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## 1Purpose

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

## 2Specification

 #include
 void g01adc (Integer k, const double x[], const Integer ifreq[], double *xmean, double *xsd, double *xskew, double *xkurt, Integer *n, NagError *fail)
The function may be called by the names: g01adc, nag_stat_summary_freq or nag_summary_stats_freq.

## 3Description

The input data consist of a univariate frequency distribution, denoted by ${f}_{i}$, for $\mathit{i}=1,2,\dots ,k-1$, and the boundary values of the classes ${x}_{i}$, for $\mathit{i}=1,2,\dots ,k$. Thus the frequency associated with the interval $\left({x}_{i},{x}_{i+1}\right)$ is ${f}_{i}$, and g01adc assumes that all the values in this interval are concentrated at the point
 $yi=(xi+1+xi)/2, i=1,2,…,k-1.$
The following quantities are calculated:
1. (a)total frequency,
 $n=∑i= 1 k- 1fi.$
2. (b)mean,
 $y¯=∑i=1 k-1fiyin.$
3. (c)standard deviation,
 $s2=∑i= 1 k- 1fi (yi-y¯) 2 (n-1) , n≥ 2.$
4. (d)coefficient of skewness,
 $s3=∑i=1 k-1fi (yi-y¯) 3 (n-1)×s23 , n≥2.$
5. (e)coefficient of kurtosis,
 $s4=∑i= 1 k- 1fi (yi-y¯) 4 (n-1)×s24 - 3, n≥ 2.$
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:
1. (i)for $k>2$,
 $x1 = (3y1-y2)/2 xj = (yj-1+yj)/2, j=2,…,k-1 xk = (3yk-1-yk-2)/2$
2. (ii)for $k=2$,
 $x1=y1-a and x2=y1+a for any ​a>0 .$

None.

## 5Arguments

1: $\mathbf{k}$Integer Input
On entry: $k$, the number of class boundaries, which is one more than the number of classes of the frequency distribution.
Constraint: ${\mathbf{k}}>1$.
2: $\mathbf{x}\left[{\mathbf{k}}\right]$const double Input
On entry: the elements of x must contain the boundary values of the classes in ascending order, so that class $\mathit{i}$ is bounded by the values in ${\mathbf{x}}\left[\mathit{i}-1\right]$ and ${\mathbf{x}}\left[\mathit{i}\right]$, for $\mathit{i}=1,2,\dots ,k-1$.
Constraint: ${\mathbf{x}}\left[\mathit{i}\right]<{\mathbf{x}}\left[\mathit{i}+1\right]$, for $\mathit{i}=0,1,\dots ,k-2$.
3: $\mathbf{ifreq}\left[{\mathbf{k}}\right]$const Integer Input
On entry: the $\mathit{i}$th element of ifreq must contain the frequency associated with the $\mathit{i}$th class, for $\mathit{i}=1,2,\dots ,k-1$. ${\mathbf{ifreq}}\left[k-1\right]$ is not used by the function.
Constraints:
• ${\mathbf{ifreq}}\left[\mathit{i}-1\right]\ge 0$, for $\mathit{i}=1,2,\dots ,k-1$;
• $\sum _{i=1}^{k-1}{\mathbf{ifreq}}\left[i-1\right]>0$.
4: $\mathbf{xmean}$double * Output
On exit: the mean value, $\overline{y}$.
5: $\mathbf{xsd}$double * Output
On exit: the standard deviation, ${s}_{2}$.
6: $\mathbf{xskew}$double * Output
On exit: the coefficient of skewness, ${s}_{3}$.
7: $\mathbf{xkurt}$double * Output
On exit: the coefficient of kurtosis, ${s}_{4}$.
8: $\mathbf{n}$Integer * Output
On exit: the total frequency, $n$.
9: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_FREQ_CONS
Either ${\mathbf{ifreq}}\left[i\right]<0$ 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, ${\mathbf{k}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{k}}>1$.
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, $\mathit{I}=⟨\mathit{\text{value}}⟩$, ${\mathbf{x}}\left[\mathit{I}-2\right]=⟨\mathit{\text{value}}⟩$ and ${\mathbf{x}}\left[\mathit{I}-1\right]=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\left[\mathit{I}-2\right]\le {\mathbf{x}}\left[\mathit{I}-1\right]$.

## 7Accuracy

The method used is believed to be stable.

## 8Parallelism and Performance

The time taken by g01adc increases linearly with $k$.

## 10Example

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.