## 51 Package descriptive ¶

### 51.1 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 ’riotorto AT yahoo DOT com’.

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 discussed 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 assigned 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 ]

Categories: Descriptive statistics · Share packages · Package descriptive ·

### 51.2 Functions and Variables for data manipulation ¶

Function: build_sample
`build_sample` (list)
`build_sample` (matrix) ¶

  Builds a sample from a table of absolute frequencies. The input table can be a matrix or a list of lists, all of them of equal size. The number of columns or the length of the lists must be greater than 1. The last element of each row or list is interpreted as the absolute frequency. The output is always a sample in matrix form.

  Examples:

  Univariate frequency table.

      (%i1) load ("descriptive")$

      (%i2) sam1: build_sample([[6,1], [j,2], [2,1]]);
                                    [ 6 ]
                                    [   ]
                                    [ j ]
      (%o2)                         [   ]
                                    [ j ]
                                    [   ]
                                    [ 2 ]

      (%i3) mean(sam1);
                                    j + 4
      (%o3)                        [-----]
                                      2

      (%i4) barsplot(sam1) $

  Multivariate frequency table.

      (%i1) load ("descriptive")$

      (%i2) sam2: build_sample([[6,3,1], [5,6,2], [u,2,1],[6,8,2]]) ;
                                  [ 6  3 ]
                                  [      ]
                                  [ 5  6 ]
                                  [      ]
                                  [ 5  6 ]
      (%o2)                       [      ]
                                  [ u  2 ]
                                  [      ]
                                  [ 6  8 ]
                                  [      ]
                                  [ 6  8 ]

      (%i3) cov(sam2);
            [   2                 2                            ]
            [  u  + 158   (u + 28)     2 u + 174   11 (u + 28) ]
            [  -------- - ---------    --------- - ----------- ]
      (%o3) [     6          36            6           12      ]
            [                                                  ]
            [ 2 u + 174   11 (u + 28)            21            ]
            [ --------- - -----------            --            ]
            [     6           12                 4             ]

      (%i4) barsplot(sam2, grouping=stacked) $

  Categories: Package descriptive ·

Function: continuous_freq
`continuous_freq` (data)
`continuous_freq` (data, m) ¶

  Divides the range of data into intervals, and counts how many values fall into each one.

  A value x falls into an interval with left and right endpoints a and b if and only if `x > a` and `x <= b`, except for the first (least or leftmost) interval, for which `x >= a` and `x <= b`. That is, an interval excludes its left endpoint and includes its right endpoint, except for the first interval, which includes both the left and right endpoints.

  data must be a list of numbers, or 1-dimensional array (as created by `make_array`).

  m is optional, and equals either the number of classes (10 by default), or a list of two elements (the least and greatest values to be counted), or a list of three elements (the least and greatest values to be counted, and the number of classes), or a set containing the endpoints of the class intervals.

  It is assumed that class intervals are contiguous. That is, the right endpoint of one interval is equal to the left endpoint of the next.

  `continuous_freq` returns a list of two lists. The first list comprises all the endpoints of the class intervals, concatenated into a single list. The second list contains the class counts for the intervals corresponding to elements of the first list.

  If sample values are all equal, this function returns exactly one class of width 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);
                     9  18  27  36
      (%o3)     [[0, -, --, --, --, 9], [16, 24, 18, 17, 25]]
                     5  5   5   5

  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]]

  The first argument may be an array.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) a1 : make_array (fixnum, length (s1)) $

      (%i4) fillarray (a1, s1);
      (%o4) {Lisp Array: #(3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2\
       6 4 3 3 8 3 2 7 9 5
                     0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7\
       4 9 4 4 5 9 2
                     3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8\
       2 5 3 4 2 1 1
                     7 0 6 7)}

      (%i5) continuous_freq (a1);
                 9   9  27  18  9  27  63  36  81
      (%o5) [[0, --, -, --, --, -, --, --, --, --, 9],
                 10  5  10  5   2  5   10  5   10
                                   [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

  Categories: Package descriptive ·

Function: discrete_freq (data) ¶

  Counts absolute frequencies in discrete samples, both numeric and categorical. Its sole argument is a list, or 1-dimensional array (as created by `make_array`).

  Examples:

      (%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]]

  In the return value, the first list gives the sample values, and the second, their absolute frequencies.

  The argument may be an array.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) a1 : make_array (fixnum, length (s1)) $

      (%i4) fillarray (a1, s1);
      (%o4) {Lisp Array: #(3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2\
       6 4 3 3 8 3 2 7 9 5
                     0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7\
       4 9 4 4 5 9 2
                     3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8\
       2 5 3 4 2 1 1
                     7 0 6 7)}

      (%i5) discrete_freq (a1);
      (%o5) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
                                   [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

  Categories: Package descriptive ·

Function: standardize
`standardize` (list)
`standardize` (matrix) ¶

  Subtracts to each element of the list the sample mean and divides the result by the standard deviation. When the input is a matrix, `standardize` subtracts to each row the multivariate mean, and then divides each component by the corresponding standard deviation.

