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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_coeffs_pearson (g02ba)

## Purpose

nag_correg_coeffs_pearson (g02ba) computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data.

## Syntax

[xbar, std, ssp, r, ifail] = g02ba(x, 'n', n, 'm', m)
[xbar, std, ssp, r, ifail] = nag_correg_coeffs_pearson(x, 'n', n, 'm', m)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 22: n was made optional

## Description

The input data consist of $n$ observations for each of $m$ variables, given as an array
 $xij, i=1,2,…,nn≥2,j=1,2,…,mm≥2,$
where ${x}_{ij}$ is the $i$th observation on the $j$th variable.
The quantities calculated are:
(a) Means:
 $x-j=1n∑i=1nxij, j=1,2,…,m.$
(b) Standard deviations:
 $sj=1n- 1 ∑i= 1n xij-x-j 2, j= 1,2,…,m.$
(c) Sums of squares and cross-products of deviations from means:
 $Sjk=∑i=1n xij-x-j xik-x-k , j,k=1,2,…,m.$
(d) Pearson product-moment correlation coefficients:
 $Rjk=SjkSjjSkk , j,k= 1,2,…,m.$
If ${S}_{jj}$ or ${S}_{kk}$ is zero, ${R}_{jk}$ is set to zero.

None.

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}\left(\mathit{ldx},{\mathbf{m}}\right)$ – double array
ldx, the first dimension of the array, must satisfy the constraint $\mathit{ldx}\ge {\mathbf{n}}$.
${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must be set to ${x}_{\mathit{i}\mathit{j}}$, the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array x.
$n$, the number of observations or cases.
Constraint: ${\mathbf{n}}\ge 2$.
2:     $\mathrm{m}$int64int32nag_int scalar
Default: the second dimension of the array x.
$m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 2$.

### Output Parameters

1:     $\mathrm{xbar}\left({\mathbf{m}}\right)$ – double array
The mean value, ${\stackrel{-}{x}}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
2:     $\mathrm{std}\left({\mathbf{m}}\right)$ – double array
The standard deviation, ${s}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
3:     $\mathrm{ssp}\left(\mathit{ldssp},{\mathbf{m}}\right)$ – double array
${\mathbf{ssp}}\left(\mathit{j},\mathit{k}\right)$ is the cross-product of deviations ${S}_{\mathit{j}\mathit{k}}$, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
4:     $\mathrm{r}\left(\mathit{ldr},{\mathbf{m}}\right)$ – double array
${\mathbf{r}}\left(\mathit{j},\mathit{k}\right)$ is the product-moment correlation coefficient ${R}_{\mathit{j}\mathit{k}}$ between the $\mathit{j}$th and $\mathit{k}$th variables, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
5:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<2$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{m}}<2$.
${\mathbf{ifail}}=3$
 On entry, $\mathit{ldx}<{\mathbf{n}}$, or $\mathit{ldssp}<{\mathbf{m}}$, or $\mathit{ldr}<{\mathbf{m}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

nag_correg_coeffs_pearson (g02ba) does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large $n$.

The time taken by nag_correg_coeffs_pearson (g02ba) depends on $n$ and $m$.
The function uses a two-pass algorithm.

## Example

This example reads in a set of data consisting of five observations on each of three variables. The means, standard deviations, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for all three variables are then calculated and printed.
```function g02ba_example

fprintf('g02ba example results\n\n');

x = [ 2,  3, 3;
4,  6, 4;
9,  9, 0;
0, 12, 2;
12, -1, 5];
[n,m] = size(x);
fprintf('Number of variables (columns) = %d\n', m);
fprintf('Number of cases     (rows)    = %d\n\n', n);
disp('Data matrix is:-');
disp(x);

[xbar, std, ssp, r, ifail] = g02ba( ...
x);

fprintf('Variable   Mean     St. dev.\n');
fprintf('%5d%11.4f%11.4f\n',[[1:m]' xbar std]');
fprintf('\nSums of squares and cross-products of deviations\n');
disp(ssp)
fprintf('Correlation coefficients\n');
disp(r);

```
```g02ba example results

Number of variables (columns) = 3
Number of cases     (rows)    = 5

Data matrix is:-
2     3     3
4     6     4
9     9     0
0    12     2
12    -1     5

Variable   Mean     St. dev.
1     5.4000     4.9800
2     5.8000     5.0695
3     2.8000     1.9235

Sums of squares and cross-products of deviations
99.2000  -57.6000    6.4000
-57.6000  102.8000  -29.2000
6.4000  -29.2000   14.8000

Correlation coefficients
1.0000   -0.5704    0.1670
-0.5704    1.0000   -0.7486
0.1670   -0.7486    1.0000

```