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# 45. descriptive

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## 45.1 Introduction to descriptive

Package `descriptive` contains a set of functions for making descriptive statistical computations and graphing. Together with the source code there are three data sets in your Maxima tree: `pidigits.data`, `wind.data` and `biomed.data`.

Any statistics manual can be used as a reference to the functions in package `descriptive`.

For comments, bugs or suggestions, please contact me at 'mario AT edu DOT xunta DOT es'.

Here is a simple example on how the descriptive functions in `descriptive` do they work, depending on the nature of their arguments, lists or matrices,

    (%i1) load (descriptive)$
    (%i2) /* univariate sample */   mean ([a, b, c]);
                                c + b + a
    (%o2)                       ---------
                                    3
    (%i3) matrix ([a, b], [c, d], [e, f]);
                                [ a  b ]
                                [      ]
    (%o3)                       [ c  d ]
                                [      ]
                                [ e  f ]
    (%i4) /* multivariate sample */ mean (%);
                          e + c + a  f + d + b
    (%o4)                [---------, ---------]
                              3          3

Note that in multivariate samples the mean is calculated for each column.

In case of several samples with possible different sizes, the Maxima function `map` can be used to get the desired results for each sample,

    (%i1) load (descriptive)$
    (%i2) map (mean, [[a, b, c], [d, e]]);
                            c + b + a  e + d
    (%o2)                  [---------, -----]
                                3        2

In this case, two samples of sizes 3 and 2 were stored into a list.

Univariate samples must be stored in lists like

    (%i1) s1 : [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5];
    (%o1)           [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]

and multivariate samples in matrices as in

    (%i1) s2 : matrix ([13.17, 9.29], [14.71, 16.88], [18.50, 16.88],
                 [10.58, 6.63], [13.33, 13.25], [13.21,  8.12]);
                            [ 13.17  9.29  ]
                            [              ]
                            [ 14.71  16.88 ]
                            [              ]
                            [ 18.5   16.88 ]
    (%o1)                   [              ]
                            [ 10.58  6.63  ]
                            [              ]
                            [ 13.33  13.25 ]
                            [              ]
                            [ 13.21  8.12  ]

In this case, the number of columns equals the random variable dimension and the number of rows is the sample size.

Data can be introduced by hand, but big samples are usually stored in plain text files. For example, file `pidigits.data` contains the first 100 digits of number `%pi`:

          3
          1
          4
          1
          5
          9
          2
          6
          5
          3 ...

In order to load these digits in Maxima,

    (%i1) s1 : read_list (file_search ("pidigits.data"))$
    (%i2) length (s1);
    (%o2)                          100

On the other hand, file `wind.data` contains daily average wind speeds at 5 meteorological stations in the Republic of Ireland (This is part of a data set taken at 12 meteorological stations. The original file is freely downloadable from the StatLib Data Repository and its analysis is discused in Haslett, J., Raftery, A. E. (1989) Space-time Modelling with Long-memory Dependence: Assessing Ireland's Wind Power Resource, with Discussion. Applied Statistics 38, 1-50). This loads the data:

    (%i1) s2 : read_matrix (file_search ("wind.data"))$
    (%i2) length (s2);
    (%o2)                          100
    (%i3) s2 [%]; /* last record */
    (%o3)            [3.58, 6.0, 4.58, 7.62, 11.25]

Some samples contain non numeric data. As an example, file `biomed.data` (which is part of another bigger one downloaded from the StatLib Data Repository) contains four blood measures taken from two groups of patients, `A` and `B`, of different ages,

    (%i1) s3 : read_matrix (file_search ("biomed.data"))$
    (%i2) length (s3);
    (%o2)                          100
    (%i3) s3 [1]; /* first record */
    (%o3)            [A, 30, 167.0, 89.0, 25.6, 364]

The first individual belongs to group `A`, is 30 years old and his/her blood measures were 167.0, 89.0, 25.6 and 364.

One must take care when working with categorical data. In the next example, symbol `a` is asigned a value in some previous moment and then a sample with categorical value `a` is taken,

    (%i1) a : 1$
    (%i2) matrix ([a, 3], [b, 5]);
                                [ 1  3 ]
    (%o2)                       [      ]
                                [ b  5 ]

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@ref{Category: Descriptive statistics} · @ref{Category: Share packages} · @ref{Category: Package descriptive}

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## 45.2 Functions and Variables for data manipulation

Function: continuous_freq (list)
Function: continuous_freq (list, m)

  The argument of `continuous_freq` must be a list of numbers. Divides the range in intervals and counts how many values are inside them. The second argument is optional and either equals the number of classes we want, 10 by default, or equals a list containing the class limits and the number of classes we want, or a list containing only the limits. Argument list must be a list of (2 or 3) real numbers. If sample values are all equal, this function returns only one class of amplitude 2.

  Examples:

  Optional argument indicates the number of classes we want. The first list in the output contains the interval limits, and the second the corresponding counts: there are 16 digits inside the interval `[0, 1.8]`, 24 digits in `(1.8, 3.6]`, and so on.

