# Descriptive statistics

## corr - Pearson correlation of array elements

Experimental

### Description

Returns the Pearson correlation of the elements of array along dimension dim if the corresponding element in mask is true.

The Pearson correlation between two rows (or columns), say x and y, of array is defined as:

 corr(x, y) = cov(x, y) / sqrt( var(x) * var(y))


### Syntax

result = corr(array, dim [, mask])

### Class

Generic subroutine

### Arguments

array: Shall be a rank-1 or a rank-2 array of type integer, real, or complex.

dim: Shall be a scalar of type integer with a value in the range from 1 to n, where n is the rank of array.

mask (optional): Shall be of type logical and either a scalar or an array of the same shape as array.

### Return value

If array is of rank 1 and of type real or complex, the result is of type real and has the same kind as array. If array is of rank 2 and of type real or complex, the result is of the same type and kind as array. If array is of type integer, the result is of type real(dp).

If array is of rank 1 and of size larger than 1, a scalar equal to 1 is returned. Otherwise, IEEE NaN is returned. If array is of rank 2, a rank-2 array with the corresponding correlations is returned.

If mask is specified, the result is the Pearson correlation of all elements of array corresponding to true elements of mask. If every element of mask is false, the result is IEEE NaN.

### Example

program demo_corr
use stdlib_stats, only: corr
implicit none
real :: x(1:6) = [ 1., 2., 3., 4., 5., 6. ]
real :: y(1:2, 1:3) = reshape([ -1., 40., -3., 4., 10., 6. ], [ 2, 3])
print *, corr(x, 1)           !returns 1.
print *, corr(y, 2)           !returns reshape([ 1., -.32480, -.32480, 1. ], [ 2, 3])
end program demo_corr


## cov - covariance of array elements

Experimental

### Description

Returns the covariance of the elements of array along dimension dim if the corresponding element in mask is true.

Per default, the covariance is defined as:

 cov(array) = 1/(n-1) sum_i (array(i) - mean(array) * (array(i) - mean(array)))


where n is the number of elements.

The scaling can be changed with the logical argument corrected. If corrected is .false., then the sum is scaled with n, otherwise with n-1.

### Syntax

result = cov(array, dim [, mask [, corrected]])

### Class

Generic subroutine

### Arguments

array: Shall be a rank-1 or a rank-2 array of type integer, real, or complex.

dim: Shall be a scalar of type integer with a value in the range from 1 to n, where n is the rank of array.

mask (optional): Shall be of type logical and either a scalar or an array of the same shape as array.

corrected (optional): Shall be a scalar of type logical. If corrected is .true. (default value), the sum is scaled with n-1. If corrected is .false., then the sum is scaled with n.

### Return value

If array is of rank 1 and of type real or complex, the result is of type real corresponding to the type of array. If array is of rank 2 and of type real or complex, the result is of the same type as array. If array is of type integer, the result is of type real(dp).

If array is of rank 1, a scalar with the covariance (that is the variance) of all elements in array is returned. If array is of rank 2, a rank-2 array is returned.

If mask is specified, the result is the covariance of all elements of array corresponding to true elements of mask. If every element of mask is false, the result is IEEE NaN.

### Example

program demo_cov
use stdlib_stats, only: cov
implicit none
real :: x(1:6) = [ 1., 2., 3., 4., 5., 6. ]
real :: y(1:2, 1:3) = reshape([ 1., 2., 3., 4., 5., 6. ], [ 2, 3])
print *, cov(x, 1)                          !returns 3.5
print *, cov(x, 1, corrected = .false.)     !returns 2.9167
print *, cov(y, 1)                          !returns a square matrix of size 3 with all elements equal to 0.5
end program demo_cov


## mean - mean of array elements

Experimental

### Description

Returns the mean of all the elements of array, or of the elements of array along dimension dim if provided, and if the corresponding element in mask is true.

### Syntax

result = mean(array [, mask])

result = mean(array, dim [, mask])

### Class

Generic subroutine

### Arguments

array: Shall be an array of type integer, real, or complex.

dim: Shall be a scalar of type integer with a value in the range from 1 to n, where n is the rank of array.

mask (optional): Shall be of type logical and either a scalar or an array of the same shape as array.

