NAG Library Function Document

nag_5pt_summary_stats (g01alc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_5pt_summary_stats (g01alc) calculates a five-point summary for a single sample.

2 Specification

#include <nag.h>

#include <nagg01.h>

void	nag_5pt_summary_stats (Integer n, const double x[], double res[], NagError *fail)

3 Description

nag_5pt_summary_stats (g01alc) 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 Arguments

1: n – IntegerInput

On entry:

n

, number of observations in the sample.

Constraint:

n \geq 5

2: x[n] – const doubleInput

On entry: the sample observations,

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

3: res[ $5$ ] – doubleOutput

On exit: res contains the five-point summary.

$res [0]$: The minimum.
$res [1]$: The lower hinge.
$res [2]$: The median.
$res [3]$: The upper hinge.
$res [4]$: The maximum.

4: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 5$ .
NE_INTERNAL_ERROR: 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.

7 Accuracy

The computations are stable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_5pt_summary_stats (g01alc) is proportional to

n

10 Example

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

12