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) continuous_freq (s1, 5);
      (%o3) [[0, 1.8, 3.6, 5.4, 7.2, 9.0], [16, 24, 18, 17, 25]]

  Optional argument indicates we want 7 classes with limits -2 and 12:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) continuous_freq (s1, [-2,12,7]);
      (%o3) [[- 2, 0, 2, 4, 6, 8, 10, 12], [8, 20, 22, 17, 20, 13, 0]]

  Optional argument indicates we want the default number of classes with limits -2 and 12:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) continuous_freq (s1, [-2,12]);
                      3  4  11  18     32  39  46  53
      (%o3)  [[- 2, - -, -, --, --, 5, --, --, --, --, 12],
                      5  5  5   5      5   5   5   5
                     [0, 8, 20, 12, 18, 9, 8, 25, 0, 0]]

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Function: discrete_freq (list)

  Counts absolute frequencies in discrete samples, both numeric and categorical. Its unique argument is a list,

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) discrete_freq (s1);
      (%o3) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
                                   [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

  The first list gives the sample values and the second their absolute frequencies. Commands `? col` and `? transpose` should help you to understand the last input.

  ·

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Function: subsample (data_matrix, predicate_function)
Function: subsample (data_matrix, predicate_function, col_num1, col_num2, ...)

  This is a sort of variant of the Maxima `submatrix` function. The first argument is the data matrix, the second is a predicate function and optional additional arguments are the numbers of the columns to be taken. Its behaviour is better understood with examples.

  These are multivariate records in which the wind speed in the first meteorological station were greater than 18. See that in the lambda expression the i-th component is refered to as `v[i]`.

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) subsample (s2, lambda([v], v[1] > 18));
                    [ 19.38  15.37  15.12  23.09  25.25 ]
                    [                                   ]
                    [ 18.29  18.66  19.08  26.08  27.63 ]
      (%o3)         [                                   ]
                    [ 20.25  21.46  19.95  27.71  23.38 ]
                    [                                   ]
                    [ 18.79  18.96  14.46  26.38  21.84 ]

  In the following example, we request only the first, second and fifth components of those records with wind speeds greater or equal than 16 in station number 1 and less than 25 knots in station number 4. The sample contains only data from stations 1, 2 and 5. In this case, the predicate function is defined as an ordinary Maxima function.

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) g(x):= x[1] >= 16 and x[4] < 25$
      (%i4) subsample (s2, g, 1, 2, 5);
                           [ 19.38  15.37  25.25 ]
                           [                     ]
                           [ 17.33  14.67  19.58 ]
      (%o4)                [                     ]
                           [ 16.92  13.21  21.21 ]
                           [                     ]
                           [ 17.25  18.46  23.87 ]

  Here is an example with the categorical variables of `biomed.data`. We want the records corresponding to those patients in group `B` who are older than 38 years.

      (%i1) load (descriptive)$
      (%i2) s3 : read_matrix (file_search ("biomed.data"))$
      (%i3) h(u):= u[1] = B and u[2] > 38 $
      (%i4) subsample (s3, h);
                      [ B  39  28.0  102.3  17.1  146 ]
                      [                               ]
                      [ B  39  21.0  92.4   10.3  197 ]
                      [                               ]
                      [ B  39  23.0  111.5  10.0  133 ]
                      [                               ]
                      [ B  39  26.0  92.6   12.3  196 ]
      (%o4)           [                               ]
                      [ B  39  25.0  98.7   10.0  174 ]
                      [                               ]
                      [ B  39  21.0  93.2   5.9   181 ]
                      [                               ]
                      [ B  39  18.0  95.0   11.3  66  ]
                      [                               ]
                      [ B  39  39.0  88.5   7.6   168 ]

  Probably, the statistical analysis will involve only the blood measures,

      (%i1) load (descriptive)$
      (%i2) s3 : read_matrix (file_search ("biomed.data"))$
      (%i3) subsample (s3, lambda([v], v[1] = B and v[2] > 38),
                       3, 4, 5, 6);
                         [ 28.0  102.3  17.1  146 ]
                         [                        ]
                         [ 21.0  92.4   10.3  197 ]
                         [                        ]
                         [ 23.0  111.5  10.0  133 ]
                         [                        ]
                         [ 26.0  92.6   12.3  196 ]
      (%o3)              [                        ]
                         [ 25.0  98.7   10.0  174 ]
                         [                        ]
                         [ 21.0  93.2   5.9   181 ]
                         [                        ]
                         [ 18.0  95.0   11.3  66  ]
                         [                        ]
                         [ 39.0  88.5   7.6   168 ]

  This is the multivariate mean of `s3`,

      (%i1) load (descriptive)$
      (%i2) s3 : read_matrix (file_search ("biomed.data"))$
      (%i3) mean (s3);
             65 B + 35 A  317          6 NA + 8144.999999999999
      (%o3) [-----------, ---, 87.178, ------------------------,
                 100      10                     100
                                                          3 NA + 19587
                                                  18.123, ------------]
                                                              100

  Here, the first component is meaningless, since `A` and `B` are categorical, the second component is the mean age of individuals in rational form, and the fourth and last values exhibit some strange behaviour. This is because symbol `NA` is used here to indicate non available data, and the two means are nonsense. A possible solution would be to take out from the matrix those rows with `NA` symbols, although this deserves some loss of information.

