naginterfaces.library.stat.summary_​2var

naginterfaces.library.stat.summary_2var(x1, x2, wt=None)

summary_2var computes the means, standard deviations, corrected sums of squares and products, maximum and minimum values, and the product-moment correlation coefficient for two variables. Unequal weighting may be given.

For full information please refer to the NAG Library document for g01ab

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01abf.html

Parameters

x1float, array-like, shape $\left(n\right)$

The observations from the first sample, $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

x2float, array-like, shape $\left(n\right)$

The observations from the second sample, $y_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

wtNone or float, array-like, shape $\left(n\right)$, optional

If weights are being supplied then the elements of $wt$ must contain the weights associated with the observations, $w_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

If weights are not supplied then $wt$ may be None.

Returns

nvalint

Is used to indicate the number of valid observations, $m$; see Notes (g).

resfloat, ndarray, shape $\left(13\right)$

The elements of $res$ contain the following results:

$res\left[0\right]$	mean of the first sample, $\overline{x}$;
$res\left[1\right]$	mean of the second sample, $\overline{y}$;
$res\left[2\right]$	standard deviation of the first sample, $s_1$;
$res\left[3\right]$	standard deviation of the second sample, $s_2$;
$res\left[4\right]$	corrected sum of squares of the first sample, $c_{11}$;
$res\left[5\right]$	corrected sum of products of the two samples, $c_{12}$;
$res\left[6\right]$	corrected sum of squares of the second sample, $c_{22}$;
$res\left[7\right]$	product-moment correlation coefficient, $R$;
$res\left[8\right]$	minimum of the first sample;
$res\left[9\right]$	maximum of the first sample;
$res\left[10\right]$	minimum of the second sample;
$res\left[11\right]$	maximum of the second sample;
$res\left[12\right]$	sum of weights, ${\sum }_{i=1}^n{w}_i$ ($=n$, if $\text{iwt}=0$, on entry).

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $3$)

The number of valid cases, $m$, is $0$.

(errno $3$)

On entry, $wt[⟨value⟩]=⟨value⟩$.

Constraint: $wt[\text{i}-1]\ge 0.0$, for $\text{i}=1,2,\dots ,n$

Warns

NagAlgorithmicWarning

(errno $2$)

The number of valid cases, $m$, is $1$. In this case standard deviation and product-moment correlation coefficient cannot be calculated.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

The data consist of two samples of $n$ observations, denoted by $x_i$, and $y_i$, for $i=1,2,\dots ,n$, with corresponding weights $w_i$, for $\text{i}=1,2,\dots ,n$.

If no specific weighting is given, then each $w_i$ is set to $1.0$ in summary_2var.

The quantities calculated are:

1. The sum of weights,

   $$W=\sum _{i=1}^n{w}_i\text{.}$$

2. The means,

   $$\overline{x}=\frac{{\sum }_{i=1}^n{w}_i{x}_i}{W}\text{,}\overline{y}=\frac{{\sum }_{i=1}^n{w}_i{y}_i}{W}\text{.}$$

3. The corrected sums of squares and products

   $$\begin{array}{c}\begin{array}{c}{c}_{11}={\sum }_{i=1}^n{w}_i{(x_i-\overline{x})}^2\\ \\ c_{21}=c_{12}={\sum }_{i=1}^n{w}_i(x_i-\overline{x})(y_i-\overline{y})\\ \\ c_{22}={\sum }_{i=1}^n{w}_i{(y_i-\overline{y})}^2\text{.}\end{array}\end{array}$$

4. The standard deviations

   $$s_j=\sqrt{\frac{c_{jj}}{d}}\text{, where}j=1,2\text{and}d=W-\frac{{\sum }_{i=1}^n{w}_i^2}{W}\text{.}$$

5. The product-moment correlation coefficient

   $$R=\frac{c_{12}}{\sqrt{c_{11}{c}_{22}}}\text{.}$$

6. The minimum and maximum elements in each of the two samples.

7. The number of pairs of observations, $m$, for which $w_i>0$, i.e., the number of valid observations. The quantities in (d) and (e) above will only be computed if $m\ge 2$. All other items are computed if $m\ge 1$.