Statistics

with Mathematica at UMBC

Contents

1. Summary Statistics
2. Graphs & Exploratory Analysis
3. Multivariate: Correlation & Regression
4. Generating Random Numbers

The Mathematica programming language has basic commands and subroutines which implement all basic statistics (and many other things). A general introduction to Mathematica use and how Mathematica can be accessed at UMBC can be found in a separate document.

The most natural way to represent a statistical data set is as a list, for example mydata = {1.2, 2.1, 1.1, 2.3, 0.2, 1.2}.

Summary Statistics

The Mean, Median, StandardDeviation, Variance and Quantile commands implement basic summary statistics.

Examples:

• mydata = {-3.1, 3.6, 0.5, 0.5, 4.8, -2.2, 2.0, -2.2, -4.1, -4.6}
• Mean[mydata]
  (returns -0.48)
• Median[mydata]
  (returns an answer of -0.85 = (-2.2 + 0.5)/2)
• Commonest[mydata]
  (returns {-2.2, 0.5} - similar to mode function)
• Quantile[mydata,1/2]
  (returns -2.2, not the median as Quantile normally returns an element of the list)
• Quantile[mydata,1/4]
  (returns -3.1)
• StandardDeviation[mydata]
  (returns 3.25672)

Note that if the data is all integers, the results will be a fraction, not a floating point number. Thus Mean[{1,2,3,4}] returns 5/2, while Mean[{1.0,2,3,4}] and N[Mean[{1,2,3,4}]] both return 2.5.

The use of Mathematica for Basic Statistics is well documented on the Wolfram.com website at [http://reference.wolfram.com/mathematica/tutorial/BasicStatistics.html].

Graphs & Exploratory Analysis

Mathematica and its various standard packages implement a variety of graphs useful for exploratory data analysis.

• Histograms: The Histogram command can be loaded with a standard package - before Mathematica 6.0 it was in the package Graphics`Graphics`, and in Mathematica 6.0 it is in the package Histograms`.
  • Needs["Histograms`"]
  • Histogram[mydata]
    (automatically chooses bins)
  • Histogram[mydata,HistogramCategories->5]
    (chooses five evenly spaced bins)
  • Histogram[mydata,HistogramCategories->{-5,-3,-1,1,3,5}]
    (use the specified bins)
  Documentation is found at [http://reference.wolfram.com/mathematica/Histograms/ref/HistogramCategories.html]
• Box Plots: The BoxWhiskerPlot command can be loaded with a standard package - before Mathematica 6.0 it was in the package Statistics`StatisticsPlots`, and in Mathematica 6.0 it is in the package StatisticalPlots`.
  • Needs["StatisticalPlots`"]
  • BoxWhiskerPlot[mydata]
    
  Documentation is found at [http://reference.wolfram.com/mathematica/StatisticalPlots/ref/BoxWhiskerPlot.html]
• Scatter Plots: A scatterplot relating two measured variables can be done with the ListPlot command.
  • ListPlot[{{x0,y0},{x1,y1},{x2,y2},{x3,y3}}]

Multivariate: Correlation & Regression

• Scatter Plots: A scatterplot relating two measured variables can be done with the ListPlot command.
  • ListPlot[{{x0,y0},{x1,y1},{x2,y2},{x3,y3}}]
• Linear Regression:
  • Fit[{{x0,y0},{x1,y1},{x2,y2},{x3,y3}},{1,x},x]
• Covariance: (new in MMA 6)
  • Covariance[{x0,x1,x2,x3},{y0,y1,y2,y3}]
• Correlation: Pearson's product-moment correlation coefficient (new in MMA 6)
  • Correlation[{x0,x1,x2,x3},{y0,y1,y2,y3}]

Generating Random Numbers