      (%i1) load (descriptive)$
      (%i2) s3 : read_matrix (file_search ("biomed.data"))$
      (%i3) g(v):= v[4] # NA and v[6] # NA $
      (%i4) mean (subsample (s3, g, 3, 4, 5, 6));
      (%o4) [79.4923076923077, 86.2032967032967, 16.93186813186813,
                                                                  2514
                                                                  ----]
                                                                   13

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## 45.3 Functions and Variables for descriptive statistics

Function: mean (list)
Function: mean (matrix)

  This is the sample mean, defined as

                             n
                           ====
                   _   1   \
                   x = -    >    x
                       n   /      i
                           ====
                           i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) mean (s1);
                                     471
      (%o3)                          ---
                                     100
      (%i4) %, numer;
      (%o4)                         4.71
      (%i5) s2 : read_matrix (file_search ("wind.data"))$
      (%i6) mean (s2);
      (%o6)     [9.9485, 10.1607, 10.8685, 15.7166, 14.8441]

  ·

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Function: var (list)
Function: var (matrix)

  This is the sample variance, defined as

                           n
                         ====
                 2   1   \          _ 2
                s  = -    >    (x - x)
                     n   /       i
                         ====
                         i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) var (s1), numer;
      (%o3)                   8.425899999999999

  See also function `var1`.

  ·

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Function: var1 (list)
Function: var1 (matrix)

  This is the sample variance, defined as

                           n
                         ====
                     1   \          _ 2
                    ---   >    (x - x)
                    n-1  /       i
                         ====
                         i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) var1 (s1), numer;
      (%o3)                    8.5110101010101
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) var1 (s2);
      (%o5) [17.39586540404041, 15.13912778787879, 15.63204924242424,
                                  32.50152569696971, 24.66977392929294]

  See also function `var`.

  ·

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Function: std (list)
Function: std (matrix)

  This is the the square root of function `var`, the variance with denominator n.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) std (s1), numer;
      (%o3)                   2.902740084816414
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) std (s2);
      (%o5) [4.149928523480858, 3.871399812729241, 3.933920277534866,
                                  5.672434260526957, 4.941970881136392]

  See also functions `var` and `std1`.

  ·

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Function: std1 (list)
Function: std1 (matrix)

  This is the the square root of function `var1`, the variance with denominator n-1.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) std1 (s1), numer;
      (%o3)                   2.917363553109228
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) std1 (s2);
      (%o5) [4.170835096721089, 3.89090320978032, 3.953738641137555,
                                  5.701010936401517, 4.966867617451963]

  See also functions `var1` and `std`.

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Function: noncentral_moment (list, k)
Function: noncentral_moment (matrix, k)

  The non central moment of order k, defined as

                             n
                           ====
                       1   \      k
                       -    >    x
                       n   /      i
                           ====
                           i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) noncentral_moment (s1, 1), numer; /* the mean */
      (%o3)                         4.71
      (%i5) s2 : read_matrix (file_search ("wind.data"))$
      (%i6) noncentral_moment (s2, 5);
      (%o6) [319793.8724761505, 320532.1923892463,
            391249.5621381556, 2502278.205988911, 1691881.797742255]

  See also function `central_moment`.

  ·

  @ref{Category: Package descriptive}

Function: central_moment (list, k)
Function: central_moment (matrix, k)

  The central moment of order k, defined as

                          n
                        ====
                    1   \          _ k
                    -    >    (x - x)
                    n   /       i
                        ====
                        i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) central_moment (s1, 2), numer; /* the variance */
      (%o3)                   8.425899999999999
      (%i5) s2 : read_matrix (file_search ("wind.data"))$
      (%i6) central_moment (s2, 3);
      (%o6) [11.29584771375004, 16.97988248298583, 5.626661952750102,
                                   37.5986572057918, 25.85981904394192]

  See also functions `central_moment` and `mean`.

  ·

  @ref{Category: Package descriptive}

Function: cv (list)
Function: cv (matrix)

  The variation coefficient is the quotient between the sample standard deviation (`std`) and the `mean`,

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) cv (s1), numer;
      (%o3)                   .6193977819764815
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) cv (s2);
      (%o5) [.4192426091090204, .3829365309260502, 0.363779605385983,
                                  .3627381836021478, .3346021393989506]

  See also functions `std` and `mean`.

  ·

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Function: smin (list)
Function: smin (matrix)

  This is the minimum value of the sample list. When the argument is a matrix, `smin` returns a list containing the minimum values of the columns, which are associated to statistical variables.

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) smin (s1);
      (%o3)                           0
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) smin (s2);
      (%o5)             [0.58, 0.5, 2.67, 5.25, 5.17]

  See also function `smax`.

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Function: smax (list)
Function: smax (matrix)

  This is the maximum value of the sample list. When the argument is a matrix, `smax` returns a list containing the maximum values of the columns, which are associated to statistical variables.

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) smax (s1);
      (%o3)                           9
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) smax (s2);
      (%o5)          [20.25, 21.46, 20.04, 29.63, 27.63]

  See also function `smin`.

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Function: range (list)
Function: range (matrix)

  The range is the difference between the extreme values.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) range (s1);
      (%o3)                           9
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) range (s2);
      (%o5)          [19.67, 20.96, 17.37, 24.38, 22.46]

  ·

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Function: quantile (list, p)
Function: quantile (matrix, p)

  This is the p-quantile, with p a number in [0, 1], of the sample list. Although there are several definitions for the sample quantile (Hyndman, R. J., Fan, Y. (1996) Sample quantiles in statistical packages. American Statistician, 50, 361-365), the one based on linear interpolation is implemented in package `descriptive`.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) /* 1st and 3rd quartiles */
               [quantile (s1, 1/4), quantile (s1, 3/4)], numer;
      (%o3)                      [2.0, 7.25]
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) quantile (s2, 1/4);
      (%o5)    [7.2575, 7.477500000000001, 7.82, 11.28, 11.48]

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Function: median (list)
Function: median (matrix)

  Once the sample is ordered, if the sample size is odd the median is the central value, otherwise it is the mean of the two central values.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) median (s1);
                                      9
      (%o3)                           -
                                      2
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) median (s2);
      (%o5)         [10.06, 9.855, 10.73, 15.48, 14.105]

  The median is the 1/2-quantile.

