simple_statistics

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_IDO, int ido,

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[],

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_IDO, int ido (Input)
Processing option.
The argument ido must be one of 0, 1, 2, or 3. If ido = 0 (the default), all of the observations are input during one invocation. If ido = 1, 2, or 3, blocks of rows of the data can be processed sequentially in separate invocations of imsls_f_simple_statistics; with this option, it is not a requirement that all observations be memory resident, thus enabling one to handle large data sets.

ido	Action
0	This is the only invocation; all the data are input at once. (Default)
1	This is the first invocation with this data; additional calls will be made. Initialization and updating for the n_observations observations of x will be performed.
2	This is an intermediate invocation; updating for the n_observations observations of x will be performed.
3	This is the final invocation of this function. Updating for the data in x and wrap-up computations are performed. Workspace is released. No further invocations of imsls_f_simple_statistics with ido greater than 1 should be made without first invoking imsls_f_simple_statistics with ido = 1.

Default: ido = 0

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_MEDIAN or IMSLS_MEDIAN_AND_SCALE can be specified only when ido is equal to 0.

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

where Φ−1(3/4) ≈ 0.6745 is the inverse of the standard norm

istribution 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).

Examples

Example 1

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

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 = 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

Example 2

Continuing with Example 1 data, the example below invokes the imsls_f_simple_statistics function using values of IDO greater than 0.

#include <imsls.h>

#define N_VARIABLES 5

#define N_OBSERVATIONS_BLOCK_1 2

#define N_OBSERVATIONS_BLOCK_2 8

#define N_OBSERVATIONS_BLOCK_3 3

int main()

{

float *simple_statistics;

float x1[] = { 7., 26., 6., 60., 78.5,

1., 29., 15., 52., 74.3};

float x2[] = {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};

float x3[] = { 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_BLOCK_1,

N_VARIABLES, x1,

IMSLS_IDO, 1, 0);

simple_statistics = imsls_f_simple_statistics(N_OBSERVATIONS_BLOCK_2,

N_VARIABLES, x2,

IMSLS_IDO, 2, 0);

simple_statistics = imsls_f_simple_statistics(N_OBSERVATIONS_BLOCK_3,

N_VARIABLES, x3,

IMSLS_IDO, 3, 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 94.289 659.816 111.792 763.434 616.688

Warning Errors

IMSLS_ROW_OF_X_CONTAINED_NAN	At least one row of “x” contained NaN (a missing value).
IMSLS_VAR_IN_X_CONTAINED_NAN	At least one observation for a variable in “x” contained NaN (a missing value). Missing observations were excluded from calculations for those variables.
IMSLS_CONSTANT_OBSERVATIONS	The observations on variable(s) are constant.
IMSLS_LESS_THAN_TWO_VALID_OBS	Fewer than two valid observations are present. The corresponding statistics are set to NaN (not a number), (except for the mean, which is not correct if no valid observations).
IMSLS_VARIANCE_UNDERFLOW	The variance for this variable underflows. Therefore, the variance and standard deviation are set to 0, and the skewness and kurtosis are set to NaN (not a number)
IMSLS_NEGATIVE_VARIANCE	The variance is negative for the variable. The corresponding confidence limits for the variance are set to NaN (not a number).
IMSLS_NOT_ENOUGH_OBSERVATIONS	Fewer than two valid observations are present for the variable. The corresponding statistics are set to NaN (not a number), (except for the mean, which is not correct if no valid observations are present, or is correct if one observation is present)
IMSLS_MIN_GREATER_THAN_MAX	The maximum value is less than the minimum value. The corresponding statistics are set to NaN (not a number).
IMSLS_MAX_LESS_THAN_MIN	The maximum value is less than the minimum value. The corresponding statistics are set to NaN (not a number).
IMSLS_SUM_OF_WEIGHTS_ZERO	The sum of the weights for variable is zero. The statistics, except for the minima, maxima, ranges and counts, are set to NaN (not a number).
IMSLS_ZERO_SUM_OF_WEIGHTS	The sum of the weights is zero. The statistics, except for the minima, maxima, ranges and counts, are set to NaN (not a number).
IMSLS_LESS_THAN_TWO_VALID_OBS	Fewer than two valid observations are present. The corresponding statistics are set to NaN (not a number), (except for the mean, which is not correct if no valid observations).
IMSLS_FOURTH_ORDER_UNDERFLOW	Since the range of variable is very small, the fourth order moment for this variable underflows. Therefore, the kurtosis is set to NaN (not a number).
IMSLS_HIGH_ORDER_UNDERFLOW	Since the range of variable %(I1) is very small, the higher order moments for this variable underflow. Therefore, the skewness and kurtosis are set to NaN (not a number).
IMSLS_CHI_SQUARED_STAT_ERROR	An error occurred in determining the chi-squared statistic. The lower confidence limit for the variance is set to NaN (not a number).

Fatal Errors

IMSLS_BAD_IDO_6	“ido” = #. Initial allocations must be performed by invoking the function with “ido” = 1.
IMSLS_BAD_IDO_7	“ido” = #. A new analysis may not begin until the previous analysis is terminated by invoking the function with “ido” = 3.
IMSLS_BAD_N_VARIABLES	“n_variables” = #. The number of variables must be the same in separate function invocations.