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<h1 class="chapter"> 46. descriptive </h1>

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<tr><td align="left" valign="top"><a href="#SEC176">46.1 Introduction to descriptive</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC177">46.2 Definitions for data manipulation</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC178">46.3 Definitions for descriptive statistics</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC179">46.4 Definitions for specific multivariate descriptive statistics</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC180">46.5 Definitions for statistical graphs</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
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<h2 class="section"> 46.1 Introduction to descriptive </h2>

<p>Package <code>descriptive</code> 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: <code>pidigits.data</code>, <code>wind.data</code> and <code>biomed.data</code>. They can be also downloaded from the web site <code>www.biomates.net</code>.
</p>
<p>Any statistics manual can be used as a reference to the functions in package <code>descriptive</code>.
</p>
<p>For comments, bugs or suggestions, please contact me at <var>'mario AT edu DOT xunta DOT es'</var>.
</p>
<p>Here is a simple example on how the descriptive functions in <code>descriptive</code> do they work, depending on the nature of their arguments, lists or matrices,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%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
</pre></td></tr></table>
<p>Note that in multivariate samples the mean is calculated for each column.
</p>
<p>In case of several samples with possible different sizes, the Maxima function <code>map</code> can be used to get the desired results for each sample,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) map (mean, [[a, b, c], [d, e]]);
                        c + b + a  e + d
(%o2)                  [---------, -----]
                            3        2
</pre></td></tr></table>
<p>In this case, two samples of sizes 3 and 2 were stored into a list.
</p>
<p>Univariate samples must be stored in lists like
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%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]
</pre></td></tr></table>
<p>and multivariate samples in matrices as in
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%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  ]
</pre></td></tr></table>
<p>In this case, the number of columns equals the random variable dimension and the number of rows is the sample size.
</p>
<p>Data can be introduced by hand, but big samples are usually stored in plain text files. For example, file <code>pidigits.data</code> contains the first 100 digits of number <code>%pi</code>:
</p><table><tr><td>&nbsp;</td><td><pre class="example">      3
      1
      4
      1
      5
      9
      2
      6
      5
      3 ...
</pre></td></tr></table>
<p>In order to load these digits in Maxima,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (numericalio)$
(%i2) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i3) length (s1);
(%o3)                          100
</pre></td></tr></table>
<p>On the other hand, file <code>wind.data</code> 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) <var>Space-time Modelling with Long-memory Dependence: Assessing Ireland's Wind Power Resource, with Discussion</var>. Applied Statistics 38, 1-50). This loads the data:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (numericalio)$
(%i2) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i3) length (s2);
(%o3)                          100
(%i4) s2 [%]; /* last record */
(%o4)            [3.58, 6.0, 4.58, 7.62, 11.25]
</pre></td></tr></table>
<p>Some samples contain non numeric data. As an example, file <code>biomed.data</code> (which is part of another bigger one downloaded from the StatLib Data Repository) contains four blood measures taken from two groups of patients, <code>A</code> and <code>B</code>, of different ages,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (numericalio)$
(%i2) s3 : read_matrix (file_search (&quot;biomed.data&quot;))$
(%i3) length (s3);
(%o3)                          100
(%i4) s3 [1]; /* first record */
(%o4)            [A, 30, 167.0, 89.0, 25.6, 364]
</pre></td></tr></table>
<p>The first individual belongs to group <code>A</code>, is 30 years old and his/her blood measures were 167.0, 89.0, 25.6 and 364.
</p>
<p>One must take care when working with categorical data. In the next example, symbol <code>a</code> is asigned a value in some previous moment and then a sample with categorical value <code>a</code> is taken,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) a : 1$
(%i2) matrix ([a, 3], [b, 5]);
                            [ 1  3 ]
(%o2)                       [      ]
                            [ b  5 ]
</pre></td></tr></table>


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<h2 class="section"> 46.2 Definitions for data manipulation </h2>

<dl>
<dt><u>Function:</u> <b>continuous_freq</b><i> (<var>list</var>)</i>
<a name="IDX1388"></a>
</dt>
<dt><u>Function:</u> <b>continuous_freq</b><i> (<var>list</var>, <var>m</var>)</i>
<a name="IDX1389"></a>
</dt>
<dd><p>The argument of <code>continuous_freq</code> must be a list of numbers, which will be then grouped in intervals and counted how many of them belong to each group. Optionally, function <code>continuous_freq</code> admits a second argument indicating the number of classes, 10 is default,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (numericalio)$
(%i2) load (descriptive)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) continuous_freq (s1, 5);
(%o4) [[0, 1.8, 3.6, 5.4, 7.2, 9.0], [16, 24, 18, 17, 25]]
</pre></td></tr></table>
<p>The first list contains the interval limits and the second the corresponding counts: there are 16 digits inside the interval <code>[0, 1.8]</code>, that is 0's and 1's, 24 digits in <code>(1.8, 3.6]</code>, that is 2's and 3's, and so on.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>discrete_freq</b><i> (<var>list</var>)</i>
<a name="IDX1390"></a>
</dt>
<dd><p>Counts absolute frequencies in discrete samples, both numeric and categorical. Its unique argument is a list,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;));
(%o3) [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]
(%i4) discrete_freq (s1);
(%o4) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 
                             [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]
</pre></td></tr></table>
<p>The first list gives the sample values and the second their absolute frequencies. Commands <code>? col</code> and <code>? transpose</code> should help you to understand the last input.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>subsample</b><i> (<var>data_matrix</var>, <var>logical_expression</var>)</i>
<a name="IDX1391"></a>
</dt>
<dt><u>Function:</u> <b>subsample</b><i> (<var>data_matrix</var>, <var>logical_expression</var>, <var>col_num</var>, <var>col_num</var>, ...)</i>
<a name="IDX1392"></a>
</dt>
<dd><p>This is a sort of variation of the Maxima <code>submatrix</code> function. The first argument is the name of the data matrix, the second is a quoted logical expression and optional additional arguments are the numbers of the columns to be taken. Its behaviour is better understood with examples,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) subsample (s2, '(%c[1] &gt; 18));
              [ 19.38  15.37  15.12  23.09  25.25 ]
              [                                   ]
              [ 18.29  18.66  19.08  26.08  27.63 ]
(%o4)         [                                   ]
              [ 20.25  21.46  19.95  27.71  23.38 ]
              [                                   ]
              [ 18.79  18.96  14.46  26.38  21.84 ]
</pre></td></tr></table>
<p>These are multivariate records in which the wind speeds in the first meteorological station were greater than 18. See that in the quoted logical expression the <var>i</var>-th component is refered to as <code>%c[i]</code>. Symbol <code>%c[i]</code> is used inside function <code>subsample</code>, therefore when used as a categorical variable, Maxima gets confused. 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 lesser than 25 knots in station number 4,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) subsample (s2, '(%c[1] &gt;= 16 and %c[4] &lt; 25), 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 ]
</pre></td></tr></table>
<p>Here is an example with the categorical variables of <code>biomed.data</code>. We want the records corresponding to those patients in group <code>B</code> who are older than 38 years,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s3 : read_matrix (file_search (&quot;biomed.data&quot;))$
(%i4) subsample (s3, '(%c[1] = B and %c[2] &gt; 38));
                [ 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 ]
</pre></td></tr></table>