  See also function `quantile`.

  ·

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Function: qrange (list)
Function: qrange (matrix)

  The interquartilic range is the difference between the third and first quartiles, `quantile(list,3/4) - quantile(list,1/4)`,

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) qrange (s1);
                                     21
      (%o3)                          --
                                     4
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) qrange (s2);
      (%o5) [5.385, 5.572499999999998, 6.022500000000001,
                                  8.729999999999999, 6.649999999999999]

  See also function `quantile`.

  ·

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Function: mean_deviation (list)
Function: mean_deviation (matrix)

  The mean deviation, defined as

                           n
                         ====
                     1   \          _
                     -    >    |x - x|
                     n   /       i
                         ====
                         i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) mean_deviation (s1);
                                     51
      (%o3)                          --
                                     20
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) mean_deviation (s2);
      (%o5) [3.287959999999999, 3.075342, 3.23907, 4.715664000000001,
                                                     4.028546000000002]

  See also function `mean`.

  ·

  @ref{Category: Package descriptive}

Function: median_deviation (list)
Function: median_deviation (matrix)

  The median deviation, defined as

                       n
                     ====
                 1   \
                 -    >    |x - med|
                 n   /       i
                     ====
                     i = 1

  where `med` is the median of list.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) median_deviation (s1);
                                      5
      (%o3)                           -
                                      2
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) median_deviation (s2);
      (%o5)           [2.75, 2.755, 3.08, 4.315, 3.31]

  See also function `mean`.

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Function: harmonic_mean (list)
Function: harmonic_mean (matrix)

  The harmonic mean, defined as

                        n
                     --------
                      n
                     ====
                     \     1
                      >    --
                     /     x
                     ====   i
                     i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
      (%i3) harmonic_mean (y), numer;
      (%o3)                   3.901858027632205
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) harmonic_mean (s2);
      (%o5) [6.948015590052786, 7.391967752360356, 9.055658197151745,
                                  13.44199028193692, 13.01439145898509]

  See also functions `mean` and `geometric_mean`.

  ·

  @ref{Category: Package descriptive}

Function: geometric_mean (list)
Function: geometric_mean (matrix)

  The geometric mean, defined as

                       /  n      \ 1/n
                       | /===\   |
                       |  ! !    |
                       |  ! !  x |
                       |  ! !   i|
                       | i = 1   |
                       \         /

  Example:

      (%i1) load (descriptive)$
      (%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
      (%i3) geometric_mean (y), numer;
      (%o3)                   4.454845412337012
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) geometric_mean (s2);
      (%o5) [8.82476274347979, 9.22652604739361, 10.0442675714889,
                                  14.61274126349021, 13.96184163444275]

  See also functions `mean` and `harmonic_mean`.

  ·

  @ref{Category: Package descriptive}

Function: kurtosis (list)
Function: kurtosis (matrix)

  The kurtosis coefficient, defined as

                          n
                        ====
                  1     \          _ 4
                 ----    >    (x - x)  - 3
                    4   /       i
                 n s    ====
                        i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) kurtosis (s1), numer;
      (%o3)                  - 1.273247946514421
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) kurtosis (s2);
      (%o5) [- .2715445622195385, 0.119998784429451,
           - .4275233490482861, - .6405361979019522, - .4952382132352935]

  See also functions `mean`, `var` and `skewness`.

  ·

  @ref{Category: Package descriptive}

Function: skewness (list)
Function: skewness (matrix)

  The skewness coefficient, defined as

                          n
                        ====
                  1     \          _ 3
                 ----    >    (x - x)
                    3   /       i
                 n s    ====
                        i = 1

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) skewness (s1), numer;
      (%o3)                  .009196180476450424
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) skewness (s2);
      (%o5) [.1580509020000978, .2926379232061854, .09242174416107717,
                                  .2059984348148687, .2142520248890831]

  See also functions `mean`, `var` and `kurtosis`.

  ·

  @ref{Category: Package descriptive}

Function: pearson_skewness (list)
Function: pearson_skewness (matrix)

  Pearson's skewness coefficient, defined as

                      _
                   3 (x - med)
                   -----------
                        s

  where med is the median of list.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) pearson_skewness (s1), numer;
      (%o3)                   .2159484029093895
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) pearson_skewness (s2);
      (%o5) [- .08019976629211892, .2357036272952649,
               .1050904062491204, .1245042340592368, .4464181795804519]

  See also functions `mean`, `var` and `median`.

  ·

  @ref{Category: Package descriptive}

Function: quartile_skewness (list)
Function: quartile_skewness (matrix)

  The quartile skewness coefficient, defined as

                     c    - 2 c    + c
                      3/4      1/2    1/4
                     --------------------
                         c    - c
                          3/4    1/4

  where c_p is the p-quantile of sample list.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) quartile_skewness (s1), numer;
      (%o3)                  .04761904761904762
      (%i4) s2 : read_matrix (file_search ("wind.data"))$
      (%i5) quartile_skewness (s2);
      (%o5) [- 0.0408542246982353, .1467025572005382,
             0.0336239103362392, .03780068728522298, .2105263157894735]

  See also function `quantile`.

