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Overview of the Statistics Package

  

The Statistics package is a collection of tools for mathematical statistics and data analysis. The package supports a wide range of common statistical tasks such as quantitative and graphical data analysis, simulation, and curve fitting. Also included is the Data Analysis Assistant, a graphical interface to the data analysis tools in the Statistics package. It can be accessed from the Tools>Assistants menu or using the InteractiveDataAnalysis command.

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In addition to standard data analysis tools the Statistics package provides a wide range of symbolic and numeric tools for computing with random variables. The package supports over 35 major probability distributions and provides facilities for defining new distributions.

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Much of the functionality in the Statistics package is accessible through context menus. Context-sensitive functionality is available when selecting any data container (such as a Vector, list, or Array), known probability distributions (such as Normal(1,2)), or random variables.

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Some related functionality regarding time series is available through the TimeSeriesAnalysis package.

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For additional examples detailing the uses of the Statistics package, see the following example worksheets.

Data Smoothing

Estimation

Hypothesis Testing

Probability Distributions

Robust Statistics

Statistics with DataFrames

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Below is the list of primary topics. See also Statistics[Commands] for an alphabetical list of Statistics commands.

 

Inventory of Probability Distributions

Descriptive Statistics, Data Summary and Tabulation

Probability Calculations, Random Variables

Visualization

Simulation

Regression

Estimation

Data Manipulation

Data Smoothing

Hypothesis Testing and Inference

Inventory of Probability Distributions

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Over 35 continuous and discrete probability distributions as well as tools for creating new distributions. Here is the list of relevant commands.

  

 

Bernoulli

Bernoulli distribution

Beta

beta distribution

Binomial

binomial distribution

Cauchy

Cauchy distribution

ChiSquare

chi-square distribution

DiscreteUniform

discrete uniform distribution

DiscreteValueMap

create a noninteger discrete distribution

Distribution

create new distribution

EmpiricalDistribution

empirical distribution

Erlang

Erlang distribution

Error

error (exponential power) distribution

Exponential

exponential distribution

FRatio

Fisher f-distribution

Gamma

gamma distribution

Geometric

geometric distribution

Gumbel

Gumbel distribution

Hypergeometric

hypergeometric distribution

InverseGaussian

inverse Gaussian (Wald) distribution

Laplace

Laplace distribution

Logistic

logistic distribution

LogNormal

log normal distribution

Maxwell

Maxwell distribution

Moyal

Moyal distribution

NegativeBinomial

negative binomial (Pascal) distribution

NonCentralBeta

noncentral beta distribution

NonCentralChiSquare

noncentral chi-square distribution

NonCentralFRatio

noncentral f-distribution

NonCentralStudentT

noncentral t-distribution

Normal

normal (Gaussian) distribution

Pareto

Pareto distribution

Poisson

Poisson distribution

Power

power distribution

ProbabilityTable

probability table

Rayleigh

Rayleigh distribution

StudentT

Student-t distribution

Triangular

triangular distribution

Uniform

uniform (rectangular) distribution

VonMises

von Mises distribution

Weibull

Weibull distribution

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More information is available in the Statistics[Distributions] help page.

Descriptive Statistics, Data Summary and Tabulation

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A wide range of functions for computing descriptive statistics. This includes location, dispersion and shape statistics, moments and cumulants, as well as several data summary and tabulation commands. Here is the list of available commands.

  

 

