# NAG Library Routine Document

## 1Purpose

g01aef constructs a frequency distribution of a variable, according to either user-supplied, or routine-calculated class boundary values.

## 2Specification

Fortran Interface
 Subroutine g01aef ( n, k, x, cb, xmin, xmax,
 Integer, Intent (In) :: n, k, iclass Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ifreq(k) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: cb(k) Real (Kind=nag_wp), Intent (Out) :: xmin, xmax
#include <nagmk26.h>
 void g01aef_ (const Integer *n, const Integer *k, const double x[], const Integer *iclass, double cb[], Integer ifreq[], double *xmin, double *xmax, Integer *ifail)

## 3Description

The data consists of a sample of $n$ observations of a continuous variable, denoted by ${x}_{i}$, for $\mathit{i}=1,2,\dots ,n$. Let $a=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({x}_{1},\dots ,{x}_{n}\right)$ and $b=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({x}_{1},\dots ,{x}_{n}\right)$.
g01aef constructs a frequency distribution with $k\left(>1\right)$ classes denoted by ${f}_{i}$, for $\mathit{i}=1,2,\dots ,k$.
The boundary values may be either user-supplied, or routine-calculated, and are denoted by ${y}_{j}$, for $\mathit{j}=1,2,\dots ,k-1$.
If the boundary values of the classes are to be routine-calculated, then they are determined in one of the following ways:
 (a) if $k>2$, the range of $x$ values is divided into $k-2$ intervals of equal length, and two extreme intervals, defined by the class boundary values ${y}_{1},{y}_{2},\dots ,{y}_{k-1}$; (b) if $k=2$, ${y}_{1}=\frac{1}{2}\left(a+b\right)$.
However formed, the values ${y}_{1},\dots ,{y}_{k-1}$ are assumed to be in ascending order. The class frequencies are formed with
• ${f}_{1}=\text{}$ the number of $x$ values in the interval $\left(-\infty ,{y}_{1}\right)$
• ${f}_{i}=\text{}$ the number of $x$ values in the interval $\left[{y}_{i-1},{y}_{i}\right)$, $\text{ }i=2,\dots ,k-1$
• ${f}_{k}=\text{}$ the number of $x$ values in the interval $\left[{y}_{k-1},\infty \right)$,
where [ means inclusive, and ) means exclusive. If the class boundary values are routine-calculated and $k>2$, then ${f}_{1}={f}_{k}=0$, and ${y}_{1}$ and ${y}_{k-1}$ are chosen so that ${y}_{1} and ${y}_{k-1}>b$.
If a frequency distribution is required for a discrete variable, then it is suggested that you supply the class boundary values; routine-calculated boundary values may be slightly imprecise (due to the adjustment of ${y}_{1}$ and ${y}_{k-1}$ outlined above) and cause values very close to a class boundary to be assigned to the wrong class.

None.

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}\ge 1$.
2:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of classes desired in the frequency distribution. Whether or not class boundary values are user-supplied, k must include the two extreme classes which stretch to $±\infty$.
Constraint: ${\mathbf{k}}\ge 2$.
3:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the sample of observations of the variable for which the frequency distribution is required, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. The values may be in any order.
4:     $\mathbf{iclass}$ – IntegerInput
On entry: indicates whether class boundary values are to be calculated within g01aef, or are supplied by you.
If ${\mathbf{iclass}}=0$, the class boundary values are to be calculated within the routine.
If ${\mathbf{iclass}}=1$, they are user-supplied.
Constraint: ${\mathbf{iclass}}=0$ or $1$.
5:     $\mathbf{cb}\left({\mathbf{k}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{iclass}}=0$, the elements of cb need not be assigned values, as g01aef calculates $k-1$ class boundary values.
If ${\mathbf{iclass}}=1$, the first $k-1$ elements of cb must contain the class boundary values you supplied, in ascending order.
In both cases, the element ${\mathbf{cb}}\left(k\right)$ need not be assigned, as it is not used in the routine.
On exit: the first $k-1$ elements of cb contain the class boundary values in ascending order.
Constraint: if ${\mathbf{iclass}}=1$, ${\mathbf{cb}}\left(\mathit{i}\right)<{\mathbf{cb}}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,k-2$.
6:     $\mathbf{ifreq}\left({\mathbf{k}}\right)$ – Integer arrayOutput
On exit: the elements of ifreq contain the frequencies in each class, ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$. In particular ${\mathbf{ifreq}}\left(1\right)$ contains the frequency of the class up to ${\mathbf{cb}}\left(1\right)$, ${f}_{1}$, and ${\mathbf{ifreq}}\left(k\right)$ contains the frequency of the class greater than ${\mathbf{cb}}\left(k-1\right)$, ${f}_{k}$.
7:     $\mathbf{xmin}$ – Real (Kind=nag_wp)Output
On exit: the smallest value in the sample, $a$.
8:     $\mathbf{xmax}$ – Real (Kind=nag_wp)Output
On exit: the largest value in the sample, $b$.
9:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{cb}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{cb}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{cb}}\left(〈\mathit{\text{value}}〉\right)<{\mathbf{cb}}\left(〈\mathit{\text{value}}〉\right)$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The method used is believed to be stable.

## 8Parallelism and Performance

g01aef is not threaded in any implementation.

The time taken by g01aef increases with k and n. It also depends on the distribution of the sample observations.

## 10Example

This example summarises a number of datasets. For each dataset the sample observations and optionally class boundary values are read. g01aef is then called and the frequency distribution and largest and smallest observations printed.

### 10.1Program Text

Program Text (g01aefe.f90)

### 10.2Program Data

Program Data (g01aefe.d)

### 10.3Program Results

Program Results (g01aefe.r)