  ·

  @ref{Category: Package descriptive}

Function: cov (matrix)

  The covariance matrix of the multivariate sample, defined as

                    n
                   ====
                1  \           _        _
            S = -   >    (X  - X) (X  - X)'
                n  /       j        j
                   ====
                   j = 1

  where X_j is the j-th row of the sample matrix.

  Example:

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) fpprintprec : 7$  /* change precision for pretty output */
      (%i4) cov (s2);
            [ 17.22191  13.61811  14.37217  19.39624  15.42162 ]
            [                                                  ]
            [ 13.61811  14.98774  13.30448  15.15834  14.9711  ]
            [                                                  ]
      (%o4) [ 14.37217  13.30448  15.47573  17.32544  16.18171 ]
            [                                                  ]
            [ 19.39624  15.15834  17.32544  32.17651  20.44685 ]
            [                                                  ]
            [ 15.42162  14.9711   16.18171  20.44685  24.42308 ]

  See also function `cov1`.

  ·

  @ref{Category: Package descriptive}

Function: cov1 (matrix)

  The covariance matrix of the multivariate sample, defined as

                    n
                   ====
               1   \           _        _
         S  = ---   >    (X  - X) (X  - X)'
          1   n-1  /       j        j
                   ====
                   j = 1

  where X_j is the j-th row of the sample matrix.

  Example:

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) fpprintprec : 7$ /* change precision for pretty output */
      (%i4) cov1 (s2);
            [ 17.39587  13.75567  14.51734  19.59216  15.5774  ]
            [                                                  ]
            [ 13.75567  15.13913  13.43887  15.31145  15.12232 ]
            [                                                  ]
      (%o4) [ 14.51734  13.43887  15.63205  17.50044  16.34516 ]
            [                                                  ]
            [ 19.59216  15.31145  17.50044  32.50153  20.65338 ]
            [                                                  ]
            [ 15.5774   15.12232  16.34516  20.65338  24.66977 ]

  See also function `cov`.

  ·

  @ref{Category: Package descriptive}

Function: global_variances (matrix)
Function: global_variances (matrix, logical_value)

  Function `global_variances` returns a list of global variance measures:

  - total variance: `trace(S_1)`,
  - mean variance: `trace(S_1)/p`,
  - generalized variance: `determinant(S_1)`,
  - generalized standard deviation: `sqrt(determinant(S_1))`,
  - efective variance `determinant(S_1)^(1/p)`, (defined in: Peña, D. (2002) Análisis de datos multivariantes; McGraw-Hill, Madrid.)
  - efective standard deviation: `determinant(S_1)^(1/(2*p))`.

  where p is the dimension of the multivariate random variable and S_1 the covariance matrix returned by `cov1`.

  Example:

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) global_variances (s2);
      (%o3) [105.338342060606, 21.06766841212119, 12874.34690469686,
               113.4651792608501, 6.636590811800795, 2.576158149609762]

  The function `global_variances` has an optional logical argument: `global_variances(x, true)` tells Maxima that `x` is the data matrix, making the same as `global_variances(x)`. On the other hand, `global_variances(x, false)` means that `x` is not the data matrix, but the covariance matrix, avoiding its recalculation,

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) s : cov1 (s2)$
      (%i4) global_variances (s, false);
      (%o4) [105.338342060606, 21.06766841212119, 12874.34690469686,
               113.4651792608501, 6.636590811800795, 2.576158149609762]

  See also `cov` and `cov1`.

  ·

  @ref{Category: Package descriptive}

Function: cor (matrix)
Function: cor (matrix, logical_value)

  The correlation matrix of the multivariate sample.

  Example:

      (%i1) load (descriptive)$
      (%i2) fpprintprec : 7 $
      (%i3) s2 : read_matrix (file_search ("wind.data"))$
      (%i4) cor (s2);
            [   1.0     .8476339  .8803515  .8239624  .7519506 ]
            [                                                  ]
            [ .8476339    1.0     .8735834  .6902622  0.782502 ]
            [                                                  ]
      (%o4) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
            [                                                  ]
            [ .8239624  .6902622  .7764065    1.0     .7293848 ]
            [                                                  ]
            [ .7519506  0.782502  .8323358  .7293848    1.0    ]

  Function `cor` has an optional logical argument: `cor(x,true)` tells Maxima that `x` is the data matrix, making the same as `cor(x)`. On the other hand, `cor(x,false)` means that `x` is not the data matrix, but the covariance matrix, avoiding its recalculation,

      (%i1) load (descriptive)$
      (%i2) fpprintprec : 7 $
      (%i3) s2 : read_matrix (file_search ("wind.data"))$
      (%i4) s : cov1 (s2)$
      (%i5) cor (s, false); /* this is faster */
            [   1.0     .8476339  .8803515  .8239624  .7519506 ]
            [                                                  ]
            [ .8476339    1.0     .8735834  .6902622  0.782502 ]
            [                                                  ]
      (%o5) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
            [                                                  ]
            [ .8239624  .6902622  .7764065    1.0     .7293848 ]
            [                                                  ]
            [ .7519506  0.782502  .8323358  .7293848    1.0    ]

  See also `cov` and `cov1`.