<p>Probably, the statistical analysis will involve only the blood measures,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s3 : read_matrix (file_search (&quot;biomed.data&quot;))$
(%i4) subsample (s3, '(%c[1] = B and %c[2] &gt; 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 ]
(%o4)              [                        ]
                   [ 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 ]
</pre></td></tr></table>

<p>This is the multivariate mean of <code>s3</code>,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s3 : read_matrix (file_search (&quot;biomed.data&quot;))$
(%i4) mean (s3);
       65 B + 35 A  317          6 NA + 8145.0
(%o4) [-----------, ---, 87.178, -------------, 18.123, 
           100      10                100
                                                    3 NA + 19587
                                                    ------------]
                                                        100
</pre></td></tr></table><p>Here, the first component is meaningless, since <code>A</code> and <code>B</code> 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 <code>NA</code> is used here to indicate <var>non available</var> data, and the two means are of course nonsense. A possible solution would be to take out from the matrix those rows with <code>NA</code> symbols, although this deserves some loss of information,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s3 : read_matrix (file_search (&quot;biomed.data&quot;))$
(%i4) mean (subsample (s3, '(%c[4] # NA and %c[6] # NA), 3, 4, 5, 6));
(%o4) [79.4923076923077, 86.2032967032967, 16.93186813186813, 
                                                            2514
                                                            ----]
                                                             13
</pre></td></tr></table></dd></dl>


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<h2 class="section"> 46.3 Definitions for descriptive statistics </h2>



<dl>
<dt><u>Function:</u> <b>mean</b><i> (<var>list</var>)</i>
<a name="IDX1393"></a>
</dt>
<dt><u>Function:</u> <b>mean</b><i> (<var>matrix</var>)</i>
<a name="IDX1394"></a>
</dt>
<dd><p>This is the sample mean, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                       n
                     ====
             _   1   \
             x = -    &gt;    x
                 n   /      i
                     ====
                     i = 1
</pre></td></tr></table>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) mean (s1);
                               471
(%o4)                          ---
                               100
(%i5) %, numer;
(%o5)                         4.71
(%i6) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i7) mean (s2);
(%o7)     [9.9485, 10.1607, 10.8685, 15.7166, 14.8441]
</pre></td></tr></table></dd></dl>



