g01aa calculates the mean, standard deviation, coefficients of skewness and kurtosis, and the maximum and minimum values for a set of ungrouped data. Weighting may be used.

Syntax

C#
public static void g01aa( int n, double[] x, ref int iwt, double[] wt, out double xmean, out double s2, out double s3, out double s4, out double xmin, out double xmax, out double wtsum, out int ifail )

public static void g01aa(
	int n,
	double[] x,
	ref int iwt,
	double[] wt,
	out double xmean,
	out double s2,
	out double s3,
	out double s4,
	out double xmin,
	out double xmax,
	out double wtsum,
	out int ifail
)

Visual Basic
Public Shared Sub g01aa ( _ n As Integer, _ x As Double(), _ ByRef iwt As Integer, _ wt As Double(), _ <OutAttribute> ByRef xmean As Double, _ <OutAttribute> ByRef s2 As Double, _ <OutAttribute> ByRef s3 As Double, _ <OutAttribute> ByRef s4 As Double, _ <OutAttribute> ByRef xmin As Double, _ <OutAttribute> ByRef xmax As Double, _ <OutAttribute> ByRef wtsum As Double, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g01aa ( _
	n As Integer, _
	x As Double(), _
	ByRef iwt As Integer, _
	wt As Double(), _
	<OutAttribute> ByRef xmean As Double, _
	<OutAttribute> ByRef s2 As Double, _
	<OutAttribute> ByRef s3 As Double, _
	<OutAttribute> ByRef s4 As Double, _
	<OutAttribute> ByRef xmin As Double, _
	<OutAttribute> ByRef xmax As Double, _
	<OutAttribute> ByRef wtsum As Double, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g01aa( int n, array<double>^ x, int% iwt, array<double>^ wt, [OutAttribute] double% xmean, [OutAttribute] double% s2, [OutAttribute] double% s3, [OutAttribute] double% s4, [OutAttribute] double% xmin, [OutAttribute] double% xmax, [OutAttribute] double% wtsum, [OutAttribute] int% ifail )

Visual C++

public:
static void g01aa(
	int n, 
	array<double>^ x, 
	int% iwt, 
	array<double>^ wt, 
	[OutAttribute] double% xmean, 
	[OutAttribute] double% s2, 
	[OutAttribute] double% s3, 
	[OutAttribute] double% s4, 
	[OutAttribute] double% xmin, 
	[OutAttribute] double% xmax, 
	[OutAttribute] double% wtsum, 
	[OutAttribute] int% ifail
)

F#
static member g01aa : n : int * x : float[] * iwt : int byref * wt : float[] * xmean : float byref * s2 : float byref * s3 : float byref * s4 : float byref * xmin : float byref * xmax : float byref * wtsum : float byref * ifail : int byref -> unit

static member g01aa : 
        n : int * 
        x : float[] * 
        iwt : int byref * 
        wt : float[] * 
        xmean : float byref * 
        s2 : float byref * 
        s3 : float byref * 
        s4 : float byref * 
        xmin : float byref * 
        xmax : float byref * 
        wtsum : float byref * 
        ifail : int byref -> unit

Parameters

n: Type: System..::..Int32
On entry: $n$ , the number of observations.

Constraint: $n \geq 1$ .

x: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: the sample observations, $x_{i}$ , for $i = 1, 2, \dots, n$ .

iwt

Type: System..::..Int32%

On entry: indicates whether weights are to be supplied by you or not. In the latter case, the weights will be assumed equal and assigned the value

1.0

in the method.

$iwt = 0$: Indicates no user-supplied weights.
$iwt = 1$: Indicates user-supplied weights are required, and they will be supplied in the array wt.

On exit: iwt is used to indicate the number of valid observations,

m

; see (g) in [Description] above.

wt: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: if $iwt = 1$ , the elements of wt must contain the weights associated with the observations, $w_{i}$ , for $i = 1, 2, \dots, n$ .
If $iwt = 0$ , the elements of wt need not be set.

On exit: if $iwt = 1$ , the elements of wt are unchanged.
If $iwt = 0$ , each element of wt will be assigned the value $1.0$ .

xmean: Type: System..::..Double%
On exit: the mean, $\bar{x}$ .

s2: Type: System..::..Double%
On exit: the standard deviation, $s_{2}$ .

s3: Type: System..::..Double%
On exit: the coefficient of skewness, $s_{3}$ .

s4: Type: System..::..Double%
On exit: the coefficient of kurtosis, $s_{4}$ .

xmin: Type: System..::..Double%
On exit: the smallest value in the sample.

xmax: Type: System..::..Double%
On exit: the largest value in the sample.

wtsum: Type: System..::..Double%
On exit: the sum of the weights in the array wt, that is $\sum_{i = 1}^{n} w_{i}$ . This will be n if iwt was $0$ on entry.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

The data consist of a single sample of

n

observations, denoted by

x_{i}

, with corresponding weights,

w_{i}

, for

i = 1, 2, \dots, n

If no specific weighting is required, then each

w_{i}

is set to

1

The quantities computed are:

(a)

The sum of the weights

W = \sum_{i = 1}^{n} w_{i} .

(b)

Mean

\bar{x} = \frac{\sum_{i = 1}^{n} w_{i} x_{i}}{W} .

(c)

Standard deviation

s_{2} = \sqrt{\frac{\sum_{i = 1}^{n} w_{i} {(x_{i} - \bar{x})}^{2}}{d}},   where d = W - \frac{\sum_{i = 1}^{n} w_{i}^{2}}{W} .

(d)

Coefficient of skewness

s_{3} = \frac{\sum_{i = 1}^{n} w_{i} {(x_{i} - \bar{x})}^{3}}{d \times s_{2}^{3}} .

(e)

Coefficient of kurtosis

s_{4} = \frac{\sum_{i = 1}^{n} w_{i} {(x_{i} - \bar{x})}^{4}}{d \times s_{2}^{4}} - 3 .

(f)

Maximum and minimum elements of the sample.

(g)

The number of observations for which

w_{i} > 0

, i.e., the number of valid observations. Suppose

m

observations are valid, then the quantities in (c), (d) and (e) will be computed if

m \geq 2

, and will be based on

m - 1

degrees of freedom. The other quantities are evaluated provided

m \geq 1

References

None.

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

On entry,

n < 1

$ifail = 2$: The number of valid cases, $m$ , is $1$ . In this case, standard deviation and coefficients of skewness and of kurtosis cannot be calculated.