# NAG Library Function Document

## 1Purpose

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

## 2Specification

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

## 3Description

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 $m=\frac{n}{2}$ 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:    $\mathbf{x}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the sample observations, ${x}_{1},{x}_{2},\dots ,{x}_{n}$.
3:    $\mathbf{res}\left[5\right]$doubleOutput
On exit: res contains the five-point summary.
${\mathbf{res}}\left[0\right]$
The minimum.
${\mathbf{res}}\left[1\right]$
The lower hinge.
${\mathbf{res}}\left[2\right]$
The median.
${\mathbf{res}}\left[3\right]$
The upper hinge.
${\mathbf{res}}\left[4\right]$
The maximum.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT_ARG_LT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 5$.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The computations are stable.

## 8Parallelism and Performance

nag_5pt_summary_stats (g01alc) is not threaded in any implementation.

The time taken by nag_5pt_summary_stats (g01alc) is proportional to $n$.

## 10Example

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

### 10.1Program Text

Program Text (g01alce.c)

### 10.2Program Data

Program Data (g01alce.d)

### 10.3Program Results

Program Results (g01alce.r)

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