<dl>
<dt><u>Function:</u> <b>var</b><i> (<var>list</var>)</i>
<a name="IDX1395"></a>
</dt>
<dt><u>Function:</u> <b>var</b><i> (<var>matrix</var>)</i>
<a name="IDX1396"></a>
</dt>
<dd><p>This is the sample variance, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                     n
                   ====
           2   1   \          _ 2
          s  = -    &gt;    (x - x)
               n   /       i
                   ====
                   i = 1
</pre></td></tr></table>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) var (s1), numer;
(%o4)                   8.425899999999999
</pre></td></tr></table>
<p>See also function <code>var1</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>var1</b><i> (<var>list</var>)</i>
<a name="IDX1397"></a>
</dt>
<dt><u>Function:</u> <b>var1</b><i> (<var>matrix</var>)</i>
<a name="IDX1398"></a>
</dt>
<dd><p>This is the sample variance, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                     n
                   ====
               1   \          _ 2
              ---   &gt;    (x - x)
              n-1  /       i
                   ====
                   i = 1
</pre></td></tr></table>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) var1 (s1), numer;
(%o4)                    8.5110101010101
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) var1 (s2);
(%o6) [17.39586540404041, 15.13912778787879, 15.63204924242424, 
                            32.50152569696971, 24.66977392929294]
</pre></td></tr></table>
<p>See also function <code>var</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>std</b><i> (<var>list</var>)</i>
<a name="IDX1399"></a>
</dt>
<dt><u>Function:</u> <b>std</b><i> (<var>matrix</var>)</i>
<a name="IDX1400"></a>
</dt>
<dd><p>This is the the square root of function <code>var</code>, the variance with denominator <em>n</em>.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) std (s1), numer;
(%o4)                   2.902740084816414
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) std (s2);
(%o6) [4.149928523480858, 3.871399812729241, 3.933920277534866, 
                            5.672434260526957, 4.941970881136392]
</pre></td></tr></table>
<p>See also functions <code>var</code> and <code>std1</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>std1</b><i> (<var>list</var>)</i>
<a name="IDX1401"></a>
</dt>
<dt><u>Function:</u> <b>std1</b><i> (<var>matrix</var>)</i>
<a name="IDX1402"></a>
</dt>
<dd><p>This is the the square root of function <code>var1</code>, the variance with denominator <em>n-1</em>.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) std1 (s1), numer;
(%o4)                   2.917363553109228
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) std1 (s2);
(%o6) [4.17083509672109, 3.89090320978032, 3.953738641137555, 
                            5.701010936401517, 4.966867617451963]
</pre></td></tr></table>
<p>See also functions <code>var1</code> and <code>std</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>noncentral_moment</b><i> (<var>list</var>, <var>k</var>)</i>
<a name="IDX1403"></a>
</dt>
<dt><u>Function:</u> <b>noncentral_moment</b><i> (<var>matrix</var>, <var>k</var>)</i>
<a name="IDX1404"></a>
</dt>
<dd><p>The non central moment of order <em>k</em>, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                       n
                     ====
                 1   \      k
                 -    &gt;    x
                 n   /      i
                     ====
                     i = 1
</pre></td></tr></table>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) noncentral_moment (s1, 1), numer; /* the mean */
(%o4)                         4.71
(%i6) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i7) noncentral_moment (s2, 5);
(%o7) [319793.8724761506, 320532.1923892463, 391249.5621381556, 
                            2502278.205988911, 1691881.797742255]
</pre></td></tr></table>
<p>See also function <code>central_moment</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>central_moment</b><i> (<var>list</var>, <var>k</var>)</i>
<a name="IDX1405"></a>
</dt>
<dt><u>Function:</u> <b>central_moment</b><i> (<var>matrix</var>, <var>k</var>)</i>
<a name="IDX1406"></a>
</dt>
<dd><p>The central moment of order <em>k</em>, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                    n
                  ====
              1   \          _ k
              -    &gt;    (x - x)
              n   /       i
                  ====
                  i = 1
</pre></td></tr></table>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) central_moment (s1, 2), numer; /* the variance */
(%o4)                   8.425899999999999
(%i6) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i7) central_moment (s2, 3);
(%o7) [11.29584771375004, 16.97988248298583, 5.626661952750102, 
                             37.5986572057918, 25.85981904394192]
</pre></td></tr></table>
<p>See also functions <code>central_moment</code> and <code>mean</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>cv</b><i> (<var>list</var>)</i>
<a name="IDX1407"></a>
</dt>
<dt><u>Function:</u> <b>cv</b><i> (<var>matrix</var>)</i>
<a name="IDX1408"></a>
</dt>
<dd><p>The variation coefficient is the quotient between the sample standard deviation (<code>std</code>) and the <code>mean</code>,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) cv (s1), numer;
(%o4)                   .6193977819764815
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) cv (s2);
(%o6) [.4192426091090204, .3829365309260502, 0.363779605385983, 
                            .3627381836021478, .3346021393989506]
</pre></td></tr></table>
<p>See also functions <code>std</code> and <code>mean</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>mini</b><i> (<var>list</var>)</i>
<a name="IDX1409"></a>
</dt>
<dt><u>Function:</u> <b>mini</b><i> (<var>matrix</var>)</i>
<a name="IDX1410"></a>
</dt>
<dd><p>This is the minimum value of the sample <var>list</var>,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) mini (s1);
(%o4)                           0
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) mini (s2);
(%o6)             [0.58, 0.5, 2.67, 5.25, 5.17]
</pre></td></tr></table>
<p>See also function <code>maxi</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>maxi</b><i> (<var>list</var>)</i>
<a name="IDX1411"></a>
</dt>
<dt><u>Function:</u> <b>maxi</b><i> (<var>matrix</var>)</i>
<a name="IDX1412"></a>
</dt>
<dd><p>This is the maximum value of the sample <var>list</var>,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) maxi (s1);
(%o4)                           9
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) maxi (s2);
(%o6)          [20.25, 21.46, 20.04, 29.63, 27.63]
</pre></td></tr></table>
<p>See also function <code>mini</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>range</b><i> (<var>list</var>)</i>
<a name="IDX1413"></a>
</dt>
<dt><u>Function:</u> <b>range</b><i> (<var>matrix</var>)</i>
<a name="IDX1414"></a>
</dt>
<dd><p>The range is the difference between the extreme values.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) range (s1);
(%o4)                           9
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) range (s2);
(%o6)          [19.67, 20.96, 17.37, 24.38, 22.46]
</pre></td></tr></table></dd></dl>



<dl>
<dt><u>Function:</u> <b>quantile</b><i> (<var>list</var>, <var>p</var>)</i>
<a name="IDX1415"></a>
</dt>
<dt><u>Function:</u> <b>quantile</b><i> (<var>matrix</var>, <var>p</var>)</i>
<a name="IDX1416"></a>
</dt>
<dd><p>This is the <var>p</var>-<code>quantile</code>, with <var>p</var> a number in <em>[0, 1]</em>, of the sample <var>list</var>.
Although there are several definitions for the sample quantile (Hyndman, R. J., Fan, Y. (1996) <var>Sample quantiles in statistical packages</var>. American Statistician, 50, 361-365), the one based on linear interpolation is implemented in package <code>descriptive</code>.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) /* 1st and 3rd quartiles */ [quantile (s1, 1/4), quantile (s1, 3/4)], numer;
(%o4)                      [2.0, 7.25]
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) quantile (s2, 1/4);
(%o6)    [7.2575, 7.477500000000001, 7.82, 11.28, 11.48]
</pre></td></tr></table></dd></dl>



