# NAG Library Routine Document

## 1Purpose

g01alf calculates a five-point summary for a single sample.

## 2Specification

Fortran Interface
 Subroutine g01alf ( n, x, iwrk, res,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iwrk(n) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: res(5)
#include nagmk26.h
 void g01alf_ (const Integer *n, const double x[], Integer iwrk[], double res[], Integer *ifail)

## 3Description

g01alf calculates the minimum, lower hinge, median, upper hinge and the maximum of a sample of $n$ observations.
The data consist of a single sample of $n$ observations denoted by ${x}_{i}$ and let ${z}_{i}$, for $i=1,2,\dots ,n$, represent the sample observations sorted into ascending order.
Let $m=\frac{n}{2}$ if $n$ is even and $\frac{\left(n+1\right)}{2}$ if $n$ is odd,
and $k=\frac{m}{2}$ if $m$ is even and $\frac{\left(m+1\right)}{2}$ if $m$ is odd.
Then we have
 Minimum $\text{}={z}_{1}$, Maximum $\text{}={z}_{n}$, Median $\text{}={z}_{m}$ if $n$ is odd, $\text{}=\frac{{z}_{m}+{z}_{m+1}}{2}$ if $n$ is even, $\phantom{\frac{1}{2}}$ Lower hinge $\text{}={z}_{k}$ if $m$ is odd, $\text{}=\frac{{z}_{k}+{z}_{k+1}}{2}$ if $m$ is even, $\phantom{\frac{1}{2}}$ Upper hinge $\text{}={z}_{n-k+1}$ if $m$ is odd, $\text{}=\frac{{z}_{n-k}+{z}_{n-k+1}}{2}$ if $m$ is even.$\phantom{\frac{1}{2}}$

## 4References

Erickson B H and Nosanchuk T A (1985) Understanding Data Open University Press, Milton Keynes
Tukey J W (1977) Exploratory Data Analysis Addison–Wesley

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, number of observations in the sample.
Constraint: ${\mathbf{n}}\ge 5$.
2:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the sample observations, ${x}_{1},{x}_{2},\dots ,{x}_{n}$.
3:     $\mathbf{iwrk}\left({\mathbf{n}}\right)$ – Integer arrayWorkspace
4:     $\mathbf{res}\left(5\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: res contains the five-point summary.
${\mathbf{res}}\left(1\right)$
The minimum.
${\mathbf{res}}\left(2\right)$
The lower hinge.
${\mathbf{res}}\left(3\right)$
The median.
${\mathbf{res}}\left(4\right)$
The upper hinge.
${\mathbf{res}}\left(5\right)$
The maximum.
5:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 5$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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 computations are stable.

## 8Parallelism and Performance

g01alf is not threaded in any implementation.

The time taken by g01alf is proportional to $n$.

## 10Example

This example calculates a five-point summary for a sample of $12$ observations.

### 10.1Program Text

Program Text (g01alfe.f90)

### 10.2Program Data

Program Data (g01alfe.d)

### 10.3Program Results

Program Results (g01alfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017