NAG Library Routine Document

G01ALF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G01ALF calculates a five-point summary for a single sample.

2 Specification

SUBROUTINE G01ALF (

N, X, IWRK, RES, IFAIL)

INTEGER	N, IWRK(N), IFAIL
REAL (KIND=nag_wp)	X(N), RES(5)

3 Description

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}

n

is even and

\frac{(n + 1)}{2}

n

is odd,

and

k = \frac{m}{2}

m

is even and

\frac{(m + 1)}{2}

m

is odd.

Then we have

Minimum	$= z_{1}$ ,
Maximum	$= z_{n}$ ,
Median	$= z_{m}$	if $n$ is odd,
	$= \frac{z_{m} + z_{m + 1}}{2}$	if $n$ is even,
Lower hinge	$= z_{k}$	if $m$ is odd,
	$= \frac{z_{k} + z_{k + 1}}{2}$	if $m$ is even,
Upper hinge	$= z_{n - k + 1}$	if $m$ is odd,
	$= \frac{z_{n - k} + z_{n - k + 1}}{2}$	if $m$ is even.

4 References

Erickson B H and Nosanchuk T A (1985) Understanding Data Open University Press, Milton Keynes

Tukey J W (1977) Exploratory Data Analysis Addison–Wesley

5 Parameters

1: N – INTEGERInput

On entry:

n

, number of observations in the sample.

Constraint:

N \geq 5

2: X(N) – REAL (KIND=nag_wp) arrayInput

On entry: the sample observations,

x_{1}, x_{2}, \dots, x_{n}

3: IWRK(N) – INTEGER arrayWorkspace

4: RES( $5$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: RES contains the five-point summary.

$RES (1)$: The minimum.
$RES (2)$: The lower hinge.
$RES (3)$: The median.
$RES (4)$: The upper hinge.
$RES (5)$: The maximum.

5: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ 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 parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,

N < 5

7 Accuracy

The computations are stable.

8 Further Comments

The time taken by G01ALF is proportional to

n

9 Example

This example calculates a five-point summary for a sample of

12