Statistical Functions ¶
Array API specification for statistical functions.
A conforming implementation of the array API standard must provide and support the following functions adhering to the following conventions.

Positional parameters must be positionalonly parameters. Positionalonly parameters have no externallyusable name. When a function accepting positionalonly parameters is called, positional arguments are mapped to these parameters based solely on their order.

Optional parameters must be keywordonly arguments.

Broadcasting semantics must follow the semantics defined in Broadcasting .

Unless stated otherwise, functions must support the data types defined in Data Types .

Unless stated otherwise, functions must adhere to the type promotion rules defined in Type Promotion Rules .

Unless stated otherwise, floatingpoint operations must adhere to IEEE 7542019.
Objects in API ¶
max(x, /, *, axis=None, keepdims=False) ¶
Calculates the maximum value of the input array
x
.
Note
When the number of elements over which to compute the maximum value is zero, the maximum value is implementationdefined. Specificationcompliant libraries may choose to error, return a sentinel value (e.g., if
x
is a floatingpoint input array, return
NaN
), or return the minimum possible value for the input array
x
data type (e.g., if
x
is a floatingpoint array, return
infinity
).
Parameters ¶

x : <array>

input array. Should have a numeric data type.


axis : Optional[ Union[ int, Tuple[ int, … ] ] ]

axis or axes along which maximum values must be computed. By default, the maximum value must be computed over the entire array. If a tuple of integers, maximum values must be computed over multiple axes. Default:
None
.


keepdims : bool

If
True
, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see Broadcasting ). Otherwise, ifFalse
, the reduced axes (dimensions) must not be included in the result. Default:False
.

Returns ¶

out : <array>

if the maximum value was computed over the entire array, a zerodimensional array containing the maximum value; otherwise, a nonzerodimensional array containing the maximum values. The returned array must have the same data type as
x
.

mean(x, /, *, axis=None, keepdims=False) ¶
Calculates the arithmetic mean of the input array
x
.
Special Cases ¶
For a floatingpoint input array
x
, let
N
equal the number of elements over which to compute the arithmetic mean and

if
N
is0
, the arithmetic mean isNaN
.
Parameters ¶

x : <array>

input array. Should have a floatingpoint data type.


axis : Optional[ Union[ int, Tuple[ int, … ] ] ]

axis or axes along which arithmetic means must be computed. By default, the mean must be computed over the entire array. If a tuple of integers, arithmetic means must be computed over multiple axes. Default:
None
.


keepdims : bool

If
True
, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see Broadcasting ). Otherwise, ifFalse
, the reduced axes (dimensions) must not be included in the result. Default:False
.

Returns ¶

out : <array>

if the arithmetic mean was computed over the entire array, a zerodimensional array containing the arithmetic mean; otherwise, a nonzerodimensional array containing the arithmetic means. The returned array must have the same data type as
x
.Note
While this specification recommends that this function only accept input arrays having a floatingpoint data type, specificationcompliant array libraries may choose to accept input arrays having an integer data type. While mixed data type promotion is implementationdefined, if the input array
x
has an integer data type, the returned array must have the default floatingpoint data type.

min(x, /, *, axis=None, keepdims=False) ¶
Calculates the minimum value of the input array
x
.
Note
When the number of elements over which to compute the minimum value is zero, the minimum value is implementationdefined. Specificationcompliant libraries may choose to error, return a sentinel value (e.g., if
x
is a floatingpoint input array, return
NaN
), or return the maximum possible value for the input array
x
data type (e.g., if
x
is a floatingpoint array, return
+infinity
).
Parameters ¶

x : <array>

input array. Should have a numeric data type.


axis : Optional[ Union[ int, Tuple[ int, … ] ] ]

axis or axes along which minimum values must be computed. By default, the minimum value must be computed over the entire array. If a tuple of integers, minimum values must be computed over multiple axes. Default:
None
.


keepdims : bool

If
True
, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see Broadcasting ). Otherwise, ifFalse
, the reduced axes (dimensions) must not be included in the result. Default:False
.