### Return value

If array is of type real or complex, the result is of the same type as array. If array is of type integer, the result is of type real(dp).

If dim is absent, a scalar with the mean of all elements in array is returned. Otherwise, an array of rank n-1, where n equals the rank of array, and a shape similar to that of array with dimension dim dropped is returned.

If mask is specified, the result is the mean of all elements of array corresponding to true elements of mask. If every element of mask is false, the result is IEEE NaN.

### Example

program demo_mean
use stdlib_stats, only: mean
implicit none
real :: x(1:6) = [ 1., 2., 3., 4., 5., 6. ]
real :: y(1:2, 1:3) = reshape([ 1., 2., 3., 4., 5., 6. ], [ 2, 3])
print *, mean(x)                                  !returns 3.5
print *, mean(y)                                  !returns 3.5
print *, mean(y, 1)                               !returns [ 1.5, 3.5, 5.5 ]
print *, mean(y, 1,y > 3.)                        !returns [ NaN, 4.0, 5.5 ]
end program demo_mean


## median - median of array elements

Experimental

### Description

Returns the median of all the elements of array, or of the elements of array along dimension dim if provided, and if the corresponding element in mask is true.

The median of the elements of array is defined as the "middle" element, after that the elements are sorted in an increasing order, e.g. array_sorted = sort(array). If n = size(array) is an even number, the median is:

median(array) = array_sorted( floor( (n + 1) / 2.))


and if n is an odd number, the median is:

median(array) = mean( array_sorted( floor( (n + 1) / 2.):floor( (n + 1) / 2.) + 1 ) )


The current implementation is a quite naive implementation that relies on sorting the whole array, using the subroutine ord_sort provided by the stdlib_sorting module.

### Syntax

result = median(array [, mask])

result = median(array, dim [, mask])

### Class

Generic subroutine

### Arguments

array: Shall be an array of type integer or real.

dim: Shall be a scalar of type integer with a value in the range from 1 to n, where n is the rank of array.

mask (optional): Shall be of type logical and either a scalar or an array of the same shape as array.

### Return value

If array is of type real, the result is of type real with the same kind as array. If array is of type real and contains IEEE NaN, the result is IEEE NaN. If array is of type integer, the result is of type real(dp).

If dim is absent, a scalar with the median of all elements in array is returned. Otherwise, an array of rank n-1, where n equals the rank of array, and a shape similar to that of array with dimension dim dropped is returned.

If mask is specified, the result is the median of all elements of array corresponding to true elements of mask. If every element of mask is false, the result is IEEE NaN.

### Example

program demo_median
use stdlib_stats, only: median
implicit none
real :: x(1:6) = [ 1., 2., 3., 4., 5., 6. ]
real :: y(1:2, 1:3) = reshape([ 1., 2., 3., 4., 5., 6. ], [ 2, 3])
print *, median(x)                                  !returns 3.5
print *, median(y)                                  !returns 3.5
print *, median(y, 1)                               !returns [ 1.5, 3.5, 5.5 ]
print *, median(y, 1,y > 3.)                        !returns [ NaN, 4.0, 5.5 ]
end program demo_median


## moment - central moments of array elements

Experimental

### Description

Returns the k-th order central moment of all the elements of array, or of the elements of array along dimension dim if provided, and if the corresponding element in mask is true.

If a scalar or an array center is provided, the function returns the k-th order moment about 'center', of all the elements of array, or of the elements of array along dimension dim if provided, and if the corresponding element in mask is true.

The k-th order central moment is defined as :

 moment(array) = 1/n sum_i (array(i) - mean(array))^k


where n is the number of elements.