  Categories: Package descriptive ·

Function: subsample
`subsample` (data_matrix, predicate_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.

  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 referred 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);
             13 B + 7 A  317
      (%o3) [----------, ---, 87.178, 0.06 NA + 81.44999999999999,
                 20      10
                                                          3 NA + 19587
                                      18.122999999999998, ------------]
                                                              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

  Categories: Package descriptive ·

Function: transform_sample (matrix, varlist, exprlist) ¶

  Transforms the sample matrix, where each column is called according to varlist, following expressions in exprlist.

  Examples:

  The second argument assigns names to the three columns. With these names, a list of expressions define the transformation of the sample.

      (%i1) load ("descriptive")$
      (%i2) data: matrix([3,2,7],[3,7,2],[8,2,4],[5,2,4]) $

      (%i3) transform_sample(data, [a,b,c], [c, a*b, log(a)]);
                                     [ 7  6   log(3) ]
                                     [               ]
                                     [ 2  21  log(3) ]
      (%o3)                          [               ]
                                     [ 4  16  log(8) ]
                                     [               ]
                                     [ 4  10  log(5) ]

  Add a constant column and remove the third variable.

      (%i1) load ("descriptive")$
      (%i2) data: matrix([3,2,7],[3,7,2],[8,2,4],[5,2,4]) $
      (%i3) transform_sample(data, [a,b,c], [makelist(1,k,length(data)),a,b]);

                                        [ 1  3  2 ]
                                        [         ]
                                        [ 1  3  7 ]
      (%o3)                             [         ]
                                        [ 1  8  2 ]
                                        [         ]
                                        [ 1  5  2 ]

  Categories: Package descriptive ·

### 51.3 Functions and Variables for descriptive statistics ¶

Function: mean
`mean` (x)
`mean` (x, w) ¶

  Returns the sample mean. x must be a list or matrix.

  When x is a list, `mean` returns the sample mean of x.

  When x is a matrix, `mean` returns a list comprising the sample mean of each column.

  w is an optional per-datum weight. w must either be 1, in which case every datum x[i] is given equal weight, or a list of the same length as x, in which case the weight for x[i] is given by w[i]. The elements of w must be nonnegative and not all zero; it is not required that they sum to 1.

  The unweighted sample mean is defined as

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

  The weighted sample mean is defined as

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

  where Z is the sum of the weights,

                         n
                        ====
                        \
                   Z =   >    w
                        /      i
                        ====
                        i = 1

  Examples:

  Sample mean of a list.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$

      (%i3) mean (s1);
                                     471
      (%o3)                          ---
                                     100

  Sample mean of each column of a matrix.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$

      (%i3) mean (s2);
      (%o3) [9.9485, 10.160700000000004, 10.868499999999997,
                                15.716600000000001, 14.844100000000001]

  Weighted sample mean of a list.

      (%i1) load ("descriptive")$

      (%i2) mean ([a, b, c, d], [1, 2, 3, 4]);
                             4 d + 3 c + 2 b + a
      (%o2)                  -------------------
                                     10

  Weighted sample mean of each column of a matrix.

      (%i1) load ("descriptive")$

      (%i2) mm: matrix ([p, q, r], [s, t, u]);
                                 [ p  q  r ]
      (%o2)                      [         ]
                                 [ s  t  u ]

      (%i3) mean (mm, [vv, ww]);
                    s ww + p vv  t ww + q vv  u ww + r vv
      (%o3)        [-----------, -----------, -----------]
                      ww + vv      ww + vv      ww + vv

  Categories: Package descriptive ·

Function: var
`var` (x)
`var` (x, w) ¶

  Returns the sample variance. x must be a list or matrix.

  When x is a list, `var` returns the sample variance of x.

  When x is a matrix, `var` returns a list comprising the sample variance of each column.

  w is an optional per-datum weight. w must either be 1, in which case every datum x[i] is given equal weight, or a list of the same length as x, in which case the weight for x[i] is given by w[i]. The elements of w must be nonnegative and not all zero; it is not required that they sum to 1.

  The unweighted sample variance is defined as

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

  The weighted sample variance is defined as

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

  where Z is the sum of the weights,

                         n
                        ====
                        \
                   Z =   >    w
                        /      i
                        ====
                        i = 1

  Example:

  Sample variance of a list.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$

      (%i3) var (s1), numer;
      (%o3)                   8.425899999999999

  Sample variance of each column of a matrix.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$

      (%i3) var (s2);
      (%o3) [17.22190675000001, 14.987736510000005,
             15.475728749999998, 32.17651044000001, 24.423076190000007]

  Weighted sample variance of a list.

