39.3 Functions and Variables for descriptive statistics

Function: mean (list) ¶

Function: mean (matrix) ¶

This is the sample mean, defined as

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

Example:

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

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

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

(%i5) s2 : read_matrix (file_search ("wind.data"))$

(%i6) mean (s2);
(%o6)     [9.9485, 10.1607, 10.8685, 15.7166, 14.8441]

Function: var (list) ¶

Function: var (matrix) ¶

This is the sample variance, defined as

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

Example:

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

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

See also function var1.

Function: var1 (list) ¶

Function: var1 (matrix) ¶

This is the sample variance, defined as

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

Example:

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

(%i3) var1 (s1), numer;
(%o3)                    8.5110101010101

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

(%i5) var1 (s2);
(%o5) [17.39586540404041, 15.13912778787879, 15.63204924242424, 
                            32.50152569696971, 24.66977392929294]

See also function var.

Function: std (list) ¶

Function: std (matrix) ¶

This is the the square root of function var, the variance with denominator \(n\).

Example:

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

(%i3) std (s1), numer;
(%o3)                   2.902740084816414

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

(%i5) std (s2);
(%o5) [4.149928523480858, 3.871399812729241, 3.933920277534866, 
                            5.672434260526957, 4.941970881136392]

See also functions var and std1.

Function: std1 (list) ¶

Function: std1 (matrix) ¶

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

Example:

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

(%i3) std1 (s1), numer;
(%o3)                   2.917363553109228

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

(%i5) std1 (s2);
(%o5) [4.170835096721089, 3.89090320978032, 3.953738641137555, 
                            5.701010936401517, 4.966867617451963]

See also functions var1 and std.

Function: noncentral_moment (list, k) ¶

Function: noncentral_moment (matrix, k) ¶

The non central moment of order \(k\), defined as

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

Example:

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

(%i3) noncentral_moment (s1, 1), numer; /* the mean */
(%o3)                         4.71

(%i5) s2 : read_matrix (file_search ("wind.data"))$

(%i6) noncentral_moment (s2, 5);
(%o6) [319793.8724761505, 320532.1923892463,
      391249.5621381556, 2502278.205988911, 1691881.797742255]

See also function central_moment.

Function: central_moment (list, k) ¶

Function: central_moment (matrix, k) ¶

The central moment of order \(k\), defined as

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

Example:

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

(%i3) central_moment (s1, 2), numer; /* the variance */
(%o3)                   8.425899999999999

(%i5) s2 : read_matrix (file_search ("wind.data"))$

(%i6) central_moment (s2, 3);
(%o6) [11.29584771375004, 16.97988248298583, 5.626661952750102,
                             37.5986572057918, 25.85981904394192]

See also functions central_moment and mean.

Function: cv (list) ¶

Function: cv (matrix) ¶

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

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

(%i3) cv (s1), numer;
(%o3)                   .6193977819764815

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

(%i5) cv (s2);
(%o5) [.4192426091090204, .3829365309260502, 0.363779605385983, 
                            .3627381836021478, .3346021393989506]

See also functions std and mean.

Function: mini (list) ¶

Function: mini (matrix) ¶

This is the minimum value of the sample list,

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

(%i3) mini (s1);
(%o3)                           0

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

(%i5) mini (s2);
(%o5)             [0.58, 0.5, 2.67, 5.25, 5.17]

See also function maxi.

Function: maxi (list) ¶

Function: maxi (matrix) ¶

This is the maximum value of the sample list,

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

(%i3) maxi (s1);
(%o3)                           9

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

(%i5) maxi (s2);
(%o5)          [20.25, 21.46, 20.04, 29.63, 27.63]

See also function mini.

Function: range (list) ¶

Function: range (matrix) ¶

The range is the difference between the extreme values.

Example:

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

(%i3) range (s1);
(%o3)                           9

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

(%i5) range (s2);
(%o5)          [19.67, 20.96, 17.37, 24.38, 22.46]

Function: quantile (list, p) ¶

Function: quantile (matrix, p) ¶

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

Example:

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

(%i3) /* 1st and 3rd quartiles */
         [quantile (s1, 1/4), quantile (s1, 3/4)], numer;
(%o3)                      [2.0, 7.25]

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

(%i5) quantile (s2, 1/4);
(%o5)    [7.2575, 7.477500000000001, 7.82, 11.28, 11.48]

Function: median (list) ¶

Function: median (matrix) ¶

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

Example:

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

(%i3) median (s1);
                                9
(%o3)                           -
                                2

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

(%i5) median (s2);
(%o5)         [10.06, 9.855, 10.73, 15.48, 14.105]

The median is the 1/2-quantile.

