(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 $k > 2$ ,

$\begin{array}{rcl} x_{1} & = & (3 y_{1} - y_{2}) / 2 \\ x_{j} & = & (y_{j - 1} + y_{j}) / 2, j = 2, \dots, k - 1 \\ x_{k} & = & (3 y_{k - 1} - y_{k - 2}) / 2 \end{array}$
(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: $k$ – Integer Input

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 double Input

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 Integer Input

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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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, $n$ , is less than $2$ , hence the quantities $s_{2}$ , $s_{3}$ and $s_{4}$ cannot be calculated.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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

g01adc is not threaded in any implementation.

9 Further Comments

The time taken by 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 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.

g01ad: FL CL CPP AD

NAG CL Interfaceg01adc (summary_​freq)

▸▿ 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 CL Interface
g01adc (summary_freq)