  ·

  @ref{Category: Package descriptive}

Function: list_correlations (matrix)
Function: list_correlations (matrix, logical_value)

  Function `list_correlations` returns a list of correlation measures:

  - precision matrix: the inverse of the covariance matrix S_1,

               -1     ij
              S   = (s  )
               1         i,j = 1,2,...,p

  - multiple correlation vector: (R_1^2, R_2^2, ..., R_p^2), with

               2          1
              R  = 1 - -------
               i        ii
                       s   s
                            ii

    being an indicator of the goodness of fit of the linear multivariate regression model on X_i when the rest of variables are used as regressors.

  - partial correlation matrix: with element (i, j) being

                                 ij
                                s
              r        = - ------------
               ij.rest     / ii  jj\ 1/2
                           |s   s  |
                           \       /

  Example:

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) z : list_correlations (s2)$
      (%i4) fpprintprec : 5$ /* for pretty output */
      (%i5) z[1];  /* precision matrix */
            [  .38486   - .13856   - .15626   - .10239    .031179  ]
            [                                                      ]
            [ - .13856   .34107    - .15233    .038447   - .052842 ]
            [                                                      ]
      (%o5) [ - .15626  - .15233    .47296    - .024816  - .10054  ]
            [                                                      ]
            [ - .10239   .038447   - .024816   .10937    - .034033 ]
            [                                                      ]
            [ .031179   - .052842  - .10054   - .034033   .14834   ]
      (%i6) z[2];  /* multiple correlation vector */
      (%o6)      [.85063, .80634, .86474, .71867, .72675]
      (%i7) z[3];  /* partial correlation matrix */
            [  - 1.0     .38244   .36627   .49908   - .13049 ]
            [                                                ]
            [  .38244    - 1.0    .37927  - .19907   .23492  ]
            [                                                ]
      (%o7) [  .36627    .37927   - 1.0    .10911    .37956  ]
            [                                                ]
            [  .49908   - .19907  .10911   - 1.0     .26719  ]
            [                                                ]
            [ - .13049   .23492   .37956   .26719    - 1.0   ]

  Function `list_correlations` also has an optional logical argument: `list_correlations(x,true)` tells Maxima that `x` is the data matrix, making the same as `list_correlations(x)`. On the other hand, `list_correlations(x,false)` means that `x` is not the data matrix, but the covariance matrix, avoiding its recalculation.

  See also `cov` and `cov1`.

  ·

  @ref{Category: Package descriptive}

0

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## 45.4 Functions and Variables for statistical graphs

Function: barsplot (data1, data2, …, option_1, option_2, …)
Function: barsplot_description (…)

  Plots bars diagrams for discrete statistical variables, both for one or multiple samples.

  data can be a list of outcomes representing one sample, or a matrix of m rows and n columns, representing n samples of size m each.

  Available options are:

  - box_width (default, `3/4`): relative width of rectangles. This value must be in the range `[0,1]`.
  - grouping (default, `clustered`): indicates how multiple samples are shown. Valid values are: `clustered` and `stacked`.
  - groups_gap (default, `1`): a positive integer number representing the gap between two consecutive groups of bars.
  - bars_colors (default, `[]`): a list of colors for multiple samples. When there are more samples than specified colors, the extra necesary colors are chosen at random. See `color` to learn more about them.
  - frequency (default, `absolute`): indicates the scale of the ordinates. Possible values are: `absolute`, `relative`, and `percent`.
  - ordering (default, `orderlessp`): possible values are `orderlessp` or `ordergreatp`, indicating how statistical outcomes should be ordered on the x-axis.
  - sample_keys (default, `[]`): a list with the strings to be used in the legend. When the list length is other than 0 or the number of samples, an error message is returned.
  - start_at (default, `0`): indicates where the plot begins to be plotted on the x axis.
  - All global `draw` options, except `xtics`, which is internally assigned by `barsplot`. If you want to set your own values for this option or want to build complex scenes, make use of `barsplot_description`. See example below.
  - The following local `draw` options: `key`, `color`, `fill_color`, `fill_density` and `line_width`. See also `bars`.

  Function `barsplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects. There is also a function `wxbarsplot` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Examples:

  Univariate sample in matrix form. Absolute frequencies.

      (%i1) load (descriptive)$
      (%i2) m : read_matrix (file_search ("biomed.data"))$
      (%i3) barsplot(
              col(m,2),
              title        = "Ages",
              xlabel       = "years",
              box_width    = 1/2,
              fill_density = 3/4)$

  Two samples of different sizes, with relative frequencies and user declared colors.

      (%i1) load (descriptive)$
      (%i2) l1:makelist(random(10),k,1,50)$
      (%i3) l2:makelist(random(10),k,1,100)$
      (%i4) barsplot(
              l1,l2,
              box_width    = 1,
              fill_density = 1,
              bars_colors  = [black, grey],
              frequency = relative,
              sample_keys = ["A", "B"])$

  Four non numeric samples of equal size.

      (%i1) load (descriptive)$
      (%i2) barsplot(
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              title  = "Asking for something to four groups",
              ylabel = "# of individuals",
              groups_gap   = 3,
              fill_density = 0.5,
              ordering     = ordergreatp)$

  Stacked bars.

