simple_statistics

Computes basic univariate statistics.

Synopsis

#include <imsl.h>

float *imsl_f_simple_statistics (int n_observations, int _variables, float x[], …, 0)

The type double procedure is imsl_d_simple_statistics.

Required Arguments

int n_observations (Input)
The number of observations.

int n_variables (Input)
The number of variables.

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

Return Value

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

Row	Statistic
0	the mean
1	the variance
2	the standard deviation
3	the coefficient of skewness
4	the coefficient of excess (kurtosis)
5	the minimum value
6	the maximum value
7	the range
8	the coefficient of variation (when defined) If the coefficient of variation is not defined, zero is returned.
9	the number of observations (the counts)
10	a lower confidence limit for the mean (assuming normality) The default is a 95 percent confidence interval.
11	an upper confidence limit for the mean (assuming normality)
12	a lower confidence limit for the variance (assuming normality) The default is a 95 percent confidence interval.
13	an upper confidence limit for the variance (assuming normality)

Synopsis with Optional Arguments

#include <imsl.h>

float *imsl_f_simple_statistics (int n_observations, int n_variables, float x[],

IMSL_CONFIDENCE_MEANS, float confidence_means,

IMSL_CONFIDENCE_VARIANCES, float confidence_variances,

IMSL_X_COL_DIM, int x_col_dim,

IMSL_STAT_COL_DIM, int stat_col_dim,

IMSL_MEDIAN,

IMSL_MEDIAN_AND_SCALE,

IMSL_RETURN_USER, float simple_statistics[],

Optional Arguments

IMSL_CONFIDENCE_MEANS, float confidence_means (Input)
The 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 IMSL_CONFIDENCE_MEANS is not specified, a 95 percent confidence interval is computed.

IMSL_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 IMSL_CONFIDENCE_VARIANCES is not specified, a 95 percent confidence interval is computed.

IMSL_X_COL_DIM, int x_col_dim (Input)
The column dimension of array x.
Default: x_col_dim = n_variables

IMSL_STAT_COL_DIM, int stat_col_dim (Input)
The column dimension of the returned value array, or if IMSL_RETURN_USER is specified, the column dimension of array simple_statistics.
Default: stat_col_dim = n_variables

IMSL_MEDIAN, or

IMSL_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 IMSL_MEDIAN is specified, the medians are computed and stored in one additional row (row number 14) in the returned matrix of simple statistics. If IMSL_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.

IMSL_RETURN_USER, float simple_statistics[] (Output)
Store the matrix of statistics in the user-provided array simple_statistics. If neither IMSL_MEDIAN nor IMSL_MEDIAN_AND_SCALE is specified, the matrix is 14 by n_variables. If IMSL_MEDIAN is specified, the matrix is 15 by n_variables. If IMSL_MEDIAN_AND_SCALE is specified, the matrix is 17 by n_variables.

Description

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

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

Simple Robust Estimate of Scale

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

This example uses data from Draper and Smith (1981). There are five variables and 13 observations.

#include <imsl.h>

#define N_VARIABLES 5

#define N_OBSERVATIONS 13

int 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 = imsl_f_simple_statistics(N_OBSERVATIONS,

N_VARIABLES, x, 0);

imsl_f_write_matrix("* * * Statistics * * *\n", 14, N_VARIABLES,

simple_statistics,

IMSL_ROW_LABELS, row_labels,

IMSL_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