      (%i1) load ("descriptive")$

      (%i2) var ([a - b, a, a + b], [3, 5, 7]);
                                        2
                                   134 b
      (%o2)                        ------
                                    225

  Weighted sample variance of each column of a matrix.

      (%i1) load ("descriptive")$

      (%i2) mm: matrix ([a - b, c - d], [a, c], [a + b, c + d]);
                              [ a - b  c - d ]
                              [              ]
      (%o2)                   [   a      c   ]
                              [              ]
                              [ b + a  d + c ]

      (%i3) var (mm, [3, 5, 7]);
                                    2       2
                               134 b   134 d
      (%o3)                   [------, ------]
                                225     225

  See also function `var1`.

  Categories: Package descriptive ·

Function: var1
`var1` (list)
`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) [17.395865404040414, 15.139127787878794,
             15.632049242424243, 32.50152569696971, 24.669773929292937]

  See also function `var`.

  Categories: Package descriptive ·

Function: std
`std` (x)
`std` (x, w) ¶

  Returns the sample standard deviation. x must be a list or matrix.

  When x is a list, `std` returns the sample standard deviation of x, which is defined as the square root of the sample variance, as computed by `var`.

  When x is a matrix, `std` returns a list comprising the sample standard deviation of each column.

  w is an optional per-datum weight. w must either be 1, in which case every datum x[i] is given equal weight, or a list of the same length as x, in which case the weight for x[i] is given by w[i]. The elements of w must be nonnegative and not all zero; it is not required that they sum to 1.

  Example:

  Sample standard deviation of a list.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$

      (%i3) std (s1), numer;
      (%o3)                  2.9027400848164135

  Sample standard deviation of each column of a matrix.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$

      (%i3) std (s2);
      (%o3) [4.149928523480858, 3.8713998127292415,
              3.9339202775348663, 5.672434260526957, 4.941970881136392]

  See also functions `var` and `std1`.

  Categories: Package descriptive ·

Function: std1
`std1` (list)
`std1` (matrix) ¶

  This is the square root of the 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, 3.8909032097803196,
              3.9537386411375555, 5.701010936401517, 4.966867617451963]

  See also functions `var1` and `std`.

  Categories: Package descriptive ·

Function: noncentral_moment
`noncentral_moment` (x, k)
`noncentral_moment` (x, k, w) ¶

  Returns the noncentral moment of order k. x must be a list or matrix.

  When x is a list, `noncentral_moment` returns the noncentral moment of order k of x.

  When x is a matrix, `noncentral_moment` returns a list comprising the noncentral moment of order k of each column.

  w is an optional per-datum weight. w must either be 1, in which case every datum x[i] is given equal weight, or a list of the same length as x, in which case the weight for x[i] is given by w[i]. The elements of w must be nonnegative and not all zero; it is not required that they sum to 1.

  The unweighted noncentral moment of order k is defined as

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

  The weighted noncentral moment of order k is defined as

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

  where Z is the sum of the weights,

                         n
                        ====
                        \
                   Z =   >    w
                        /      i
                        ====
                        i = 1

  Examples:

  First noncentral moment of a list. The first noncentral moment is equal to the sample mean.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$

      (%i3) noncentral_moment (s1, 1), numer;
      (%o3)                         4.71

      (%i4) mean (s1), numer;
      (%o4)                         4.71

  Fifth noncentral moment of each column of a matrix.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$

      (%i3) noncentral_moment (s2, 5);
      (%o3) [319793.87247615046, 320532.19238924625,
             391249.56213815557, 2502278.205988911, 1691881.7977422548]

  See also function `central_moment`.

  Categories: Package descriptive ·

Function: central_moment
`central_moment` (x, k)
`central_moment` (x, k, w) ¶

  Returns the central moment of order k. x must be a list or matrix.

  When x is a list, `central_moment` returns the central moment of order k of x.

  When x is a matrix, `central_moment` returns a list comprising the central moment of order k of each column.

  w is an optional per-datum weight. w must either be 1, in which case every datum x[i] is given equal weight, or a list of the same length as x, in which case the weight for x[i] is given by w[i]. The elements of w must be nonnegative and not all zero; it is not required that they sum to 1.

  The unweighted central moment of order k is defined as

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

  The weighted central moment of order k is defined as

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

  where Z is the sum of the weights,

                         n
                        ====
                        \
                   Z =   >    w
                        /      i
                        ====
                        i = 1

  Examples:

  Second central moment of a list. The second central moment is equal to the sample variance.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$

      (%i3) central_moment (s1, 2), numer;
      (%o3)                   8.425899999999999

      (%i4) var (s1), numer;
      (%o4)                   8.425899999999999

  Third central moment of each column of a matrix.

      (%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`.

  Categories: Package descriptive ·

Function: cv
`cv` (list)
`cv` (matrix) ¶

  Returns the variation coefficient, defined as the sample standard deviation `std` divided by the `mean`.