      (%i1) load (descriptive)$
      (%i2) barsplot(
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              makelist([Yes, No, Maybe][random(3)+1],k,1,50),
              title  = "Asking for something to four groups",
              ylabel = "# of individuals",
              grouping     = stacked,,
              fill_density = 0.5,
              ordering     = ordergreatp)$

  `barsplot` in a multiplot context.

      (%i1) load (descriptive)$
      (%i2) l1:makelist(random(10),k,1,50)$
      (%i3) l2:makelist(random(10),k,1,100)$
      (%i4) bp1 :
              barsplot_description(
               l1,
               box_width = 1,
               fill_density = 0.5,
               bars_colors = [blue],
               frequency = relative)$
      (%i5) bp2 :
              barsplot_description(
               l2,
               box_width = 1,
               fill_density = 0.5,
               bars_colors = [red],
               frequency = relative)$
      (%i6) draw(gr2d(bp1), gr2d(bp2))$

  For bars diagrams related options, see `bars` of package `draw`. See also functions `histogram` and `piechart`.

  ·

  @ref{Category: Package descriptive} · @ref{Category: Plotting}

Function: boxplot (data)
Function: boxplot (data, option_1, option_2, …)
Function: boxplot_description (…)

  This function plots box-and-whishker diagrams. Argument data can be a list, which is not of great interest, since these diagrams are mainly used for comparing different samples, or a matrix, so it is possible to compare two or more components of a multivariate statistical variable. But it is also allowed data to be a list of samples with possible different sample sizes, in fact this is the only function in package `descriptive` that admits this type of data structure.

  Available options are:

  - box_width (default, `3/4`): relative width of boxes. This value must be in the range `[0,1]`.
  - box_orientation (default, `vertical`): possible values: `vertical` and `horizontal`.
  - All `draw` options, except `points_joined`, `point_size`, `point_type`, `xtics`, `ytics`, `xrange`, and `yrange`, which are internally assigned by `boxplot`. If you want to set your own values for this options or want to build complex scenes, make use of `boxplot_description`.
  - The following local `draw` options: `key`, `color`, and `line_width`.

  Function `boxplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects. There is also a function `wxboxplot` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Examples:

  Box-and-whishker diagram from a multivariate sample.

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix(file_search("wind.data"))$
      (%i3) boxplot(s2,
              box_width  = 0.2,
              title      = "Windspeed in knots",
              xlabel     = "Stations",
              color      = red,
              line_width = 2)$

  Box-and-whishker diagram from three samples of different sizes.

      (%i1) load (descriptive)$
      (%i2) A :
             [[6, 4, 6, 2, 4, 8, 6, 4, 6, 4, 3, 2],
              [8, 10, 7, 9, 12, 8, 10],
              [16, 13, 17, 12, 11, 18, 13, 18, 14, 12]]$
      (%i3) boxplot (A, box_orientation = horizontal)$

  ·

  @ref{Category: Package descriptive} · @ref{Category: Plotting}

Function: histogram (list)
Function: histogram (list, option_1, option_2, …)
Function: histogram (one_column_matrix)
Function: histogram (one_column_matrix, option_1, option_2, …)
Function: histogram (one_row_matrix)
Function: histogram (one_row_matrix, option_1, option_2, …)
Function: histogram_description (…)

  This function plots an histogram from a continuous sample. Sample data must be stored in a list of numbers or a one dimensional matrix.

  Available options are:

  - nclasses (default, `10`): number of classes of the histogram, or a list indicating the limits of the classes and the number of them, or only the limits.
  - frequency (default, `absolute`): indicates the scale of the ordinates. Possible values are: `absolute`, `relative`, and `percent`.
  - htics (default, `auto`): format of the histogram tics. Possible values are: `auto`, `endpoints`, `intervals`, or a list of labels.
  - All global `draw` options, except `xrange`, `yrange`, and `xtics`, which are internally assigned by `histogram`. If you want to set your own values for these options, make use of `histogram_description`. See examples bellow.
  - The following local `draw` options: `key`, `color`, `fill_color`, `fill_density` and `line_width`. See also `bars`.

  Function `histogram_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects. There is also a function `wxhistogram` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Examples:

  A simple with eight classes:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) histogram (
                 s1,
                 nclasses     = 8,
                 title        = "pi digits",
                 xlabel       = "digits",
                 ylabel       = "Absolute frequency",
                 fill_color   = grey,
                 fill_density = 0.6)$

  Setting the limits of the histogram to -2 and 12, with 3 classes. Also, we introduce predefined tics:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) histogram (
                 s1,
                 nclasses     = [-2,12,3],
                 htics        = ["A", "B", "C"],
                 terminal     = png,
                 fill_color   = "#23afa0",
                 fill_density = 0.6)$

  We make use of `histogram_description` for setting the `xrange` and adding an explicit curve into the scene:

      (%i1) load (descriptive)$
      (%i2) ( load("distrib"),
              m: 14, s: 2,
              s2: random_normal(m, s, 1000) ) $
      (%i3) draw2d(
              grid   = true,
              xrange = [5, 25],
              histogram_description(
                s2,
                nclasses     = 9,
                frequency    = relative,
                fill_density = 0.5),
              explicit(pdf_normal(x,m,s), x, m - 3*s, m + 3* s))$

  ·

  @ref{Category: Package descriptive} · @ref{Category: Plotting}

Function: piechart (list)
Function: piechart (list, option_1, option_2, …)
Function: piechart (one_column_matrix)
Function: piechart (one_column_matrix, option_1, option_2, …)
Function: piechart (one_row_matrix)
Function: piechart (one_row_matrix, option_1, option_2, …)
Function: piechart_description (…)

  Similar to `barsplot`, but plots sectors instead of rectangles.

