Chapter 1: Basic Statistics

simple_statistics

Computes basic univariate statistics.

Synopsis

#include <imsls.h>

float *imsls_f_simple_statistics (int n_observations, int n_variables, float x[], ..., 0)

The type double function is imsls_d_simple_statistics.

Required Arguments

int n_observations   (Input)
Number of observations.

int n_variables   (Input)
Number of variables.

float x[]   (Input)
Array of size n_observations × n_variables containing the data matrix.

Return Value

A pointer to an array containing some simple statistics for each of the columns in x. If IMSLS_MEDIAN and IMSLS_MEDIAN_AND_SCALE are not used as optional arguments, the size of the matrix is 14 × n_variables. The columns of this matrix correspond to the columns of x, and the rows contain the following statistics:

Row

Statistic

0

mean

1

variance

2

standard deviation

3

coefficient of skewness

4

coefficient of excess (kurtosis)

5

minimum value

6

maximum value

7

range

8

coefficient of variation (when defined)  If the coefficient of variation is not defined, 0 is returned.

9

number of observations (the counts)

10

lower confidence limit for the mean (assuming normality)  The default is a 95-percent confidence interval.

11

upper confidence limit for the mean (assuming normality)

12

lower confidence limit for the variance (assuming normality)
The default is a 95-percent confidence interval.

13

upper confidence limit for the variance (assuming normality))

Synopsis with Optional Arguments

#include <imsls.h>

float *imsls_f_simple_statistics (int n_observations, int n_variables, float x[],
IMSLS_CONFIDENCE_MEANS, float confidence_means,
IMSLS_CONFIDENCE_VARIANCES, float confidence_variances,
IMSLS_X_COL_DIM, int x_col_dim,
IMSLS_STAT_COL_DIM, int stat_col_dim,
IMSLS_MEDIAN, or
IMSLS_MEDIAN_AND_SCALE,
IMSLS_MISSING_LISTWISE, or
IMSLS_MISSING_ELEMENTWISE,
IMSLS_FREQUENCIES, float frequencies[],
IMSLS_WEIGHTS, float weights[],
IMSLS_RETURN_USER, float simple_statistics[],
0)

Optional Arguments

IMSLS_CONFIDENCE_MEANS, float confidence_means   (Input)
Confidence level for a two-sided interval estimate of the means (assuming normality) in percent. Argument confidence_means must be between 0.0 and 100.0 and is often 90.0, 95.0, or 99.0. For a one-sided confidence interval with confidence level c, set confidence_means = 100.0 − 2(100 − c). If IMSLS_CONFIDENCE_MEANS is not specified, a 95-percent confidence interval is computed.

IMSLS_CONFIDENCE_VARIANCES, float confidence_variances   (Input)
The confidence level for a two-sided interval estimate of the variances (assuming normality) in percent. The confidence intervals are symmetric in probability (rather than in length). For a one-sided confidence interval with confidence level c, set confidence_means  = 100.0 − 2(100 − c). If IMSLS_CONFIDENCE_VARIANCES is not specified, a 95-percent confidence interval is computed.

IMSLS_X_COL_DIM, int x_col_dim   (Input)
Column dimension of array x.
Default: x_col_dim = n_variables

IMSLS_STAT_COL_DIM, int stat_col_dim   (Input)
Column dimension of the returned value array, or if IMSLS_RETURN_USER is specified, the column dimension of array simple_statistics.
Default: stat_col_dim = n_variables

IMSLS_MEDIAN, or

IMSLS_MEDIAN_AND_SCALE
Exactly one of these optional arguments can be specified in order to indicate the additional simple robust statistics to be computed. If IMSLS_MEDIAN is specified, the medians are computed and stored in one additional row (row number 14) in the returned matrix of simple statistics. If IMSLS_MEDIAN_AND_SCALE is specified, the medians, the medians of the absolute deviations from the medians, and a simple robust estimate of scale are computed, then stored in three additional rows (rows 14, 15, and 16) in the returned matrix of simple statistics.

IMSLS_MISSING_LISTWISE, or

IMSLS_MISSING_ELEMENTWISE
If IMSLS_MISSING_ELEMENTWISE is specified, all non missing data for any variable is used in computing the statistics for that variable. If IMSLS_MISSING_LISTWISE is specified and if an observation (row of x) contains a missing value, the observation is excluded from computations for all variables. The default is IMSLS_MISSING_LISTWISE. In either case, if weights and/or frequencies are specified and the value of the weight and/or frequency is missing, the observation is excluded from computations for all variables.