<dl>
<dt><u>Function:</u> <b>median</b><i> (<var>list</var>)</i>
<a name="IDX1417"></a>
</dt>
<dt><u>Function:</u> <b>median</b><i> (<var>matrix</var>)</i>
<a name="IDX1418"></a>
</dt>
<dd><p>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.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) median (s1);
                                9
(%o4)                           -
                                2
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) median (s2);
(%o6)         [10.06, 9.855, 10.73, 15.48, 14.105]
</pre></td></tr></table>
<p>The median is the 1/2-<code>quantile</code>.
</p>
<p>See also function <code>quantile</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>qrange</b><i> (<var>list</var>)</i>
<a name="IDX1419"></a>
</dt>
<dt><u>Function:</u> <b>qrange</b><i> (<var>matrix</var>)</i>
<a name="IDX1420"></a>
</dt>
<dd><p>The interquartilic range is the difference between the third and first quartiles, <code>quantile(list,3/4) - quantile(list,1/4)</code>,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) qrange (s1);
                               21
(%o4)                          --
                               4
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) qrange (s2);
(%o6) [5.385, 5.572499999999998, 6.0225, 8.729999999999999, 
                                               6.650000000000002]
</pre></td></tr></table>
<p>See also function <code>quantile</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>mean_deviation</b><i> (<var>list</var>)</i>
<a name="IDX1421"></a>
</dt>
<dt><u>Function:</u> <b>mean_deviation</b><i> (<var>matrix</var>)</i>
<a name="IDX1422"></a>
</dt>
<dd><p>The mean deviation, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                     n
                   ====
               1   \          _
               -    &gt;    |x - x|
               n   /       i
                   ====
                   i = 1
</pre></td></tr></table>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) mean_deviation (s1);
                               51
(%o4)                          --
                               20
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) mean_deviation (s2);
(%o6) [3.287959999999999, 3.075342, 3.23907, 4.715664000000001, 
                                               4.028546000000002]
</pre></td></tr></table>
<p>See also function <code>mean</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>median_deviation</b><i> (<var>list</var>)</i>
<a name="IDX1423"></a>
</dt>
<dt><u>Function:</u> <b>median_deviation</b><i> (<var>matrix</var>)</i>
<a name="IDX1424"></a>
</dt>
<dd><p>The median deviation, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                 n
               ====
           1   \
           -    &gt;    |x - med|
           n   /       i
               ====
               i = 1
</pre></td></tr></table><p>where <code>med</code> is the median of <var>list</var>.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) median_deviation (s1);
                                5
(%o4)                           -
                                2
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) median_deviation (s2);
(%o6)           [2.75, 2.755, 3.08, 4.315, 3.31]
</pre></td></tr></table>
<p>See also function <code>mean</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>harmonic_mean</b><i> (<var>list</var>)</i>
<a name="IDX1425"></a>
</dt>
<dt><u>Function:</u> <b>harmonic_mean</b><i> (<var>matrix</var>)</i>
<a name="IDX1426"></a>
</dt>
<dd><p>The harmonic mean, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                  n
               --------
                n
               ====
               \     1
                &gt;    --
               /     x
               ====   i
               i = 1
</pre></td></tr></table>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
(%i4) harmonic_mean (y), numer;
(%o4)                   3.901858027632205
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) harmonic_mean (s2);
(%o6) [6.948015590052786, 7.391967752360356, 9.055658197151745, 
                            13.44199028193692, 13.01439145898509]
</pre></td></tr></table>
<p>See also functions <code>mean</code> and <code>geometric_mean</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>geometric_mean</b><i> (<var>list</var>)</i>
<a name="IDX1427"></a>
</dt>
<dt><u>Function:</u> <b>geometric_mean</b><i> (<var>matrix</var>)</i>
<a name="IDX1428"></a>
</dt>
<dd><p>The geometric mean, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                 /  n      \ 1/n
                 | /===\   |
                 |  ! !    |
                 |  ! !  x |
                 |  ! !   i|
                 | i = 1   |
                 \         /
</pre></td></tr></table>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
(%i4) geometric_mean (y), numer;
(%o4)                   4.454845412337012
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) geometric_mean (s2);
(%o6) [8.82476274347979, 9.22652604739361, 10.0442675714889, 
                            14.61274126349021, 13.96184163444275]
</pre></td></tr></table>
<p>See also functions <code>mean</code> and <code>harmonic_mean</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>kurtosis</b><i> (<var>list</var>)</i>
<a name="IDX1429"></a>
</dt>
<dt><u>Function:</u> <b>kurtosis</b><i> (<var>matrix</var>)</i>
<a name="IDX1430"></a>
</dt>
<dd><p>The kurtosis coefficient, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                    n
                  ====
            1     \          _ 4
           ----    &gt;    (x - x)  - 3
              4   /       i
           n s    ====
                  i = 1
</pre></td></tr></table>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) kurtosis (s1), numer;
(%o4)                  - 1.273247946514421
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) kurtosis (s2);
(%o6) [- .2715445622195385, 0.119998784429451, 
   - .4275233490482866, - .6405361979019522, - .4952382132352935]
</pre></td></tr></table>
<p>See also functions <code>mean</code>, <code>var</code> and <code>skewness</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>skewness</b><i> (<var>list</var>)</i>
<a name="IDX1431"></a>
</dt>
<dt><u>Function:</u> <b>skewness</b><i> (<var>matrix</var>)</i>
<a name="IDX1432"></a>
</dt>
<dd><p>The skewness coefficient, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">                    n
                  ====
            1     \          _ 3
           ----    &gt;    (x - x)
              3   /       i
           n s    ====
                  i = 1
</pre></td></tr></table>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) skewness (s1), numer;
(%o4)                  .009196180476450306
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) skewness (s2);
(%o6) [.1580509020000979, .2926379232061854, .09242174416107717, 
                            .2059984348148687, .2142520248890832]
</pre></td></tr></table>
<p>See also functions <code>mean</code>, <code>var</code> and <code>kurtosis</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>pearson_skewness</b><i> (<var>list</var>)</i>
<a name="IDX1433"></a>
</dt>
<dt><u>Function:</u> <b>pearson_skewness</b><i> (<var>matrix</var>)</i>
<a name="IDX1434"></a>
</dt>
<dd><p>Pearson's skewness coefficient, defined as 
</p><table><tr><td>&nbsp;</td><td><pre class="example">                _
             3 (x - med)
             -----------
                  s
</pre></td></tr></table><p>where <var>med</var> is the median of <var>list</var>.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) pearson_skewness (s1), numer;
(%o4)                   .2159484029093895
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) pearson_skewness (s2);
(%o6) [- .08019976629211892, .2357036272952649, 
         .1050904062491204, .1245042340592368, .4464181795804519]
</pre></td></tr></table>
<p>See also functions <code>mean</code>, <code>var</code> and <code>median</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>quartile_skewness</b><i> (<var>list</var>)</i>
<a name="IDX1435"></a>
</dt>
<dt><u>Function:</u> <b>quartile_skewness</b><i> (<var>matrix</var>)</i>
<a name="IDX1436"></a>
</dt>
<dd><p>The quartile skewness coefficient, defined as 
</p><table><tr><td>&nbsp;</td><td><pre class="example">               c    - 2 c    + c
                3/4      1/2    1/4
               --------------------
                   c    - c
                    3/4    1/4
</pre></td></tr></table><p>where <em>c_p</em> is the <var>p</var>-quantile of sample <var>list</var>.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) quartile_skewness (s1), numer;
(%o4)                  .04761904761904762
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) quartile_skewness (s2);
(%o6) [- 0.0408542246982353, .1467025572005382, 
       0.0336239103362392, .03780068728522298, 0.210526315789474]
</pre></td></tr></table>
<p>See also function <code>quantile</code>.
</p></dd></dl>