  Examples:

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$

      (%i3) cv (s1), numer;
      (%o3)                  0.6162930116383044

      (%i4) s2 : read_matrix (file_search ("wind.data"))$

      (%i5) cv (s2);
      (%o5) [0.4171411291632767, 0.38101703748061055,
            0.3619561372346568, 0.3609199356430116, 0.3329249251309538]

  See also functions `std` and `mean`.

  Categories: Package descriptive ·

Function: smin
`smin` (list)
`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.

  Examples:

      (%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`.

  Categories: Package descriptive ·

Function: smax
`smax` (list)
`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.

  Examples:

      (%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`.

  Categories: Package descriptive ·

Function: range
`range` (list)
`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, 17.369999999999997, 24.38, 22.46]

  Categories: Package descriptive ·

Function: quantile
`quantile` (list, p)
`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 Package descriptive

  Examples:

  Input is a list. First and third quartiles are computed.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$

      (%i3) [quantile (s1, 1/4), quantile (s1, 3/4)], numer;
      (%o3)                      [2.0, 7.25]

  Input is a matrix. First quartile is computed for each column.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$

      (%i3) quantile (s2, 1/4);
      (%o3)    [7.2575, 7.477500000000001, 7.82, 11.28, 11.48]

  Categories: Package descriptive ·

Function: median
`median` (list)
`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)   [10.059999999999999, 9.855, 10.73, 15.48, 14.105]

  The median is the 1/2-quantile.

  See also function `quantile`.

  Categories: Package descriptive ·

Function: qrange
`qrange` (x) ¶

  Returns the interquartile range, defined as the difference between the third and first quartiles: `quantile(x, 3/4) - quantile(x, 1/4)`

  x must be a list or matrix. When x is a matrix, `qrange` returns the interquartile range for each column.

  Examples:

      (%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`.

  Categories: Package descriptive ·

Function: mean_deviation
`mean_deviation` (list)
`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) [3.2879599999999987, 3.075342, 3.2390700000000003,
                                  4.715664000000001, 4.028546000000002]

  See also function `mean`.

  Categories: Package descriptive ·

Function: median_deviation
`median_deviation` (list)
`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, 2.7550000000000003, 3.08, 4.315, 3.3099999999999996]

  See also function `mean`.

  Categories: Package descriptive ·

Function: harmonic_mean
`harmonic_mean` (list)
`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)                  3.9018580276322052

      (%i4) s2 : read_matrix (file_search ("wind.data"))$

      (%i5) harmonic_mean (s2);
      (%o5) [6.948015590052786, 7.391967752360356, 9.055658197151745,
                                 13.441990281936924, 13.01439145898509]

  See also functions `mean` and `geometric_mean`.

  Categories: Package descriptive ·

Function: geometric_mean
`geometric_mean` (list)
`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, 10.044267571488904,
                                 14.612741263490207, 13.96184163444275]

  See also functions `mean` and `harmonic_mean`.

  Categories: Package descriptive ·

Function: kurtosis
`kurtosis` (list)
`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) [- 0.2715445622195385, 0.119998784429451,
      - 0.42752334904828615, - 0.6405361979019522,
      - 0.4952382132352935]

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

  Categories: Package descriptive ·

Function: skewness
`skewness` (list)
`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)                 0.009196180476450424

      (%i4) s2 : read_matrix (file_search ("wind.data"))$

      (%i5) skewness (s2);
      (%o5) [0.1580509020000978, 0.2926379232061854,
         0.09242174416107717, 0.20599843481486865, 0.21425202488908313]

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

  Categories: Package descriptive ·

Function: pearson_skewness
`pearson_skewness` (list)
`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)                  0.21594840290938955

      (%i4) s2 : read_matrix (file_search ("wind.data"))$

      (%i5) pearson_skewness (s2);
      (%o5) [- 0.08019976629211892, 0.2357036272952649,
         0.10509040624912039, 0.12450423405923679, 0.44641817958045193]

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

  Categories: Package descriptive ·

Function: quartile_skewness
`quartile_skewness` (list)
`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)                 0.047619047619047616

      (%i4) s2 : read_matrix (file_search ("wind.data"))$

      (%i5) quartile_skewness (s2);
      (%o5) [- 0.040854224698235304, 0.14670255720053824,
         0.033623910336239196, 0.03780068728522298, 0.2105263157894735]

  See also function `quantile`.

  Categories: Package descriptive ·

Function: km
`km` (list, option ...)
`km` (matrix, option ...) ¶

  Kaplan Meier estimator of the survival, or reliability, function \(S(x)=1-F(x)\).

  Data can be introduced as a list of pairs, or as a two column matrix. The first component is the observed time, and the second component a censoring index (1 = non censored, 0 = right censored).

  The optional argument is the name of the variable in the returned expression, which is x by default.

  Examples:

  Sample as a list of pairs.

