Statistical Plots Package

A wide variety of plots and charts are used to gain an overview of data from a statistical perspective. Some summarize statistical computations on the data, while others compare data in ways that highlight its properties. This package implements several plotting functions of this class, including box-and-whisker plots, Pareto plots, quantile-quantile plots, and stem-and-leaf plots. Histograms, bar and pie charts are also commonly used in statistical applications, are included in the Mathematica kernel.

The box-and-whisker plot is invaluable for gaining a quick overview of the extent of a numeric dataset. It takes the form of a box that spans the distance between two quantiles surrounding the median, typically the 25% quantile to the 75% quantile. Commonly, "whiskers," lines that extend to span either the full dataset or the dataset excluding outliers, are added. Outliers are defined as points beyond 3/2 the discrete interquantile range from the edge of the box; far outliers are points beyond three times that range.

A number of options can be applied to control the appearance of the boxes. In addition to the above options, standard Graphics options can be used.

The amount of data covered by the boxes is determined by the BoxQuantile option. This takes the form of a number from 0 to 0.5, indicating a distance from the median (i.e., 0.5). Thus, 0.25 places quantiles at 0.25 and 0.75 (or the 25% and 75% quantiles). Note that all quantiles are computed as discrete quantiles, not interpolated quantiles.

The BoxOutliers option indicates whether outliers should be drawn in a special way. If None, the whiskers are drawn to cover the entire dataset. If All, near and far outliers (as described previously) are drawn identically, and the whiskers only cover non-outlying data points. If Automatic, the near and far outliers are drawn differently, as determined by the BoxOutlierMarkers option. BoxOutlierMarkers values are specified the same as for PlotMarkers.

The BoxOrientation option allows the graph to be oriented so that the bars run in a horizontal rather than a vertical direction. This option takes the value Vertical or Horizontal, depending on the orientation you desire.

Labels can be placed for each box via the BoxLabels option. If this option is Automatic, no label is drawn when a single box is being plotted, but if multiple boxes are being drawn, they are numbered sequentially. If a list of labels is provided, they are used cyclically for the boxes.

You can adjust the spacing between the boxes via the BoxExtraSpacing option. A space of 1 is equivalent to the space used for a single box. You can provide a list for this option, in which case the additional spaces are used cyclically. This is useful when you have multiple groups of boxes to be drawn, and you want some space between each group.

The styles used to draw the boxes are determined by the BoxFillingStyle, BoxLineStyle, and BoxMedianStyle options. The BoxFillingStyle option should be a color or list of colors to be used cyclically among the boxes. The style for the lines around the boxes is determined by the BoxLineStyle option; this may also take a list of styles to be used cyclically. Finally, the median line is often drawn somewhat differently than the rest of the lines; you can specify additional style options for the median line via the BoxMedianStyle option.

The Pareto plot is a quality control plot that combines a bar chart displaying percentages of categories in the data with a line plot showing cumulative percentages of the categories.

In the most basic form, ParetoPlot takes a list of data that is assumed to consist of discrete categories. It determines the frequency of each category in the list, converts the frequencies to percentages, and creates the plot.

If you have data where the frequencies are precomputed, you can plot it directly by providing {category, frequency} pairs instead of the raw data to ParetoPlot.

ParetoPlot accepts a number of options common to bar charts and line plots, as detailed in the above table. It also accepts the usual Graphics options.

The various Bar and Chart options are drawn from BarChart. Most of these behave the same as their BarChart counterparts. Note that unless the ChartLabels option is set to Automatic, it will simply apply the labels cyclically to the bars in the order they appear; they only correspond to the categories in the Automatic case. The BarOrigin option applies to the entire plot, not just the bars.

Quantile-quantile plots are used to determine whether two datasets come from populations with a common distribution. If the points of the plot, which are formed from the quantiles of the data, are roughly on a line with a slope of 1, then the distributions are the same.

QuantilePlot first sorts the shorter of the two lists of numbers and then determines the interpolated quantiles at the equivalent position in the longer list of data. It then plots the two sets of quantiles against each other. For datasets of equal length, this is equivalent to plotting the sorted lists against each other. The plot also displays a reference line with a slope of 1.

Typical list plot operations are available to control the display. The option ReferenceLineStyle, which can modify the reference line, can also be used. If it is set to None, the reference line is not drawn; otherwise it should be set to a style or list of styles.

The pairs or matrix scatter plot allows the individual columns in a multivariate set of data to be plotted against each other. This can be used to investigate relationships between the variables. The resulting plot is a matrix of subgraphs.

The pairs scatter plot forms a matrix of scatter plots from the columns of a multivariate dataset plotted against each other. PairwiseScatterPlot by default places the first column at the lower left side of the plot, and proceeds to the right and upwards.

A variety of options can be used to control the appearance of the plot. The DataRanges option accepts a list of range specifications, which can be used to restrict the points to be plotted. The ranges are given as {min, max} pairs or All or Automatic, used cyclically for all of the columns.

Textual annotations are provided via DataLabels and DataTicks. Labels can be supplied for each column via the DataLabels option, given as a list of labels. Ticks can be specified for each column using the usual graphics ticks syntax; these are drawn at the top and bottom of the corresponding column of the matrix of plots, as well as the right and left sides of the corresponding row. Tick labels are only drawn on alternating sides for each column to prevent labels for adjacent columns from overriding.