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<h2 class="section"> 46.4 Definitions for specific multivariate descriptive statistics </h2>



<dl>
<dt><u>Function:</u> <b>cov</b><i> (<var>matrix</var>)</i>
<a name="IDX1437"></a>
</dt>
<dd><p>The covariance matrix of the multivariate sample, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">              n
             ====
          1  \           _        _
      S = -   &gt;    (X  - X) (X  - X)'
          n  /       j        j
             ====
             j = 1
</pre></td></tr></table><p>where <em>X_j</em> is the <em>j</em>-th row of the sample matrix.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) fpprintprec : 7$  /* change precision for pretty output */
(%i5) cov (s2);
      [ 17.22191  13.61811  14.37217  19.39624  15.42162 ]
      [                                                  ]
      [ 13.61811  14.98774  13.30448  15.15834  14.9711  ]
      [                                                  ]
(%o5) [ 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 ]
</pre></td></tr></table>
<p>See also function <code>cov1</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>cov1</b><i> (<var>matrix</var>)</i>
<a name="IDX1438"></a>
</dt>
<dd><p>The covariance matrix of the multivariate sample, defined as
</p><table><tr><td>&nbsp;</td><td><pre class="example">              n
             ====
         1   \           _        _
   S  = ---   &gt;    (X  - X) (X  - X)'
    1   n-1  /       j        j
             ====
             j = 1
</pre></td></tr></table><p>where <em>X_j</em> is the <em>j</em>-th row of the sample matrix.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) fpprintprec : 7$ /* change precision for pretty output */
(%i5) cov1 (s2);
      [ 17.39587  13.75567  14.51734  19.59216  15.5774  ]
      [                                                  ]
      [ 13.75567  15.13913  13.43887  15.31145  15.12232 ]
      [                                                  ]
(%o5) [ 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 ]
</pre></td></tr></table>
<p>See also function <code>cov</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>global_variances</b><i> (<var>matrix</var>)</i>
<a name="IDX1439"></a>
</dt>
<dt><u>Function:</u> <b>global_variances</b><i> (<var>matrix</var>, <var>logical_value</var>)</i>
<a name="IDX1440"></a>
</dt>
<dd><p>Function <code>global_variances</code> returns a list of global variance measures:
</p>
<ul>
<li>
<var>total variance</var>: <code>trace(S_1)</code>,
</li><li>
<var>mean variance</var>: <code>trace(S_1)/p</code>,
</li><li>
<var>generalized variance</var>: <code>determinant(S_1)</code>,
</li><li>
<var>generalized standard deviation</var>: <code>sqrt(determinant(S_1))</code>,
</li><li>
<var>efective variance</var> <code>determinant(S_1)^(1/p)</code>, (defined in: Pe&ntilde;a, D. (2002) <var>An&aacute;lisis de datos multivariantes</var>; McGraw-Hill, Madrid.)
</li><li>
<var>efective standard deviation</var>: <code>determinant(S_1)^(1/(2*p))</code>.
</li></ul>
<p>where <var>p</var> is the dimension of the multivariate random variable and <em>S_1</em> the covariance matrix returned by <code>cov1</code>.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) global_variances (s2);
(%o4) [105.338342060606, 21.06766841212119, 12874.34690469686, 
         113.4651792608502, 6.636590811800794, 2.576158149609762]
</pre></td></tr></table>
<p>Function <code>global_variances</code> has an optional logical argument: <code>global_variances(x,true)</code> tells Maxima that <code>x</code> is the data matrix, making the same as <code>global_variances(x)</code>. On the other hand, <code>global_variances(x,false)</code> means that <code>x</code> is not the data matrix, but the covariance matrix, avoiding its recalculation,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) s : cov1 (s2)$
(%i5) global_variances (s, false);
(%o5) [105.338342060606, 21.06766841212119, 12874.34690469686, 
         113.4651792608502, 6.636590811800794, 2.576158149609762]
</pre></td></tr></table>
<p>See also <code>cov</code> and <code>cov1</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>cor</b><i> (<var>matrix</var>)</i>
<a name="IDX1441"></a>
</dt>
<dt><u>Function:</u> <b>cor</b><i> (<var>matrix</var>, <var>logical_value</var>)</i>
<a name="IDX1442"></a>
</dt>
<dd><p>The correlation matrix of the multivariate sample.
</p>
<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) fpprintprec:7$
(%i4) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i5) cor (s2);
      [   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    ]
</pre></td></tr></table>
<p>Function <code>cor</code> has an optional logical argument: <code>cor(x,true)</code> tells Maxima that <code>x</code> is the data matrix, making the same as <code>cor(x)</code>. On the other hand, <code>cor(x,false)</code> means that <code>x</code> is not the data matrix, but the covariance matrix, avoiding its recalculation,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) fpprintprec:7$
(%i4) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i5) s : cov1 (s2)$
(%i6) cor (s, false); /* this is faster */
      [   1.0     .8476339  .8803515  .8239624  .7519506 ]
      [                                                  ]
      [ .8476339    1.0     .8735834  .6902622  0.782502 ]
      [                                                  ]
(%o6) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
      [                                                  ]
      [ .8239624  .6902622  .7764065    1.0     .7293848 ]
      [                                                  ]
      [ .7519506  0.782502  .8323358  .7293848    1.0    ]
</pre></td></tr></table>
<p>See also <code>cov</code> and <code>cov1</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>list_correlations</b><i> (<var>matrix</var>)</i>
<a name="IDX1443"></a>
</dt>
<dt><u>Function:</u> <b>list_correlations</b><i> (<var>matrix</var>, <var>logical_value</var>)</i>
<a name="IDX1444"></a>
</dt>
<dd><p>Function <code>list_correlations</code> returns a list of correlation measures:
</p>
<ul>
<li>
<var>precision matrix</var>: the inverse of the covariance matrix <em>S_1</em>,
<table><tr><td>&nbsp;</td><td><pre class="example">       -1     ij
      S   = (s  )             
       1         i,j = 1,2,...,p
</pre></td></tr></table>
</li><li>
<var>multiple correlation vector</var>:  <em>(R_1^2, R_2^2, ..., R_p^2)</em>, with 
<table><tr><td>&nbsp;</td><td><pre class="example">       2          1
      R  = 1 - -------
       i        ii
               s   s
                    ii
</pre></td></tr></table><p>being an indicator of the goodness of fit of the linear multivariate regression model on <em>X_i</em> when the rest of variables are used as regressors.
</p>
</li><li>
<var>partial correlation matrix</var>: with element <em>(i, j)</em> being
<table><tr><td>&nbsp;</td><td><pre class="example">                         ij
                        s
      r        = - ------------
       ij.rest     / ii  jj\ 1/2
                   |s   s  |
                   \       /
</pre></td></tr></table>
</li></ul>