The k-th order moment about center is defined as :

 moment(array) = 1/n sum_i (array(i) - center)^k


### Syntax

result = moment(array, order [, center [, mask]])

result = moment(array, order, dim [, center [, mask]])

### Class

Generic subroutine

### Arguments

array: Shall be an array of type integer, real, or complex.

order: Shall be an scalar of type integer.

dim: Shall be a scalar of type integer with a value in the range from 1 to n, where n is the rank of array.

center (optional): Shall be a scalar of the same type of result if dim is not provided. If dim is provided, center shall be a scalar or an array (with a shape similar to that of array with dimension dim dropped) of the same type of result.

mask (optional): Shall be of type logical and either a scalar or an array of the same shape as array.

### Return value

If array is of type real or complex, the result is of the same type as array. If array is of type integer, the result is of type real(dp).

If dim is absent, a scalar with the k-th (central) moment of all elements in array is returned. Otherwise, an array of rank n-1, where n equals the rank of array, and a shape similar to that of array with dimension dim dropped is returned.

If mask is specified, the result is the k-th (central) moment of all elements of array corresponding to true elements of mask. If every element of mask is false, the result is IEEE NaN.

### Example

program demo_moment
use stdlib_stats, only: moment
implicit none
real :: x(1:6) = [ 1., 2., 3., 4., 5., 6. ]
real :: y(1:2, 1:3) = reshape([ 1., 2., 3., 4., 5., 6. ], [ 2, 3])
print *, moment(x, 2)                            !returns 2.9167
print *, moment(y, 2)                            !returns 2.9167
print *, moment(y, 2, 1)                         !returns [0.25, 0.25, 0.25]
print *, moment(y, 2, 1, mask = (y > 3.))        !returns [NaN, 0., 0.25]
print *, moment(x, 2, center = 0.)               !returns 15.1667
print *, moment(y, 1, 1, center = 0.)            !returns [1.5, 3.5, 5.5]
end program demo_moment


## var - variance of array elements

Experimental

### Description

Returns the variance of all the elements of array, or of the elements of array along dimension dim if provided, and if the corresponding element in mask is true.

Per default, the variance is defined as the best unbiased estimator and is computed as:

 var(array) = 1/(n-1) sum_i (array(i) - mean(array))^2


where n is the number of elements.

The use of the term n-1 for scaling is called Bessel 's correction. The scaling can be changed with the logical argument corrected. If corrected is .false., then the sum is scaled with n, otherwise with n-1.

### Syntax

result = var(array [, mask [, corrected]])

result = var(array, dim [, mask [, corrected]])

### Class

Generic subroutine

### Arguments

array: Shall be an array of type integer, real, or complex.

dim: Shall be a scalar of type integer with a value in the range from 1 to n, where n is the rank of array.

mask (optional): Shall be of type logical and either a scalar or an array of the same shape as array.

corrected (optional): Shall be a scalar of type logical. If corrected is .true. (default value), the sum is scaled with n-1. If corrected is .false., then the sum is scaled with n.

### Return value

If array is of type real or complex, the result is of type real corresponding to the type of array. If array is of type integer, the result is of type real(dp).

If dim is absent, a scalar with the variance of all elements in array is returned. Otherwise, an array of rank n-1, where n equals the rank of array, and a shape similar to that of array with dimension dim dropped is returned.

If mask is specified, the result is the variance of all elements of array corresponding to true elements of mask. If every element of mask is false, the result is IEEE NaN.

If the variance is computed with only one single element, then the result is IEEE NaN if corrected is .true. and is 0. if corrected is .false..

### Example

program demo_var
use stdlib_stats, only: var
implicit none
real :: x(1:6) = [ 1., 2., 3., 4., 5., 6. ]
real :: y(1:2, 1:3) = reshape([ 1., 2., 3., 4., 5., 6. ], [ 2, 3])
print *, var(x)                                  !returns 3.5
print *, var(x, corrected = .false.)             !returns 2.9167
print *, var(y)                                  !returns 3.5
print *, var(y, 1)                               !returns [0.5, 0.5, 0.5]
print *, var(y, 1, y > 3.)                       !returns [NaN, NaN, 0.5]
print *, var(y, 1, y > 3., corrected=.false.)    !returns [NaN, 0., 0.25]
end program demo_var