IMSLS_FREQUENCIES, float frequencies[]   (Input)
Array of length n_observations containing the frequency for each observation.
Default: Each observation has a frequency of 1

IMSLS_WEIGHTS, float weights[]   (Input)
Array of length n_observations containing the weight for each observation.
Default: Each observation has a weight of 1

IMSLS_RETURN_USER, float simple_statistics[]   (Output)
User-supplied array containing the matrix of statistics. If neither IMSLS_MEDIAN nor IMSLS_MEDIAN_AND_SCALE is specified, the matrix is 14 × n_variables. If IMSLS_MEDIAN is specified, the matrix is 15 × n_variables. If IMSLS_MEDIAN_AND_SCALE is specified, the matrix is 17 × n_variables.

Description

For the data in each column of x, imsls_f_simple_statistics computes the sample mean, variance, minimum, maximum, and other basic statistics. This function also computes confidence intervals for the mean and variance (under the hypothesis that the sample is from a normal population).

Frequencies are interpreted as multiple occurrences of the other values in the observations. In other words, a row of x with a frequency variable having a value of 2 has the same effect as two rows with frequencies of 1. The total of the frequencies is used in computing all the statistics based on moments (mean, variance, skewness, and kurtosis). Weights are not viewed as replication factors. The sum of the weights is used only in computing the mean (the weighted mean is used in computing the central moments). Both weights and frequencies can be 0, but neither can be negative. In general, a 0 frequency means that the row is to be eliminated from the analysis; no further processing or error checking is done on the row. A weight of 0 results in the row being counted, and updates are made of the statistics.

The definitions of some of the statistics are given below in terms of a single variable x of which the i-th datum is xi.

Mean

Variance

Skewness

Excess or Kurtosis

Minimum

Maximum

Range

Coefficient of Variation

Median

Median Absolute Deviation

MAD = median {|xi − median {xj}|}

Simple Robust Estimate of Scale

MAD/Φ−1(3/4)

where Φ−1(3/4) ≈ 0.6745 is the inverse of the standard normal distribution function evaluated at 3/4. This standardizes MAD in order to make the scale estimate consistent at the normal distribution for estimating the standard deviation (Huber 1981, pp. 107−108).

Example

Data from Draper and Smith (1981) are used in this example, which includes
5 variables and 13 observations.

#include <imsls.h>

#define N_VARIABLES             5
#define N_OBSERVATIONS         13

main()
{
    float       *simple_statistics;
    float       x[] = {
         7., 26.,  6., 60.,  78.5,
         1., 29., 15., 52.,  74.3,
        11., 56.,  8., 20., 104.3,
        11., 31.,  8., 47.,  87.6,
         7., 52.,  6., 33.,  95.9,
        11., 55.,  9., 22., 109.2,
         3., 71., 17.,  6., 102.7,
         1., 31., 22., 44.,  72.5,
         2., 54., 18., 22.,  93.1,
        21., 47.,  4., 26., 115.9,
         1., 40., 23., 34.,  83.8,
        11., 66.,  9., 12., 113.3,
        10., 68.,  8., 12., 109.4};
    char        *row_labels[] = {
        "means", "variances", "std. dev", "skewness", "kurtosis",
        "minima", "maxima", "ranges", "C.V.", "counts", "lower mean",
        "upper mean", "lower var", "upper var"};

    simple_statistics = imsls_f_simple_statistics(N_OBSERVATIONS,
        N_VARIABLES, x, 0);

    imsls_f_write_matrix("* * * Statistics * * *\n", 14, N_VARIABLES,
        simple_statistics,
        IMSLS_ROW_LABELS,  row_labels,
        IMSLS_WRITE_FORMAT, "%7.3f", 0);
}

Output

                * * * Statistics * * *

                  1        2        3        4        5
means         7.462   48.154   11.769   30.000   95.423
variances    34.603  242.141   41.026  280.167  226.314
std. dev      5.882   15.561    6.405   16.738   15.044
skewness      0.688   -0.047    0.611    0.330   -0.195
kurtosis      0.075   -1.323   -1.079   -1.014   -1.342
minima        1.000   26.000    4.000    6.000   72.500
maxima       21.000   71.000   23.000   60.000  115.900
ranges       20.000   45.000   19.000   54.000   43.400
C.V.          0.788    0.323    0.544    0.558    0.158
counts       13.000   13.000   13.000   13.000   13.000
lower mean    3.907   38.750    7.899   19.885   86.332
upper mean   11.016   57.557   15.640   40.115  104.514
lower var    17.793  124.512   21.096  144.065  116.373
upper var    94.289  659.817  111.792  763.434  616.688

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