<p>Example:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) z : list_correlations (s2)$
(%i5) fpprintprec : 5$ /* for pretty output */
(%i6) z[1];  /* precision matrix */
      [  .38486   - .13856   - .15626   - .10239    .031179  ]
      [                                                      ]
      [ - .13856   .34107    - .15233    .038447   - .052842 ]
      [                                                      ]
(%o6) [ - .15626  - .15233    .47296    - .024816  - .10054  ]
      [                                                      ]
      [ - .10239   .038447   - .024816   .10937    - .034033 ]
      [                                                      ]
      [ .031179   - .052842  - .10054   - .034033   .14834   ]
(%i7) z[2];  /* multiple correlation vector */
(%o7)       [.85063, .80634, .86474, .71867, .72675]
(%i8) z[3];  /* partial correlation matrix */
       [  - 1.0     .38244   .36627   .49908   - .13049 ]
       [                                                ]
       [  .38244    - 1.0    .37927  - .19907   .23492  ]
       [                                                ]
(%o8)  [  .36627    .37927   - 1.0    .10911    .37956  ]
       [                                                ]
       [  .49908   - .19907  .10911   - 1.0     .26719  ]
       [                                                ]
       [ - .13049   .23492   .37956   .26719    - 1.0   ]
</pre></td></tr></table>
<p>Function <code>list_correlations</code> also has an optional logical argument: <code>list_correlations(x,true)</code> tells Maxima that <code>x</code> is the data matrix, making the same as <code>list_correlations(x)</code>. On the other hand, <code>list_correlations(x,false)</code> means that <code>x</code> is not the data matrix, but the covariance matrix, avoiding its recalculation.
</p>
<p>See also <code>cov</code> and <code>cov1</code>.
</p></dd></dl>



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<h2 class="section"> 46.5 Definitions for statistical graphs </h2>



<dl>
<dt><u>Function:</u> <b>dataplot</b><i> (<var>list</var>)</i>
<a name="IDX1445"></a>
</dt>
<dt><u>Function:</u> <b>dataplot</b><i> (<var>list</var>, <var>option_1</var>, <var>option_2</var>, ...)</i>
<a name="IDX1446"></a>
</dt>
<dt><u>Function:</u> <b>dataplot</b><i> (<var>matrix</var>)</i>
<a name="IDX1447"></a>
</dt>
<dt><u>Function:</u> <b>dataplot</b><i> (<var>matrix</var>, <var>option_1</var>, <var>option_2</var>, ...)</i>
<a name="IDX1448"></a>
</dt>
<dd><p>Funtion <code>dataplot</code> permits direct visualization of sample data, both univariate (<var>list</var>) and multivariate (<var>matrix</var>). Giving values to the following <var>options</var> some aspects of the plot can be controlled:
</p>
<ul>
<li>
<code>'outputdev</code>, default <code>&quot;x&quot;</code>, indicates the output device; correct values are <code>&quot;x&quot;</code>, <code>&quot;eps&quot;</code> and <code>&quot;png&quot;</code>, for the screen, postscript and png format files, respectively.