AbsoluteDeviation

compute the average absolute deviation

AutoCorrelation

autocorrelations

CentralMoment

central moments

Correlation

correlation/correlation matrix

Covariance

covariance/covariance matrix

CrossCorrelation

cross-correlations

Cumulant

cumulants

DataSummary

seven summary statistics

Decile

deciles

ExpectedValue

compute expected values

FivePointSummary

five-point summary

FrequencyTable

frequency table

GeometricMean

geometric mean

HarmonicMean

harmonic mean

HodgesLehmann

Hodges-Lehmann statistic

InterquartileRange

interquartile range

Kurtosis

kurtosis

MakeProcedure

generate a procedure for calculating statistical quantities

Mean

arithmetic mean

MeanDeviation

average absolute deviation from the mean

Median

median

MedianDeviation

compute the median absolute deviation

Mode

mode

Moment

moments

OrderStatistic

order statistics

PCA

principal component analysis

Percentile

percentiles

PrincipalComponentAnalysis

principal component analysis

QuadraticMean

quadratic mean

Quantile

quantiles

Quartile

quartiles

Range

range

RousseeuwCrouxQn

Rousseeuw and Croux' Qn

RousseeuwCrouxSn

Rousseeuw and Croux' Sn

Skewness

skewness

StandardDeviation

standard deviation

StandardError

standard error of the sampling distribution

StandardizedMoment

standardized moments

TrimmedMean

trimmed mean

Variance

variance

Variation

coefficient of variation

WinsorizedMean

winsorized mean

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More information is available in the Statistics[DescriptiveStatistics] help page.

Probability Calculations, Random Variables

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Tools for creating and manipulating random variables as well as functions for computing their densities, moments, generating functions and other quantities. Here is the list of available commands.

  

 

AbsoluteDeviation

compute the average absolute deviation

CDF

cumulative distribution function

CentralMoment

central moments

CGF

cumulant generating function

CharacteristicFunction

characteristic function

Cumulant

cumulants

CumulantGeneratingFunction

cumulant generating function

CumulativeDistributionFunction

cumulative distribution function

Decile

deciles

ExpectedValue

compute expected values

FailureRate

hazard (failure) rate

GeometricMean

geometric mean

HarmonicMean

harmonic mean

HazardRate

hazard (failure) rate

HodgesLehmann

Hodges-Lehmann statistic

InterquartileRange

interquartile range

InverseSurvivalFunction

inverse survival function

Kurtosis

kurtosis

MakeProcedure

generate a procedure for calculating statistical quantities

Mean

arithmetic mean

MeanDeviation

average absolute deviation from the mean

Median

median

MedianDeviation

compute the median absolute deviation

MGF

moment generating function

MillsRatio

Mills ratio

Mode

mode

Moment

moments

MomentGeneratingFunction

moment generating function

OrderStatistic

order statistics

PDF

probability density function

Percentile

percentiles

Probability

compute the probability of an event

ProbabilityDensityFunction

probability density function

ProbabilityFunction

probability function

QuadraticMean

quadratic mean

Quantile

quantiles

Quartile

quartiles

RandomVariable

create new random variable

RousseeuwCrouxQn

Rousseeuw and Croux' Qn

RousseeuwCrouxSn

Rousseeuw and Croux' Sn

Skewness

skewness

StandardDeviation

standard deviation

StandardError

standard error of the sampling distribution

StandardizedMoment

standardized moments

Support

support set of a random variable

SurvivalFunction

survival function

Variance

variance

Variation

coefficient of variation

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More information is available in the Statistics[RandomVariables] help page.

Visualization

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Various statistical plots such as box plots, bar charts, histograms, probability plots, scatterplots, etc. Here is the list of available commands.

  

 

AgglomeratedPlot

generate agglomerated plots

AreaChart

create area charts from data

BarChart

create bar charts from data

Biplot

generate biplots

BoxPlot

create box plots from data

BubblePlot

generate bubble plots

ColumnGraph

create column graphs from data

CumulativeSumChart

generate cumulative sum charts

DensityPlot

plot the density of a random variable

ErrorPlot

generate error plots

FrequencyPlot

generate frequency plots

GridPlot

generate a grid of plots

HeatMap

generate heat maps

Histogram

generate histograms

KernelDensityPlot

plot the kernel density estimate of a data set

LineChart

generate line charts

NormalPlot

generate normal plots

PieChart

generate pie charts

PointPlot

generate point plots

ProbabilityPlot

generate probability plots

ProfileLikelihood

plot a profile of the likelihood function

ProfileLogLikelihood

plot a profile of the log likelihood function

QuantilePlot

generate quantile-quantile plots

ScatterPlot

generate scatter plots

ScatterPlot3D

generate 3D scatter plots

ScreePlot

generate scree plots for variance

SunflowerPlot

generate sunflower plots

SurfacePlot

generate surface plots

SymmetryPlot

generate symmetry plots

TreeMap

generate tree maps

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More information is available in the Statistics[Visualization] help page.

Simulation

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Optimized algorithms for simulating from all supported distributions as well as tools for creating custom random number generators, parametric and non-parametric bootstrap. Here is the list of available commands.