The PlotDirection option specifies the order in which plots are generated in the plot matrix. With the default {Right, Down}, row i and column j of the data are plotted in the i^th row and j^th column of the grid of scatter plots. The ordering used in versions prior to Version 6.0 can be obtained by setting PlotDirection->{Right, Up}.

The PlotStyle option can take either a single style primitive or a matrix of style primitives. If given a matrix, the primitives are applied to the subplots in a cyclic fashion.

Finally, the DataSpacing option allows the subgraphs to be drawn with varying amounts of space between them. This takes a number or a pair of numbers corresponding to the horizontal and vertical space between each graph. This number is scaled to the size of one of the subgraphs, which range from 0 to 1. You may provide negative numbers for the spacing, which can cause the subgraphs to be arranged in a different order. For example, if you prefer the first column to be at the upper rather than lower left, you can supply the option DataSpacing -> {0, -2}. An interesting effect can be derived by setting this option to -1, where all the subgraphs are overlaid.

In addition, all of the usual Graphics options are accepted.

The stem-and-leaf plot is generally used to visualize the distribution of real-valued data along with the magnitudes of the individual data values. Each data value is represented by a stem and leaf, where the stem is an integer multiple of a base unit and the leaf is the remainder given to some predetermined number of digits. With a base unit of 10, for instance, 17.3 could be represented as a stem of 1 and a leaf of 7. Leaves are collected onto common stems giving a display analogous to a histogram. The stems play a role similar to histogram bar positions, and the leaves are similar to the histogram bar heights. An advantage of the stem-and-leaf plot is that individual data values can be read directly from the plot. Side-by-side stem-and-leaf plots can be used to compare distributions and magnitudes of two datasets.

For this vector, multiples of 1 are taken as the stems and the fractional parts rounded to one digit are displayed as the leaves. The values 3.1 and 3.2 are displayed on the common stem 3 as leaves 1 and 2.

A number of options can be applied to control the appearance of the stem-and-leaf plot.

The value of StemExponent can be an integer or Automatic. If it is an integer x, the stem unit is taken to be 10^x. With StemExponent->Automatic, the exponent is chosen based on the magnitudes of the data.

The Leaves option value can be "Digits", "Tallies", or None. With Leaves->"Tallies", leaves are represented as tally marks. With Leaves->None, leaves are not included in the plot. The setting Leaves->None is most useful for plotting large datasets as stems and counts instead of displaying all the leaves.

The ColumnLabels option can be used to specify the labels for the columns of the plot. The ColumnLabels option value can be a list of a length equal to the number of columns in the plot or Automatic. With ColumnLabels->Automatic, the stem column is labeled Stem, leaf columns are labeled Leaves, and count columns are labeled Counts.

The IncludeEmptyStems option specifies whether stems within the data range should be included if they have no leaves. The possible values for this option are True and False.

The IncludeStemUnits option specifies whether or not a reminder of the stem units should be included in the plot. Possible values are True and False.

The IncludeStemCounts option specifies whether a column of counts should be included for each vector of real values plotted. If included, counts are displayed in the rightmost column in a stem-and-leaf plot of a single vector, and in the leftmost and rightmost columns for a side-by-side stem-and-leaf plot. Possible values are True, False, and Automatic. With IncludeStemCounts->Automatic, counts are only included if the Leaves option is set to None.

A number of options can be applied to control the appearance of the stem-and-leaf plot. The plot is constructed as a GridBox. In addition to the above options, standard GridBox options can be used. If IncludeStemUnits->True, GridBox options are applied to the grid of stems and leaves, but not to the label for stem units.

The StemExponent option has additional options than can be used to further subdivide stem units and label those divisions.

The "UnitDivisions" option specifies the number of stems a base unit should be divided into. The value of "UnitDivisions" must be a positive integer.

Alternate labeling of subdivisions can be specified via the "DivisionLabels" option. The value of "DivisionLabels" must be None or a list of a length equal to the "UnitDivisions" value. If the "DivisionLabels" value is a list, the values are appended to each numeric stem in the plot.

A number of options to the Leaves option can be used to modify how leaves are computed and displayed.

The "LeafDigits" option specifies the number of digits beyond the stem to use in computing leaves. This is also the number of digits displayed for each leaf if leaves are displayed as "Digits". The value must be a positive integer.

"LeafSpacing" indicates the number of spaces to display between leaves. "LeafSpacing" can be set to a non-negative integer or Automatic. With "LeafSpacing"->Automatic, zero spaces are used when "LeafDigits"->1, and one space is used otherwise.

"LeafWrapping" specifies the number of leaves after which leaves should be wrapped to a new line. "LeafWrapping" can be any positive integer or None, with None indicating that leaves should not be wrapped to new lines.

"RoundLeaves" specifies whether or not values should be rounded to the last leaf digit before computing leaves.

Wrapping leaves to a new line can be useful if there is a large number of leaves for one or more stems, as is the case with this dataset.

If leaves are displayed as "Tallies", the symbol used as tally markers can also be specified.

The "TallySymbol" option can be any string or symbol.

From the previous display it is clear which stems contain more values than others. Including other features, such as counts and leaf wrapping, can be useful in determining the actual magnitudes of the leaf counts for each stem.