation[x,xdist].

# 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(14)

### Data(9)

Exact input yields exact output:

Approximate input yields approximate output:

Mean for a matrix gives columnwise means:

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

Works with large arrays:

SparseArray data can be used just like dense arrays:

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:

### 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)

The mean represents the center of mass for a distribution:

The mean for distributions without a single mode:

The mean for multivariate distributions:

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:

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:

## Properties & Relations(15)

Mean is Total divided by Length:

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

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: