nag_stat_summary_1var (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.

Note: this function is scheduled to be withdrawn, please see g01aa in Advice on Replacement Calls for Withdrawn/Superseded Routines..

Syntax

[xmean, s2, s3, s4, xmin, xmax, iwt, wtsum, ifail] = g01aa(x, 'n', n, 'wt', wt)

[xmean, s2, s3, s4, xmin, xmax, iwt, wtsum, ifail] = nag_stat_withdraw_summary_1var(x, 'n', n, 'wt', wt)

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 23:

wt is no longer an output parameter; output parameters were reordered

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.

Parameters

Compulsory Input Parameters

1: $x (n)$ – double array: The sample observations, $x_{i}$ , for $i = 1, 2, \dots, n$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the arrays x, wt. (An error is raised if these dimensions are not equal.)
$n$ , the number of observations.

Constraint: $n \geq 1$ .
2: $wt (n)$ – double array: If the user wishes to supply weights then the elements of wt must contain the weights associated with the observations, $w_{i}$ , for $i = 1, 2, \dots, n$ .

Output Parameters

1: $xmean$ – double scalar: The mean, $\bar{x}$ .
2: $s2$ – double scalar: The standard deviation, $s_{2}$ .
3: $s3$ – double scalar: The coefficient of skewness, $s_{3}$ .
4: $s4$ – double scalar: The coefficient of kurtosis, $s_{4}$ .
5: $xmin$ – double scalar: The smallest value in the sample.
6: $xmax$ – double scalar: The largest value in the sample.
7: $iwt$ – int64int32nag_int scalar: iwt is used to indicate the number of valid observations, $m$ ; see (g) in Description above.
8: $wtsum$ – double scalar: 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.
9: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,

n < 1

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

$ifail = 3$: Either the number of valid cases is $0$ , or at least one weight is negative.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The method used is believed to be stable.

Further Comments

The time taken by nag_stat_summary_1var (g01aa) is approximately proportional to

n

Example

This example summarises an (optionally weighted) dataset and displays the results.

Open in the MATLAB editor: g01aa_example

function g01aa_example


fprintf('g01aa example results\n\n');

% Data
x = [193     216     112     161      92     140     38      33 ...
     279     249     473     339      60     130     20      50 ...
     257     284     447      52      67      61    150    2200];
n = size(x,2);

% Get simple statistics of data
[xmean, s2, s3, s4, xmin, xmax, iwt, wtsum, ifail] = ...
g01aa(x);

fprintf('Number of cases    %7d\n',n);
fprintf('Data as input -\n');
fprintf('%12.1f%12.1f%12.1f%12.1f%12.1f\n',x)
fprintf('\n\n');
fprintf('No. of valid cases %7d\n',iwt);
fprintf('Mean               %7.1f\n',xmean);
fprintf('Minimum            %7.1f\n',xmin);
fprintf('Maximum            %7.1f\n',xmax);
fprintf('Sum of weights     %7.1f\n',wtsum);
fprintf('Std devn           %7.1f\n',s2);
fprintf('Skewness           %7.1f\n',s3);
fprintf('Kurtosis           %7.1f\n',s4);

g01aa example results

Number of cases         24
Data as input -
       193.0       216.0       112.0       161.0        92.0
       140.0        38.0        33.0       279.0       249.0
       473.0       339.0        60.0       130.0        20.0
        50.0       257.0       284.0       447.0        52.0
        67.0        61.0       150.0      2200.0

No. of valid cases      24
Mean                 254.3
Minimum               20.0
Maximum             2200.0
Sum of weights        24.0
Std devn             433.5
Skewness               3.9
Kurtosis              14.7

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox