NAG CL Interface
g01adc (summary_​freq)

Settings help

CL Name Style:

1 Purpose

g01adc calculates the mean, standard deviation and coefficients of skewness and kurtosis for data grouped in a frequency distribution.

2 Specification

#include <nag.h>
void  g01adc (Integer k, const double x[], const Integer ifreq[], double *xmean, double *xsd, double *xskew, double *xkurt, Integer *n, NagError *fail)
The function may be called by the names: g01adc, nag_stat_summary_freq or nag_summary_stats_freq.

3 Description

The input data consist of a univariate frequency distribution, denoted by fi, for i=1,2,,k-1, and the boundary values of the classes xi, for i=1,2,,k. Thus the frequency associated with the interval (xi,xi+1) is fi, and g01adc assumes that all the values in this interval are concentrated at the point
yi=(xi+1+xi)/2,  i=1,2,,k-1.  
The following quantities are calculated:
  1. (a)total frequency,
    n=i= 1 k- 1fi.  
  2. (b)mean,
    y¯=i=1 k-1fiyin.  
  3. (c)standard deviation,
    s2=i= 1 k- 1fi (yi-y¯) 2 (n-1) ,   n 2.  
  4. (d)coefficient of skewness,
    s3=i=1 k-1fi (yi-y¯) 3 (n-1)×s23 ,  n2.  
  5. (e)coefficient of kurtosis,
    s4=i= 1 k- 1fi (yi-y¯) 4 (n-1)×s24 - 3,   n 2.  
The function has been developed primarily for groupings of a continuous variable. If, however, the function is to be used on the frequency distribution of a discrete variable, taking the values y1,,yk-1, then the boundary values for the classes may be defined as follows:
  1. (i)for k>2,
    x1 = (3y1-y2)/2 xj = (yj-1+yj)/2,  j=2,,k-1 xk = (3yk-1-yk-2)/2  
  2. (ii)for k=2,
    x1=y1-a   and   x2=y1+a   for any ​a>0 .  

4 References


5 Arguments

1: k Integer Input
On entry: k, the number of class boundaries, which is one more than the number of classes of the frequency distribution.
Constraint: k>1.
2: x[k] const double Input
On entry: the elements of x must contain the boundary values of the classes in ascending order, so that class i is bounded by the values in x[i-1] and x[i], for i=1,2,,k-1.
Constraint: x[i]<x[i+1], for i=0,1,,k-2.
3: ifreq[k] const Integer Input
On entry: the ith element of ifreq must contain the frequency associated with the ith class, for i=1,2,,k-1. ifreq[k-1] is not used by the function.
  • ifreq[i-1]0, for i=1,2,,k-1;
  • i=1 k-1ifreq[i-1]>0.
4: xmean double * Output
On exit: the mean value, y¯.
5: xsd double * Output
On exit: the standard deviation, s2.
6: xskew double * Output
On exit: the coefficient of skewness, s3.
7: xkurt double * Output
On exit: the coefficient of kurtosis, s4.
8: n Integer * Output
On exit: the total frequency, n.
9: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
Either ifreq[i]<0 for some i, or the sum of frequencies is zero.
The total frequency, n, is less than 2, hence the quantities s2, s3 and s4 cannot be calculated.
On entry, k=value.
Constraint: k>1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, I=value, x[I-2]=value and x[I-1]=value.
Constraint: x[I-2]x[I-1].

7 Accuracy

The method used is believed to be stable.

8 Parallelism and Performance

g01adc is not threaded in any implementation.

9 Further Comments

The time taken by g01adc increases linearly with k.

10 Example

In the example program, NPROB determines the number of sets of data to be analysed. For each analysis, the boundary values of the classes and the frequencies are read. After g01adc has been successfully called, the input data and calculated quantities are printed. In the example, there is one set of data, with 14 classes.

10.1 Program Text

Program Text (g01adce.c)

10.2 Program Data

Program Data (g01adce.d)

10.3 Program Results

Program Results (g01adce.r)