      (%i1) load ("descriptive")$

      (%i2) S: km([[2,1], [3,1], [5,0], [8,1]]);
                             charfun((3 <= x) and (x < 8))
      (%o2) charfun(x < 0) + -----------------------------
                                           2
         3 charfun((2 <= x) and (x < 3))
       + -------------------------------
                        4
       + charfun((0 <= x) and (x < 2))

      (%i3) load ("draw")$

      (%i4) draw2d(
        line_width = 3, grid = true,
        explicit(S, x, -0.1, 10))$

  Estimate survival probabilities.

      (%i1) load ("descriptive")$
      (%i2) S(t):= ''(km([[2,1], [3,1], [5,0], [8,1]], t)) $

      (%i3) S(6);
                                      1
      (%o3)                           -
                                      2

  Categories: Package descriptive ·

Function: cdf_empirical
`cdf_empirical` (list, option ...)
`cdf_empirical` (matrix, option ...) ¶

  Empirical distribution function \(F(x)\).

  Data can be introduced as a list of numbers, or as an one column matrix.

  The optional argument is the name of the variable in the returned expression, which is x by default.

  Example:

  Empirical distribution function.

      (%i1) load ("descriptive")$

      (%i2) F(x):= ''(cdf_empirical([1,3,3,5,7,7,7,8,9]));
      (%o2) F(x) := (charfun(x >= 9) + charfun(x >= 8)
       + 3 charfun(x >= 7) + charfun(x >= 5) + 2 charfun(x >= 3)
       + charfun(x >= 1))/9

      (%i3) F(6);
                                      4
      (%o3)                           -
                                      9

      (%i4) load("draw")$

      (%i5) draw2d(
         line_width = 3,
         grid       = true,
         explicit(F(z), z, -2, 12)) $

  Categories: Package descriptive ·

Function: cov
`cov` (X)
`cov` (X, w) ¶

  Returns the sample covariance matrix. X must be a matrix.

  The sample covariance matrix has the same number of rows and columns, both equal to the number of columns of X; each diagonal element X[i, i] is equal to the sample variance of the i’th column, and each off-diagonal element X[i, j] is equal to the sample covariance of the i’th and j’th columns.

  w is an optional per-datum weight. w must either be 1, in which case every datum X[i] is given equal weight, or a list of the same length as X, in which case the weight for X[i] is given by w[i]. The elements of w must be nonnegative and not all zero; it is not required that they sum to 1.

  The unweighted sample covariance is 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.

  The weighted sample covariance is defined as

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

  where Z is the sum of the weights,

                         n
                        ====
                        \
                   Z =   >    w
                        /      i
                        ====
                        i = 1

  Example:

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) fpprintprec : 7$

      (%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`.

  Categories: 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$

      (%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`.

  Categories: Package descriptive ·

Function: global_variances
`global_variances` (matrix)
`global_variances` (matrix, options ...) ¶

  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))`,
  - effective variance `determinant(S_1)^(1/p)`, (defined in: Peña, D. (2002) Análisis de datos multivariantes; McGraw-Hill, Madrid.)
  - effective 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`.

  Option:

  - `'data`, default `'true`, indicates whether the input matrix contains the sample data, in which case the covariance matrix `cov1` must be calculated, or not, and then the covariance matrix (symmetric) must be given, instead of the data.

  Examples:

  Calculate the `global_variances` from sample data.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$

      (%i3) global_variances (s2);
      (%o3) [105.33834206060595, 21.06766841212119, 12874.34690469686,
             113.46517926085015, 6.636590811800794, 2.5761581496097623]

  Calculate the `global_variances` from the covariance matrix.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) s : cov1 (s2)$

      (%i4) global_variances (s, data=false);
      (%o4) [105.33834206060595, 21.06766841212119, 12874.34690469686,
             113.46517926085015, 6.636590811800794, 2.5761581496097623]

  See also `cov` and `cov1`.

  Categories: Package descriptive ·

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

  The correlation matrix of the multivariate sample.

  Option:

  - `'data`, default `'true`, indicates whether the input matrix contains the sample data, in which case the covariance matrix `cov1` must be calculated, or not, and then the covariance matrix (symmetric) must be given, instead of the data.

  Examples:

      (%i1) load ("descriptive")$
      (%i2) fpprintprec : 7 $
      (%i3) s2 : read_matrix (file_search ("wind.data"))$

      (%i4) cor (s2);
            [    1.0     0.8476339  0.8803515  0.8239624  0.7519506 ]
            [                                                       ]
            [ 0.8476339     1.0     0.8735834  0.6902622  0.782502  ]
            [                                                       ]
      (%o4) [ 0.8803515  0.8735834     1.0     0.7764065  0.8323358 ]
            [                                                       ]
            [ 0.8239624  0.6902622  0.7764065     1.0     0.7293848 ]
            [                                                       ]
            [ 0.7519506  0.782502   0.8323358  0.7293848     1.0    ]

  Calculate the correlation matrix from the covariance matrix.

      (%i1) load ("descriptive")$
      (%i2) fpprintprec : 7 $
      (%i3) s2 : read_matrix (file_search ("wind.data"))$
      (%i4) s : cov1 (s2)$

      (%i5) cor (s, data=false); /* this is faster */
            [    1.0     0.8476339  0.8803515  0.8239624  0.7519506 ]
            [                                                       ]
            [ 0.8476339     1.0     0.8735834  0.6902622  0.782502  ]
            [                                                       ]
      (%o5) [ 0.8803515  0.8735834     1.0     0.7764065  0.8323358 ]
            [                                                       ]
            [ 0.8239624  0.6902622  0.7764065     1.0     0.7293848 ]
            [                                                       ]
            [ 0.7519506  0.782502   0.8323358  0.7293848     1.0    ]

  See also `cov` and `cov1`.

  Categories: Package descriptive ·

Function: list_correlations
`list_correlations` (matrix)
`list_correlations` (matrix, options ...) ¶

  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  |
                           \       /

  Option:

  - `'data`, default `'true`, indicates whether the input matrix contains the sample data, in which case the covariance matrix `cov1` must be calculated, or not, and then the covariance matrix (symmetric) must be given, instead of the data.

  Example:

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) z : list_correlations (s2)$
      (%i4) fpprintprec : 5$

      (%i5) precision_matrix: z[1];
      (%o5)
          [  0.38486   - 0.13856   - 0.15626   - 0.10239    0.031179  ]
          [                                                           ]
          [ - 0.13856   0.34107    - 0.15233    0.038447   - 0.052842 ]
          [                                                           ]
          [ - 0.15626  - 0.15233    0.47296    - 0.024816  - 0.10054  ]
          [                                                           ]
          [ - 0.10239   0.038447   - 0.024816   0.10937    - 0.034033 ]
          [                                                           ]
          [ 0.031179   - 0.052842  - 0.10054   - 0.034033   0.14834   ]

      (%i6) multiple_correlation_vector: z[2];
      (%o6)     [0.85063, 0.80634, 0.86474, 0.71867, 0.72675]

      (%i7) partial_correlation_matrix: z[3];
            [   - 1.0     0.38244   0.36627   0.49908   - 0.13049 ]
            [                                                     ]
            [  0.38244     - 1.0    0.37927  - 0.19907   0.23492  ]
            [                                                     ]
      (%o7) [  0.36627    0.37927    - 1.0    0.10911    0.37956  ]
            [                                                     ]
            [  0.49908   - 0.19907  0.10911    - 1.0     0.26719  ]
            [                                                     ]
            [ - 0.13049   0.23492   0.37956   0.26719     - 1.0   ]

  See also `cov` and `cov1`.

  Categories: Package descriptive ·

Function: principal_components
`principal_components` (matrix)
`principal_components` (matrix, options ...) ¶

  Calculates the principal components of a multivariate sample. Principal components are used in multivariate statistical analysis to reduce the dimensionality of the sample.

  Option:

  - `'data`, default `'true`, indicates whether the input matrix contains the sample data, in which case the covariance matrix `cov1` must be calculated, or not, and then the covariance matrix (symmetric) must be given, instead of the data.

  The output of function `principal_components` is a list with the following results:

  - variances of the principal components,
  - percentage of total variance explained by each principal component,
  - rotation matrix.

  Examples:

  In this sample, the first component explains 83.13 per cent of total variance.

      (%i1) load ("descriptive")$
      (%i2) s2 : read_matrix (file_search ("wind.data"))$
      (%i3) fpprintprec:4 $
      (%i4) res: principal_components(s2);
      0 errors, 0 warnings
      (%o4) [[87.57, 8.753, 5.515, 1.889, 1.613],
      [83.13, 8.31, 5.235, 1.793, 1.531],

      [ .4149  .03379   - .4757  - 0.581   - .5126 ]
      [                                            ]
      [ 0.369  - .3657  - .4298   .7237    - .1469 ]
      [                                            ]
      [ .3959  - .2178  - .2181  - .2749    .8201  ]]
      [                                            ]
      [ .5548   .7744    .1857    .2319    .06498  ]
      [                                            ]
      [ .4765  - .4669   0.712   - .09605  - .1969 ]

      (%i5) /* accumulated percentages  */
          block([ap: copy(res[2])],
            for k:2 thru length(ap) do ap[k]: ap[k]+ap[k-1],
            ap);
      (%o5)                 [83.13, 91.44, 96.68, 98.47, 100.0]
      (%i6) /* sample dimension */
            p: length(first(res));
      (%o6)                                  5
      (%i7) /* plot percentages to select number of
               principal components for further work */
           draw2d(
              fill_density = 0.2,
              apply(bars, makelist([k, res[2][k], 1/2], k, p)),
              points_joined = true,
              point_type    = filled_circle,
              point_size    = 3,
              points(makelist([k, res[2][k]], k, p)),
              xlabel = "Variances",
              ylabel = "Percentages",
              xtics  = setify(makelist([concat("PC",k),k], k, p))) $

  In case the covariance matrix is known, it can be passed to the function, but option `data=false` must be used.