</li><li>
<code>'maintitle</code>, default <code>&quot;&quot;</code>, is the main title between double quotes.

</li><li>
<code>'axisnames</code>, default <code>[&quot;x&quot;,&quot;y&quot;,&quot;z&quot;]</code>, is a list with the names of axis <code>x</code>, <code>y</code> and <code>z</code>.

</li><li>
<code>'joined</code>, default <code>false</code>, a logical value to select points in 2D to be joined or isolated.

</li><li>
<code>'picturescales</code>, default <code>[1.0, 1.0]</code>, scaling factors for the size of the plot.

</li><li>
<code>'threedim</code>, default <code>true</code>, tells Maxima whether to plot a three column matrix with a 3D diagram or a multivariate scatterplot. See examples bellow.

</li><li>
<code>'axisrot</code>, default <code>[60, 30]</code>, changes the point of view when <code>'threedim</code> is set to <code>true</code> and  data are stored in a three column matrix. The first number is the rotation angle of the <var>x</var>-axis, and the second number is the rotation angle of the <var>z</var>-axis, both measured in degrees.

</li><li>
<code>'nclasses</code>, default <code>10</code>, is the number of classes for the histograms in the diagonal of multivariate scatterplots.

</li><li>
<code>'pointstyle</code>, default <code>1</code>, is an integer to indicate how to display sample points.

</li></ul>

<p>For example, with the following input a simple plot of the first twenty digits of <code>%pi</code> is requested and the output stored in an eps file.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) dataplot (makelist (s1[k], k, 1, 20), 'pointstyle = 3)$
</pre></td></tr></table>
<p>Note that one dimensional data are plotted as a time series. In the next case, same more data with different settings,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) dataplot (makelist (s1[k], k, 1, 50), 'maintitle = &quot;First pi digits&quot;,
 'axisnames = [&quot;digit order&quot;, &quot;digit value&quot;], 'pointstyle = 2,
 'joined = true)$
</pre></td></tr></table>
<p>Function <code>dataplot</code> can be used to plot points in the plane. The next example is a scatterplot of the pairs of wind speeds corresponding to the first and fifth meteorological stations,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) dataplot (submatrix (s2, 2, 3, 4), 'pointstyle = 2,
 'maintitle = &quot;Pairs of wind speeds measured in knots&quot;,
 'axisnames = [&quot;Wind speed in A&quot;, &quot;Wind speed in E&quot;])$
</pre></td></tr></table>
<p>If points are stored in a two column matrix, <code>dataplot</code> can plot them directly, but if they are formatted as a list of pairs, their must be transformed to a matrix as in the following example.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) x : [[-1, 2], [5, 7], [5, -3], [-6, -9], [-4, 6]]$
(%i3) dataplot (apply ('matrix, x), 'maintitle = &quot;Points&quot;,
 'joined = true, 'axisnames = [&quot;&quot;, &quot;&quot;], 'picturescales = [0.5, 1.0])$
</pre></td></tr></table>
<p>Points in three dimensional space can be seen as a projection on the plane. In this example, plots of wind speeds corresponding to  three meteorological stations are requested, first in a 3D plot and then in a multivariate scatterplot.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) /* 3D plot */ dataplot (submatrix (s2, 4, 5), 'pointstyle = 2,
 'maintitle = &quot;Pairs of wind speeds measured in knots&quot;,
 'axisnames = [&quot;Station A&quot;, &quot;Station B&quot;, &quot;Station C&quot;])$
(%i5) /* Multivariate scatterplot */ dataplot (submatrix (s2, 4, 5),
 'nclasses = 6, 'threedim = false)$
</pre></td></tr></table><p>Note that in the last example, the number of classes in the histograms of the diagonal is set to 6, and that option <code>'threedim</code> is set to <code>false</code>.
</p>
<p>For more than three dimensions only multivariate scatterplots are possible, as in
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) dataplot (s2)$
</pre></td></tr></table></dd></dl>



<dl>
<dt><u>Function:</u> <b>histogram</b><i> (<var>list</var>)</i>
<a name="IDX1449"></a>
</dt>
<dt><u>Function:</u> <b>histogram</b><i> (<var>list</var>, <var>option_1</var>, <var>option_2</var>, ...)</i>
<a name="IDX1450"></a>
</dt>
<dt><u>Function:</u> <b>histogram</b><i> (<var>one_column_matrix</var>)</i>
<a name="IDX1451"></a>
</dt>
<dt><u>Function:</u> <b>histogram</b><i> (<var>one_column_matrix</var>, <var>option_1</var>, <var>option_2</var>, ...)</i>
<a name="IDX1452"></a>
</dt>
<dd><p>This function plots an histogram. Sample data must be stored in a list of numbers or a one column matrix. Giving values to the following <var>options</var> some aspects of the plot can be controlled:
</p>
<ul>
<li>
<code>'outputdev</code>, default <code>&quot;x&quot;</code>, indicates the output device; correct values are <code>&quot;x&quot;</code>, <code>&quot;eps&quot;</code> and <code>&quot;png&quot;</code>, for the screen, postscript and png format files, respectively.

</li><li>
<code>'maintitle</code>, default <code>&quot;&quot;</code>, is the main title between double quotes.

</li><li>
<code>'axisnames</code>, default <code>[&quot;x&quot;, &quot;Fr.&quot;]</code>, is a list with the names of axis <code>x</code> and <code>y</code>.

</li><li>
<code>'picturescales</code>, default <code>[1.0, 1.0]</code>, scaling factors for the size of the plot.

</li><li>
<code>'nclasses</code>, default <code>10</code>, is the number of classes or bars.

</li><li>
<code>'relbarwidth</code>, default <code>0.9</code>, a decimal number between 0 and 1 to control bars width.