  

 

Bootstrap

compute bootstrap statistics

KernelDensitySample

sample a kernel density estimate

Sample

generate random sample

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More information is available in the Statistics[Simulation] help page.

Regression

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Tools for fitting linear and nonlinear models to data points and performing regression analysis. Here is the list of available commands.

  

 

ExponentialFit

fit an exponential function to data

Fit

fit a model function to data

LinearFit

fit a linear model function to data

LogarithmicFit

fit a logarithmic function to data

Lowess

produce lowess smoothed functions

NonlinearFit

fit a nonlinear model function to data

OneWayANOVA

generate a one-way ANOVA table

PolynomialFit

fit a polynomial to data

PowerFit

fit a power function to data

PredictiveLeastSquares

fit a predictive linear model function to data

RepeatedMedianEstimator

robust linear regression

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More information is available in the Statistics[Regression] help page.

Estimation

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Tools for manipulating likelihood functions, maximum likelihood estimation, kernel density estimation, bootstrap. Here is the list of available commands.

  

 

FisherInformation

Fisher information

Information

statistical information

KernelDensity

estimate the probability density of a data set

Likelihood

likelihood function

LikelihoodRatioStatistic

compute the likelihood ratio statistic

LogLikelihood

log likelihood function

MaximumLikelihoodEstimate

compute the maximum likelihood estimate

Score

statistical score

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More information is available in the Statistics[Estimation] help page.

Data Manipulation

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Tools for manipulating statistical data. Here is the list of available commands.

  

 

Count

compute number/total weight of observations

CountMissing

compute number/total weight of missing values

CumulativeProduct

compute cumulative products

CumulativeSum

compute cumulative sums

EvaluateToFloat

evaluate data using floating-point arithmetic

Excise

remove data items based on density

Join

join data samples

OrderByRank

order data items according to their ranks

Rank

rank data items according to their numeric values

Remove

remove data items satisfying a condition

RemoveInRange

remove data items which belong to the given range

RemoveNonNumeric

remove non-numeric values

Scale

center and/or scale a set of data

Select

select data items satisfying a condition

SelectInRange

select data items which belong to the given range

SelectNonNumeric

select non-numeric values

Shuffle

apply random permutation to a data sample

Sort

sort numeric data

SplitByColumn

split matrix data into submatrices

Tally

compute data frequencies

TallyInto

compute cumulative data frequencies

Trim

trim data set

Winsorize

winsorize data set

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More information is available in the Statistics[DataManipulation] help page.

Data Smoothing

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Data smoothing functions including moving averages, exponential smoothing, linear filters, etc. Here is the list of available commands.

  

 

ExponentialSmoothing

apply exponential smoothing to a data set

LinearFilter

apply linear filter to a data set

MovingAverage

compute moving averages for a data set

MovingMedian

compute moving medians for a data set

MovingStatistic

compute moving statistics for a data set

WeightedMovingAverage

compute weighted moving averages for a data set

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More information is available in the Statistics[DataSmoothing] help page.

Hypothesis Testing and Inference

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Common tools for performing hypothesis testing and inference, including several parametric and non-parametric tests.  Here is the list of available commands.

  

 

ChiSquareGoodnessOfFitTest

apply the chi-square test for goodness-of-fit

ChiSquareIndependenceTest

apply the chi-square test for independence in a matrix

ChiSquareSuitableModelTest

apply the chi-square suitable model test

OneSampleChiSquareTest

apply the one sample chi-square test for the population standard deviation

OneSampleTTest

apply the one sample t-test for the population mean

OneSampleZTest

apply the one sample z-test for the population mean

ShapiroWilkWTest

apply Shapiro and Wilk's W-test for normality

TwoSampleFTest

apply the two sample F-test for population variances

TwoSamplePairedTTest

apply the paired t-test for population means

TwoSampleTTest

apply the two sample t-test for population means

TwoSampleZTest

apply the two sample z-test for population means

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More information is available at the Statistics[Tests] help page.

See Also

Data Smoothing Example Worksheet

Estimation Example Worksheet

Hypothesis Testing Example Worksheet

Probability Distributions Example Worksheet

Robust Statistics Example Worksheet

Statistics with DataFrames Example Worksheet

Statistics[Computation]

TimeSeriesAnalysis

 


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