NAG Library Routine Document

G01ADF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

G01ADF calculates the mean, standard deviation and coefficients of skewness and kurtosis for data grouped in a frequency distribution.

2 Specification

SUBROUTINE G01ADF (

K, X, IFREQ, XMEAN, S2, S3, S4, N, IFAIL)

INTEGER	K, IFREQ(K), N, IFAIL
REAL (KIND=nag_wp)	X(K), XMEAN, S2, S3, S4

3 Description

The input data consist of a univariate frequency distribution, denoted by

f_{i}

, for

i = 1, 2, \dots, k - 1

, and the boundary values of the classes

x_{i}

, for

i = 1, 2, \dots, k

. Thus the frequency associated with the interval

(x_{i}, x_{i + 1})

f_{i}

, and G01ADF assumes that all the values in this interval are concentrated at the point

y_{i} = (x_{i + 1} + x_{i}) / 2, i = 1, 2, \dots, k - 1 .

The following quantities are calculated:

(a)

total frequency,

n = \sum_{i = 1}^{k - 1} f_{i} .

(b)

mean,

\bar{y} = \frac{\sum_{i = 1}^{k - 1} f_{i} y_{i}}{n} .

(c)

standard deviation,

s_{2} = \sqrt{\frac{\sum_{i = 1}^{k - 1} f_{i} {(y_{i} - \bar{y})}^{2}}{(n - 1)}}, n \geq 2 .

(d)

coefficient of skewness,

s_{3} = \frac{\sum_{i = 1}^{k - 1} f_{i} {(y_{i} - \bar{y})}^{3}}{(n - 1) \times s_{2}^{3}}, n \geq 2 .

(e)

coefficient of kurtosis,

s_{4} = \frac{\sum_{i = 1}^{k - 1} f_{i} {(y_{i} - \bar{y})}^{4}}{(n - 1) \times s_{2}^{4}} - 3, n \geq 2 .

The routine has been developed primarily for groupings of a continuous variable. If, however, the routine is to be used on the frequency distribution of a discrete variable, taking the values

y_{1}, \dots, y_{k - 1}

, then the boundary values for the classes may be defined as follows:

(i)

for

k > 2

\begin{array}{l} x_{1} = (3 y_{1} - y_{2}) / 2 \\ x_{j} = (y_{j - 1} + y_{j}) / 2, & j = 2, \dots, k - 1 \\ x_{k} = (3 y_{k - 1} - y_{k - 2}) / 2 \end{array}

(ii)

for

k = 2

x_{1} = y_{1} - a and x_{2} = y_{1} + a for any ​ a > 0 .

4 References

None.

5 Parameters

1: $K$ – INTEGERInput

On entry:

k

, the number of class boundaries, which is one more than the number of classes of the frequency distribution.

Constraint:

K > 1

2: $X (K)$ – REAL (KIND=nag_wp) arrayInput

On entry: the elements of X must contain the boundary values of the classes in ascending order, so that class

i

is bounded by the values in

X (i)

and

X (i + 1)

, for

i = 1, 2, \dots, k - 1

Constraint:

X (i) < X (i + 1)

, for

i = 1, 2, \dots, k - 1

3: $IFREQ (K)$ – INTEGER arrayInput

On entry: the

i

th element of IFREQ must contain the frequency associated with the

i

th class, for

i = 1, 2, \dots, k - 1

IFREQ (k)

is not used by the routine.

Constraints:

$IFREQ (i) \geq 0$ , for $i = 1, 2, \dots, k - 1$ ;
$\sum_{i = 1}^{k - 1} IFREQ (i) > 0$ .

4: $XMEAN$ – REAL (KIND=nag_wp)Output

On exit: the mean value,

\bar{y}

5: $S2$ – REAL (KIND=nag_wp)Output

On exit: the standard deviation,

s_{2}

6: $S3$ – REAL (KIND=nag_wp)Output

On exit: the coefficient of skewness,

s_{3}

7: $S4$ – REAL (KIND=nag_wp)Output

On exit: the coefficient of kurtosis,

s_{4}

8: $N$ – INTEGEROutput

On exit: the total frequency,

n

9: $IFAIL$ – 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,

K \leq 1

$IFAIL = 2$

On entry,

the boundary values of the classes in X are not in ascending order.

$IFAIL = 3$

On entry,

\sum_{i = 1}^{k - 1} IFREQ (i) = 0

IFREQ (i) < 0

for some

i

, for

i = 1, 2, \dots, k - 1

$IFAIL = 4$: The total frequency, $n$ , is less than $2$ , hence the quantities $s_{2}$ , $s_{3}$ and $s_{4}$ cannot be calculated.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

The method used is believed to be stable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by G01ADF increases linearly with

k

10 Example

In the example program, NPROB determines the number of sets of data to be analysed. For each analysis, the boundary values of the classes and the frequencies are read. After G01ADF has been successfully called, the input data and calculated quantities are printed. In the example, there is one set of data, with

14

classes.

NAG Library Routine DocumentG01ADF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

NAG Library Routine Document

G01ADF