</li><li>
<code>'barcolor</code>, default <code>1</code>, an integer to indicate bars color.

</li><li>
<code>'colorintensity</code>, default <code>1</code>, a decimal number between 0 and 1 to fix color intensity.

</li></ul>

<p>In the next two examples, histograms are requested for the first 100 digits of number <code>%pi</code> and for the wind speeds in the third meteorological station.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s1 : read_list (file_search (&quot;pidigits.data&quot;))$
(%i4) histogram (s1, 'maintitle = &quot;pi digits&quot;, 'axisnames = [&quot;&quot;, &quot;Absolute frequency&quot;],
 'relbarwidth = 0.2, 'barcolor = 3, 'colorintensity = 0.6)$
(%i5) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i6) histogram (col (s2, 3), 'colorintensity = 0.3)$
</pre></td></tr></table><p>Note that in the first case, <code>s1</code> is a list and in the second example, <code>col(s2,3)</code> is a matrix.
</p>
<p>See also function <code>barsplot</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>barsplot</b><i> (<var>list</var>)</i>
<a name="IDX1453"></a>
</dt>
<dt><u>Function:</u> <b>barsplot</b><i> (<var>list</var>, <var>option_1</var>, <var>option_2</var>, ...)</i>
<a name="IDX1454"></a>
</dt>
<dt><u>Function:</u> <b>barsplot</b><i> (<var>one_column_matrix</var>)</i>
<a name="IDX1455"></a>
</dt>
<dt><u>Function:</u> <b>barsplot</b><i> (<var>one_column_matrix</var>, <var>option_1</var>, <var>option_2</var>, ...)</i>
<a name="IDX1456"></a>
</dt>
<dd><p>Similar to <code>histogram</code> but for discrete, numeric or categorical, statistical variables. These are the options,
</p>
<ul>
<li>
<code>'outputdev</code>, default <code>&quot;x&quot;</code>, indicates the output device; correct values are <code>&quot;x&quot;</code>, <code>&quot;eps&quot;</code> and <code>&quot;png&quot;</code>, for the screen, postscript and png format files, respectively.

</li><li>
<code>'maintitle</code>, default <code>&quot;&quot;</code>, is the main title between double quotes.

</li><li>
<code>'axisnames</code>, default <code>[&quot;x&quot;, &quot;Fr.&quot;]</code>, is a list with the names of axis <code>x</code> and <code>y</code>.

</li><li>
<code>'picturescales</code>, default <code>[1.0, 1.0]</code>, scaling factors for the size of the plot.

</li><li>
<code>'relbarwidth</code>, default <code>0.9</code>, a decimal number between 0 and 1 to control bars width.

</li><li>
<code>'barcolor</code>, default <code>1</code>, an integer to indicate bars color.

</li><li>
<code>'colorintensity</code>, default <code>1</code>, a decimal number between 0 and 1 to fix color intensity.

</li></ul>

<p>This example plots the barchart for groups <code>A</code> and <code>B</code> of patients in sample <code>s3</code>,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s3 : read_matrix (file_search (&quot;biomed.data&quot;))$
(%i4) barsplot (col (s3, 1), 'maintitle = &quot;Groups of patients&quot;,
 'axisnames = [&quot;Group&quot;, &quot;# of individuals&quot;], 'colorintensity = 0.2)$
</pre></td></tr></table><p>The first column in sample <code>s3</code> stores the categorical values <code>A</code> and <code>B</code>, also known sometimes as factors. On the other hand, the positive integer numbers in the second column are ages, in years, which is a discrete variable, so we can plot the absolute frequencies for these values,
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s3 : read_matrix (file_search (&quot;biomed.data&quot;))$
(%i4) barsplot (col (s3, 2), 'maintitle = &quot;Ages&quot;,
 'axisnames = [&quot;Years&quot;, &quot;# of individuals&quot;], 'colorintensity = 0.2,
 'relbarwidth = 0.6)$
</pre></td></tr></table>

<p>See also function <code>histogram</code>.
</p></dd></dl>



<dl>
<dt><u>Function:</u> <b>boxplot</b><i> (<var>data</var>)</i>
<a name="IDX1457"></a>
</dt>
<dt><u>Function:</u> <b>boxplot</b><i> (<var>data</var>, <var>option_1</var>, <var>option_2</var>, ...)</i>
<a name="IDX1458"></a>
</dt>
<dd><p>This function plots box diagrams. Argument <var>data</var> 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 <var>data</var> to be a list of samples with possible different sample sizes, in fact this is the only function in package <code>descriptive</code> that admits this type of data structure. See example bellow.  These are the options,
</p>
<ul>
<li>
<code>'outputdev</code>, default <code>&quot;x&quot;</code>, indicates the output device; correct values are <code>&quot;x&quot;</code>, <code>&quot;eps&quot;</code> and <code>&quot;png&quot;</code>, for the screen, postscript and png format files, respectively.

</li><li>
<code>'maintitle</code>, default <code>&quot;&quot;</code>, is the main title between double quotes.

</li><li>
<code>'axisnames</code>, default <code>[&quot;sample&quot;, &quot;y&quot;]</code>, is a list with the names of axis <code>x</code> and <code>y</code>.

</li><li>
<code>'picturescales</code>, default <code>[1.0, 1.0]</code>, scaling factors for the size of the plot.

</li></ul>

<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) load (descriptive)$
(%i2) load (numericalio)$
(%i3) s2 : read_matrix (file_search (&quot;wind.data&quot;))$
(%i4) boxplot (s2, 'maintitle = &quot;Windspeed in knots&quot;,
 'axisnames = [&quot;Seasons&quot;, &quot;&quot;])$
(%i5) 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]]$
(%i6) boxplot (A)$
</pre></td></tr></table></dd></dl>

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