  Available options are:

  - sector_colors (default, `[]`): a list of colors for sectors. When there are more sectors than specified colors, the extra necesary colors are chosen at random. See `color` to learn more about them.
  - pie_center (default, `[0,0]`): diagram's center.
  - pie_radius (default, `1`): diagram's radius.
  - All global `draw` options, except `key`, which is internally assigned by `piechart`. If you want to set your own values for this option or want to build complex scenes, make use of `piechart_description`.
  - The following local `draw` options: `key`, `color`, `fill_density` and `line_width`. See also `ellipse`.

  Function `piechart_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects. There is also a function `wxpiechart` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Example:

      (%i1) load (descriptive)$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) piechart(
              s1,
              xrange  = [-1.1, 1.3],
              yrange  = [-1.1, 1.1],
              title   = "Digit frequencies in pi")$

  See also function `barsplot`.

  ·

  @ref{Category: Package descriptive} · @ref{Category: Plotting}

Function: scatterplot (list)
Function: scatterplot (list, option_1, option_2, …)
Function: scatterplot (matrix)
Function: scatterplot (matrix, option_1, option_2, …)
Function: scatterplot_description (…)

  Plots scatter diagrams both for univariate (list) and multivariate (matrix) samples.

  Available options are the same admitted by `histogram`.

  Function `scatterplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects. There is also a function `wxscatterplot` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Examples:

  Univariate scatter diagram from a simulated Gaussian sample.

      (%i1) load (descriptive)$
      (%i2) load (distrib)$
      (%i3) scatterplot(
              random_normal(0,1,200),
              xaxis      = true,
              point_size = 2,
              dimensions = [600,150])$

  Two dimensional scatter plot.

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) scatterplot(
             submatrix(s2, 1,2,3),
             title      = "Data from stations #4 and #5",
             point_type = diamant,
             point_size = 2,
             color      = blue)$

  Three dimensional scatter plot.

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) scatterplot(submatrix (s2, 1,2), nclasses=4)$

  Five dimensional scatter plot, with five classes histograms.

      (%i1) load (descriptive)$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) scatterplot(
              s2,
              nclasses     = 5,
              frequency    = relative,
              fill_color   = blue,
              fill_density = 0.3,
              xtics        = 5)$

  For plotting isolated or line-joined points in two and three dimensions, see `points`. See also `histogram`.

  ·

  @ref{Category: Package descriptive} · @ref{Category: Plotting}

Function: starplot (data1, data2, …, option_1, option_2, …)
Function: starplot_description (…)

  Plots star diagrams for discrete statistical variables, both for one or multiple samples.

  data can be a list of outcomes representing one sample, or a matrix of m rows and n columns, representing n samples of size m each.

  Available options are:

  - stars_colors (default, `[]`): a list of colors for multiple samples. When there are more samples than specified colors, the extra necesary colors are chosen at random. See `color` to learn more about them.
  - frequency (default, `absolute`): indicates the scale of the radii. Possible values are: `absolute` and `relative`.
  - ordering (default, `orderlessp`): possible values are `orderlessp` or `ordergreatp`, indicating how statistical outcomes should be ordered.
  - sample_keys (default, `[]`): a list with the strings to be used in the legend. When the list length is other than 0 or the number of samples, an error message is returned.
  - star_center (default, `[0,0]`): diagram's center.
  - star_radius (default, `1`): diagram's radius.
  - All global `draw` options, except `points_joined`, `point_type`, and `key`, which are internally assigned by `starplot`. If you want to set your own values for this options or want to build complex scenes, make use of `starplot_description`.
  - The following local `draw` option: `line_width`.

  Function `starplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects. There is also a function `wxstarplot` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Example:

  Plot based on absolute frequencies. Location and radius defined by the user.

      (%i1) load (descriptive)$
      (%i2) l1: makelist(random(10),k,1,50)$
      (%i3) l2: makelist(random(10),k,1,200)$
      (%i4) starplot(
              l1, l2,
              stars_colors = [blue,red],
              sample_keys = ["1st sample", "2nd sample"],
              star_center = [1,2],
              star_radius = 4,
              proportional_axes = xy,
              line_width = 2 ) $

  ·

  @ref{Category: Package descriptive} · @ref{Category: Plotting}

Function: stemplot (data)
Function: stemplot (data, option)

  Plots stem and leaf diagrams.

  Unique available option is:

  - leaf_unit (default, `1`): indicates the unit of the leaves; must be a power of 10.

  Example:

      (%i1) load (descriptive)$
      (%i2) load(distrib)$
      (%i3) stemplot(
              random_normal(15, 6, 100),
              leaf_unit = 0.1);
      -5|4
       0|37
       1|7
       3|6
       4|4
       5|4
       6|57
       7|0149
       8|3
       9|1334588
      10|07888
      11|01144467789
      12|12566889
      13|24778
      14|047
      15|223458
      16|4
      17|11557
      18|000247
      19|4467799
      20|00
      21|1
      22|2335
      23|01457
      24|12356
      25|455
      27|79
      key: 6|3 =  6.3
      (%o3)                  done

  ·

  @ref{Category: Package descriptive} · @ref{Category: Plotting}

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