See also function quantile.

Function: qrange (list) ¶

Function: qrange (matrix) ¶

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

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

(%i3) qrange (s1);
                               21
(%o3)                          --
                               4

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

(%i5) qrange (s2);
(%o5) [5.385, 5.572499999999998, 6.022500000000001, 
                            8.729999999999999, 6.649999999999999]

See also function quantile.

Function: mean_deviation (list) ¶

Function: mean_deviation (matrix) ¶

The mean deviation, defined as

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

Example:

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

(%i3) mean_deviation (s1);
                               51
(%o3)                          --
                               20

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

(%i5) mean_deviation (s2);
(%o5) [3.287959999999999, 3.075342, 3.23907, 4.715664000000001, 
                                               4.028546000000002]

See also function mean.

Function: median_deviation (list) ¶

Function: median_deviation (matrix) ¶

The median deviation, defined as

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

where med is the median of list.

Example:

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

(%i3) median_deviation (s1);
                                5
(%o3)                           -
                                2

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

(%i5) median_deviation (s2);
(%o5)           [2.75, 2.755, 3.08, 4.315, 3.31]

See also function mean.

Function: harmonic_mean (list) ¶

Function: harmonic_mean (matrix) ¶

The harmonic mean, defined as

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

Example:

(%i1) load ("descriptive")$
(%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$

(%i3) harmonic_mean (y), numer;
(%o3)                   3.901858027632205

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

(%i5) harmonic_mean (s2);
(%o5) [6.948015590052786, 7.391967752360356, 9.055658197151745, 
                            13.44199028193692, 13.01439145898509]

See also functions mean and geometric_mean.

Function: geometric_mean (list) ¶

Function: geometric_mean (matrix) ¶

The geometric mean, defined as

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

Example:

(%i1) load ("descriptive")$
(%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$

(%i3) geometric_mean (y), numer;
(%o3)                   4.454845412337012

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

(%i5) geometric_mean (s2);
(%o5) [8.82476274347979, 9.22652604739361, 10.0442675714889, 
                            14.61274126349021, 13.96184163444275]

See also functions mean and harmonic_mean.

Function: kurtosis (list) ¶

Function: kurtosis (matrix) ¶

The kurtosis coefficient, defined as

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

Example:

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

(%i3) kurtosis (s1), numer;
(%o3)                  - 1.273247946514421

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

(%i5) kurtosis (s2);
(%o5) [- .2715445622195385, 0.119998784429451, 
   - .4275233490482861, - .6405361979019522, - .4952382132352935]

See also functions mean, var and skewness.

Function: skewness (list) ¶

Function: skewness (matrix) ¶

The skewness coefficient, defined as

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

Example:

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

(%i3) skewness (s1), numer;
(%o3)                  .009196180476450424

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

(%i5) skewness (s2);
(%o5) [.1580509020000978, .2926379232061854, .09242174416107717, 
                            .2059984348148687, .2142520248890831]

See also functions mean, var and kurtosis.

Function: pearson_skewness (list) ¶

Function: pearson_skewness (matrix) ¶

Pearson’s skewness coefficient, defined as

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

where med is the median of list.

Example:

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

(%i3) pearson_skewness (s1), numer;
(%o3)                   .2159484029093895

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

(%i5) pearson_skewness (s2);
(%o5) [- .08019976629211892, .2357036272952649, 
         .1050904062491204, .1245042340592368, .4464181795804519]

See also functions mean, var and median.

Function: quartile_skewness (list) ¶

Function: quartile_skewness (matrix) ¶

The quartile skewness coefficient, defined as

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

where \(c_p\) is the p-quantile of sample list.

Example:

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

(%i3) quartile_skewness (s1), numer;
(%o3)                  .04761904761904762

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

(%i5) quartile_skewness (s2);
(%o5) [- 0.0408542246982353, .1467025572005382, 
       0.0336239103362392, .03780068728522298, .2105263157894735]

See also function quantile.