NAG Library Function Document

nag_summary_stats_freq (g01adc)

+− 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_summary_stats_freq (g01adc) calculates the mean, standard deviation and coefficients of skewness and kurtosis for data grouped in a frequency distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

void	nag_summary_stats_freq (Integer k, const double x[], const Integer ifreq[], double xmean, double xsd, double xskew, double xkurt, Integer n, NagError fail)

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 nag_summary_stats_freq (g01adc) 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 function has been developed primarily for groupings of a continuous variable. If, however, the function 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 Arguments

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] – const doubleInput

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 - 1]

and

x [i]

, for

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

Constraint:

x [i] < x [i + 1]

, for

i = 0, 1, \dots, k - 2

3: ifreq[k] – const IntegerInput

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 - 1]

is not used by the function.

Constraints:

$ifreq [i - 1] \geq 0$ , for $i = 1, 2, \dots, k - 1$ ;
$\sum_{i = 1}^{k - 1} ifreq [i - 1] > 0$ .

4: xmean – double *Output

On exit: the mean value,

\bar{y}

5: xsd – double *Output

On exit: the standard deviation,

s_{2}

6: xskew – double *Output

On exit: the coefficient of skewness,

s_{3}

7: xkurt – double *Output

On exit: the coefficient of kurtosis,

s_{4}

8: n – Integer *Output

On exit: the total frequency,

n

9: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_FREQ_CONS: Either $ifreq [i] < 0$ for some $i$ , or the sum of frequencies is zero.
NE_FREQ_SUM: The total frequency is less than $2$ .
NE_INT: On entry, $k = ⟨value⟩$ .
Constraint: $k > 1$ .
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.
NE_NOT_INCREASING: On entry, $I = ⟨value⟩$ , $x [I - 2] = ⟨value⟩$ and $x [I - 1] = ⟨value⟩$ .
Constraint: $x [I - 2] \leq x [I - 1]$ .

7 Accuracy

The method used is believed to be stable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_summary_stats_freq (g01adc) 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 nag_summary_stats_freq (g01adc) 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 Function Documentnag_summary_stats_freq (g01adc)

+− 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

NAG Library Function Document

nag_summary_stats_freq (g01adc)