      (%i1) load ("descriptive")$
      (%i2) S: matrix([1,-2,0],[-2,5,0],[0,0,2]);
                                      [  1   - 2  0 ]
                                      [             ]
      (%o2)                           [ - 2   5   0 ]
                                      [             ]
                                      [  0    0   2 ]
      (%i3) fpprintprec:4 $
      (%i4) /* the argument is a covariance matrix */
            res: principal_components(S, data=false);
      0 errors, 0 warnings
                                                        [ - .3827  0.0  .9239 ]
                                                        [                     ]
      (%o4) [[5.828, 2.0, .1716], [72.86, 25.0, 2.145], [  .9239   0.0  .3827 ]]
                                                        [                     ]
                                                        [   0.0    1.0   0.0  ]
      (%i5) /* transformation to get the principal components
               from original records */
            matrix([a1,b2,c3],[a2,b2,c2]).last(res);
                   [ .9239 b2 - .3827 a1  1.0 c3  .3827 b2 + .9239 a1 ]
      (%o5)        [                                                  ]
                   [ .9239 b2 - .3827 a2  1.0 c2  .3827 b2 + .9239 a2 ]

  Categories: Package descriptive ·

### 51.4 Functions and Variables for statistical graphs ¶

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

  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 necessary 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 Package draw options: `key`, `color_draw`, `fill_color`, `fill_density` and `line_width`. See also `barsplot`.

  There is also a function `wxbarsplot` for creating embedded histograms in interfaces wxMaxima and iMaxima. `barsplot` in a multiplot context.

  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)$

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

  Categories: Package descriptive · Plotting ·

Function: barsplot_description (…) ¶

  Function `barsplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects.

  Example: `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))$

  Categories: Package descriptive · Plotting ·

Function: boxplot (data)
`boxplot` (data, option_1, option_2, …) ¶

  This function plots box-and-whisker 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.

  The box is plotted from the first quartile to the third, with an horizontal segment situated at the second quartile or median. By default, lower and upper whiskers are plotted at the minimum and maximum values, respectively. Option range can be used to indicate that values greater than `quantile(x,3/4)+range*(quantile(x,3/4)-quantile(x,1/4))` or less than `quantile(x,1/4)-range*(quantile(x,3/4)-quantile(x,1/4))` must be considered as outliers, in which case they are plotted as isolated points, and the whiskers are located at the extremes of the rest of the sample.

  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`.
  - range (default, `inf`): positive coefficient of the interquartilic range to set outliers boundaries.
  - outliers_size (default, `1`): circle size for isolated outliers.
  - 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`.

  There is also a function `wxboxplot` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  Examples:

  Box-and-whisker 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-whisker 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)$

  Option range can be used to handle outliers.

      (%i1)  load ("descriptive")$
       B: [[7, 15, 5, 8, 6, 5, 7, 3, 1],
           [10, 8, 12, 8, 11, 9, 20],
           [23, 17, 19, 7, 22, 19]] $
       boxplot (B, range=1)$
       boxplot (B, range=1.5, box_orientation = horizontal)$
       draw2d(
          boxplot_description(
             B,
             range            = 1.5,
             line_width       = 3,
             outliers_size    = 2,
             color            = red,
             background_color = light_gray),
          xtics = {["Low",1],["Medium",2],["High",3]}) $

  Categories: Package descriptive · Plotting ·

Function: boxplot_description (…) ¶

  Function `boxplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects.

  Categories: Package descriptive · Plotting ·

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

  Constructs and displays a histogram from a data sample. Data must be stored as a list of numbers, or a matrix of one row or one column.

  Optional arguments:

  - `nclasses` (default, 10): the number of classes (also called bins) in the histogram, or a list of two numbers (the least and greatest values included in the histogram), or a list of three numbers (the least and greatest values included in the histogram, and the number of classes), or a set containing the endpoints of the class intervals, or a symbol specifying the name of one of three algorithms to automatically determine the number of classes: `fd` (Ref. [1]), `scott` (Ref. [2]), or `sturges` (Ref. [3]).

    A class interval excludes its left endpoint and includes its right endpoint, except for the first interval, which includes both the left and right endpoints. It is assumed that class intervals are contiguous. That is, the right endpoint of one interval is equal to the left endpoint of the next.

  - `frequency` (default, `absolute`): indicates the scale of the vertical axis. Possible values are: `absolute` (heights of bars add up to number of data), `relative` (heights of bars add up to 1), `percent` (heights of bars add up to 100), and `density` (total area of histogram is 1).
  - `htics` (default, `auto`): format of tic marks on the horizontal axis. Possible values are: `auto` (tics are placed automatically), `endpoints` (tics are placed at the divisions between classes), `intervals` (classes are labeled with the corresponding intervals), or a list of labels, one for each class.
  - 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`.
  - The following local Package draw options: `key`, `fill_color`, `fill_density`, and `line_width`. Note that the outlines of bars, as well as the interior of bars when `fill_density` is nonzero, are drawn with `fill_color`, not `color`.

  `histogram` honors the global option `histogram_skyline`. When `histogram_skyline` is `true`, `histogram` and `histogram_description` construct "skyline" plots, which shows the outline of the histogram bars, instead of drawing all the vertical segments. Otherwise (the default), histograms are displayed with bars showing vertical segments.

  There is also a function `wxhistogram` for creating embedded histograms in interfaces wxMaxima and iMaxima.

  See also `continuous_freq`, which, like `histogram`, counts data in intervals, but returns the counts instead of displaying a graphic representation.

  See also `barsplot`.

  Examples:

  A simple histogram 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)$

  Bounds for varying class widths.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) histogram (s1, nclasses = {0,3,6,7,11})$

  Freedman-Diaconis formula for the number of classes.

      (%i1) load ("descriptive")$
      (%i2) s1 : read_list (file_search ("pidigits.data"))$
      (%i3) histogram(s1, nclasses=fd) $

  References:

  [1] Freedman, D., and Diaconis, P. (1981) On the histogram as a density estimator: L_2 theory. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 57, 453-476.

  [2] Scott, D. W. (1979) On optimal and data-based histograms. Biometrika 66, 605-610.

  [3] Sturges, H. A. (1926) The choice of a class interval. Journal of the American Statistical Association 21, 65-66.

  Categories: Package descriptive · Plotting ·

Function: histogram_description (…) ¶

  Creates a graphic object which represents a histogram. Such an object is suitable for creating complex scenes together with other graphic objects, to be displayed by `draw2d`.

  `histogram_description` takes the same arguments as the stand-alone function `histogram`. See `histogram` for more information.

  Example:

  We make use of `histogram_description` for setting `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    = density,
                fill_density = 0.5),
              explicit(pdf_normal(x,m,s), x, m - 3*s, m + 3* s))$

  Categories: Package descriptive · Plotting ·

Option variable: histogram_skyline ¶

  Default value: `false`

  When `histogram_skyline` is `true`, `histogram` and `histogram_description` construct "skyline" plots, which shows the outline of the histogram bars, instead of drawing all the vertical segments.

  The outline is drawn with the current `fill_color` (not the current `color`). The interior of the histogram is filled with `fill_color`, but only if `fill_density` is nonzero.

  Otherwise, histograms are displayed with bars showing vertical segments.

  Examples:

  Construct a skyline histogram, and an ordinary histogram for comparison, on the same plot.

      (%i1) load ("descriptive") $
      (%i2) L: read_list (file_search ("pidigits.data")) $
      (%i3) histogram_skyline: true $
      (%i4) skyline_hist: histogram_description (L) $
      (%i5) histogram_skyline: false $
      (%i6) ordinary_hist: histogram_description (L) $
      (%i7) draw (gr2d (skyline_hist), gr2d (ordinary_hist)) $

  Continuing the preceding example. Set display options for `fill_color` and `fill_density`.

      (%i8) histogram_skyline: true $
      (%i9) skyline_hist: histogram_description (L, fill_color = blue, fill_density = 0.2) $
      (%i10) histogram_skyline: false $
      (%i11) ordinary_hist: histogram_description (L, fill_color = blue, fill_density = 0.2) $
      (%i12) draw (gr2d (skyline_hist), gr2d (ordinary_hist)) $

  Categories: Package descriptive · Plotting ·

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

  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 necessary 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`

  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`.

  Categories: Package descriptive · Plotting ·

Function: piechart_description (…) ¶

  Function `piechart_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects.

  Categories: Package descriptive · Plotting ·

Function: scatterplot
`scatterplot` (list)
`scatterplot` (list, option_1, option_2, …)
`scatterplot` (matrix)
`scatterplot` (matrix, option_1, option_2, …) ¶

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

  Available options are the same admitted by `histogram`.

  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`.

  Categories: Package descriptive · Plotting ·

Function: scatterplot_description (…) ¶

  Function `scatterplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects.

  Categories: Package descriptive · Plotting ·

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

  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 necessary 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`.

  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 ) $

  Categories: Package descriptive · Plotting ·

Function: starplot_description (…) ¶

  Function `starplot_description` creates a graphic object suitable for creating complex scenes, together with other graphic objects.

  Categories: Package descriptive · Plotting ·

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

  Plots stem and leaf diagrams.

  The only 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

  Categories: Package descriptive · Plotting ·

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