# NAG Library Routine Document

## 1Purpose

g01ahf performs a Normal probability plot on a character printing device, with a chosen number of character positions in each direction.

## 2Specification

Fortran Interface
 Subroutine g01ahf ( x, nobs, xbar, xstd,
 Integer, Intent (In) :: nobs, nstepx, nstepy, istand, lwork Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: iwork(nobs) Real (Kind=nag_wp), Intent (In) :: x(nobs) Real (Kind=nag_wp), Intent (Out) :: nscores(nobs), xsort(nobs), xbar, xstd
#include nagmk26.h
 void g01ahf_ (const double x[], const Integer *nobs, const Integer *nstepx, const Integer *nstepy, const Integer *istand, Integer iwork[], double nscores[], const Integer *lwork, double xsort[], double *xbar, double *xstd, Integer *ifail)

## 3Description

In a Normal probability plot, the data $\left(x\right)$ are plotted against Normal scores $\left(y\right)$. The degree of linearity in the resultant plot provides a visual indication of the Normality of distribution of a set of residuals from some fitting process, such as multiple regression.
The data values are sorted into descending order prior to plotting, and may also be standardized to zero mean and unit standard deviation, if requested.
The plot is produced on a character printing device, using a chosen number of character positions in each direction. The output is directed to the current advisory message unit number (see the Users' Note for your implementation). This number may be changed by an appropriate call to x04abf before calling g01ahf.
Axes are drawn and annotated and data points are plotted on the nearest character position. An appropriate step size for each axis is computed from the list
• $\left(0.1,0.15,0.2,0.25,0.4,0.5,0.6,0.75,0.8\right)×\text{}$ power of $10$.
Points are plotted using the digits $1$ to $9$ to indicate the equivalent number of observations at a particular character position, a letter A–Z for $10–35$ occurrences, or * if there are $36$ or more coincident occurrences. Zero axes are marked if included in the plotting area.

None.

## 5Arguments

1:     $\mathbf{x}\left({\mathbf{nobs}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the vector of data values.
Constraint: all data values must not be equal.
2:     $\mathbf{nobs}$ – IntegerInput
On entry: the number of data values.
Constraint: ${\mathbf{nobs}}\ge 2$.
3:     $\mathbf{nstepx}$ – IntegerInput
On entry: the number of steps (character positions) to be plotted in the $x$-direction. If the supplied value of nstepx is less than $10$, the value $10$ will be used by g01ahf. The maximum value for nstepx is the number of character positions available on the chosen output device less $15$, up to a maximum of $133$. If nstepx exceeds $133$ on input, the value $133$ will be used by the routine.
4:     $\mathbf{nstepy}$ – IntegerInput
On entry: the number of steps (character positions) to be plotted in the $y$-direction. If the supplied value of nstepy is less than $10$, the value $10$ will be used by g01ahf. There is no maximum value for nstepy, but you should bear in mind that (${\mathbf{nstepy}}+5$) records (lines) of output are generated by the routine.
5:     $\mathbf{istand}$ – IntegerInput
On entry: indicates whether the residuals are to be standardized prior to plotting.
If ${\mathbf{istand}}>0$, the elements of x are standardized to zero mean and unit standard deviation.
6:     $\mathbf{iwork}\left({\mathbf{nobs}}\right)$ – Integer arrayWorkspace
7:     $\mathbf{nscores}\left({\mathbf{nobs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the Normal scores in ascending magnitude.
8:     $\mathbf{lwork}$ – IntegerInput
On entry: this argument is no longer referenced, but is included for backwards compatability.
9:     $\mathbf{xsort}\left({\mathbf{nobs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the data values, sorted into descending order, and standardized if istand was positive on entry.
10:   $\mathbf{xbar}$ – Real (Kind=nag_wp)Output
On exit: the mean of the data values.
11:   $\mathbf{xstd}$ – Real (Kind=nag_wp)Output
On exit: the standard deviation of the data values.
12:   $\mathbf{ifail}$ – 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, ${\mathbf{nobs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nobs}}\ge 2$.
${\mathbf{ifail}}=2$
All the supplied data values are equal.
${\mathbf{ifail}}=-99$
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

Accuracy is limited by the number of plotting positions available.

## 8Parallelism and Performance

g01ahf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

For details of timing see g01agf and g01daf.
No blank records are output before or after the plot.
You must make sure that it is permissible to write records containing nstepx characters to the current advisory message unit.

## 10Example

The data are residuals from a linear regression. The $25$ values are standardized and plotted against the Normal scores, and are seen to follow a straight line fairly closely, indicating that Normality assumptions are justified.

### 10.1Program Text

Program Text (g01ahfe.f90)

### 10.2Program Data

Program Data (g01ahfe.d)

### 10.3Program Results

Program Results (g01ahfe.r)

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