naginterfaces.library.stat.summary_​freq

naginterfaces.library.stat.summary_freq(x, ifreq)

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

For full information please refer to the NAG Library document for g01ad

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/g01/g01adf.html

Parameters

xfloat, array-like, shape $\left(k\right)$

The elements of $x$ must contain the boundary values of the classes in ascending order, so that class $\text{i}$ is bounded by the values in $x[\text{i}-1]$ and $x\left[\text{i}\right]$, for $\text{i}=1,2,\dots ,k-1$.

ifreqint, array-like, shape $\left(k\right)$

The $\text{i}$th element of $ifreq$ must contain the frequency associated with the $\text{i}$th class, for $\text{i}=1,2,\dots ,k-1$. $ifreq[k-1]$ is not used by the function.

Returns

xmeanfloat

The mean value, $\overline{y}$.

s2float

The standard deviation, $s_2$.

s3float

The coefficient of skewness, $s_3$.

s4float

The coefficient of kurtosis, $s_4$.

nint

The total frequency, $n$.

Raises

NagValueError

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k>1$.

(errno $2$)

On entry, $\text{I}=⟨value⟩$, $x[\text{I}-2]=⟨value⟩$ and $x[\text{I}-1]=⟨value⟩$.

Constraint: $x[\text{I}-2]\le x[\text{I}-1]$.

(errno $3$)

Either $ifreq\left[i\right]<0$ for some $i$, or the sum of frequencies is zero.

Warns

NagAlgorithmicWarning

(errno $4$)

The total frequency, $n$, is less than $2$, hence the quantities $s_2$, $s_3$ and $s_4$ cannot be calculated.

Notes

The input data consist of a univariate frequency distribution, denoted by $f_i$, for $\text{i}=1,2,\dots ,k-1$, and the boundary values of the classes $x_i$, for $\text{i}=1,2,\dots ,k$. Thus the frequency associated with the interval $(x_i,x_{i+1})$ is $f_i$, and summary_freq assumes that all the values in this interval are concentrated at the point

$$y_i=(x_{i+1}+x_i)/2\text{,}i=1,2,\dots ,k-1\text{.}$$

The following quantities are calculated:

1. total frequency,

   $$n=\sum _{i=1}^{k-1}{f}_i\text{.}$$

2. mean,

   $$\overline{y}=\frac{{\sum }_{i=1}^{k-1}{f}_i{y}_i}{n}\text{.}$$

3. standard deviation,

   $$s_2=\sqrt{\frac{{\sum }_{i=1}^{k-1}{f}_i{(y_i-\overline{y})}^2}{(n-1)}}\text{,}n\ge 2\text{.}$$

4. coefficient of skewness,

   $$s_3=\frac{{\sum }_{i=1}^{k-1}{f}_i{(y_i-\overline{y})}^3}{(n-1)\times s_2^3}\text{,}n\ge 2\text{.}$$

5. coefficient of kurtosis,

   $$s_4=\frac{{\sum }_{i=1}^{k-1}{f}_i{(y_i-\overline{y})}^4}{(n-1)\times s_2^4}-3\text{,}n\ge 2\text{.}$$

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. for $k>2$,

   $$\begin{array}{c}\begin{array}{ccc}{x}_1& =& (3y_1-y_2)/2\\ x_j& =& (y_{j-1}+y_j)/2\text{,}j=2,\dots ,k-1\\ x_k& =& (3y_{k-1}-y_{k-2})/2\end{array}\end{array}$$

2. for $k=2$,

   $$x_1=y_1-a\text{and}{x}_2=y_1+a\text{for any}a>0\text{.}$$