Returns ¶

out : <array>

if the minimum value was computed over the entire array, a zerodimensional array containing the minimum value; otherwise, a nonzerodimensional array containing the minimum values. The returned array must have the same data type as
x
.

prod(x, /, *, axis=None, dtype=None, keepdims=False) ¶
Calculates the product of input array
x
elements.
Special Cases ¶
For an input array
x
, let
N
equal the number of elements over which to compute the product and

if
N
is0
, the product is1
(i.e., the empty product).
Parameters ¶

x : <array>

input array. Should have a numeric data type.


axis : Optional[ Union[ int, Tuple[ int, … ] ] ]

axis or axes along which products must be computed. By default, the product must be computed over the entire array. If a tuple of integers, products must be computed over multiple axes. Default:
None
.


dtype : Optional[ <dtype> ]

data type of the returned array. If
None
,
if the default data type corresponding to the data type “kind” (integer or floatingpoint) of
x
has a smaller range of values than the data type ofx
(e.g.,x
has data typeint64
and the default data type isint32
, orx
has data typeuint64
and the default data type isint64
), the returned array must have the same data type asx
. 
if
x
has a floatingpoint data type, the returned array must have the default floatingpoint data type. 
if
x
has a signed integer data type (e.g.,int16
), the returned array must have the default integer data type. 
if
x
has an unsigned integer data type (e.g.,uint16
), the returned array must have an unsigned integer data type having the same number of bits as the default integer data type (e.g., if the default integer data type isint32
, the returned array must have auint32
data type).
If the data type (either specified or resolved) differs from the data type of
x
, the input array should be cast to the specified data type before computing the product. Default:None
.Note
This keyword argument is intended to help prevent data type overflows.



keepdims : bool

If
True
, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see Broadcasting ). Otherwise, ifFalse
, the reduced axes (dimensions) must not be included in the result. Default:False
.

Returns ¶

out : <array>

if the product was computed over the entire array, a zerodimensional array containing the product; otherwise, a nonzerodimensional array containing the products. The returned array must have a data type as described by the
dtype
parameter above.

std(x, /, *, axis=None, correction=0.0, keepdims=False) ¶
Calculates the standard deviation of the input array
x
.
Special Cases ¶
For a floatingpoint input array
x
, let
N
equal the number of elements over which to compute the standard deviation and

if
N  correction
is less than or equal to0
, the standard deviation isNaN
.
Parameters ¶

x : <array>

input array. Should have a floatingpoint data type.


axis : Optional[ Union[ int, Tuple[ int, … ] ] ]

axis or axes along which standard deviations must be computed. By default, the standard deviation must be computed over the entire array. If a tuple of integers, standard deviations must be computed over multiple axes. Default:
None
.


correction : Union[ int, float ]

degrees of freedom adjustment. Setting this parameter to a value other than
0
has the effect of adjusting the divisor during the calculation of the standard deviation according toNc
whereN
corresponds to the total number of elements over which the standard deviation is computed andc
corresponds to the provided degrees of freedom adjustment. When computing the standard deviation of a population, setting this parameter to0
is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the corrected sample standard deviation, setting this parameter to1
is the standard choice (i.e., the provided array contains data sampled from a larger population; this is commonly referred to as Bessel’s correction). Default:0
.


keepdims : bool

If
True
, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see Broadcasting ). Otherwise, ifFalse
, the reduced axes (dimensions) must not be included in the result. Default:False
.

Returns ¶

out : <array>

if the standard deviation was computed over the entire array, a zerodimensional array containing the standard deviation; otherwise, a nonzerodimensional array containing the standard deviations. The returned array must have the same data type as
x
.Note
While this specification recommends that this function only accept input arrays having a floatingpoint data type, specificationcompliant array libraries may choose to accept input arrays having an integer data type. While mixed data type promotion is implementationdefined, if the input array
x
has an integer data type, the returned array must have the default floatingpoint data type.

sum(x, /, *, axis=None, dtype=None, keepdims=False) ¶
Calculates the sum of the input array
x
.
Special Cases ¶
For an input array
x
, let
N
equal the number of elements over which to compute the sum and

if
N
is0
, the sum is0
(i.e., the empty sum).
Parameters ¶

x : <array>

input array. Should have a numeric data type.


axis : Optional[ Union[ int, Tuple[ int, … ] ] ]

axis or axes along which sums must be computed. By default, the sum must be computed over the entire array. If a tuple of integers, sums must be computed over multiple axes. Default:
None
.


dtype : Optional[ <dtype> ]

data type of the returned array. If
None
,
if the default data type corresponding to the data type “kind” (integer or floatingpoint) of
x
has a smaller range of values than the data type ofx
(e.g.,x
has data typeint64
and the default data type isint32
, orx
has data typeuint64
and the default data type isint64
), the returned array must have the same data type asx
. 
if
x
has a floatingpoint data type, the returned array must have the default floatingpoint data type. 
if
x
has a signed integer data type (e.g.,int16
), the returned array must have the default integer data type. 
if
x
has an unsigned integer data type (e.g.,uint16
), the returned array must have an unsigned integer data type having the same number of bits as the default integer data type (e.g., if the default integer data type isint32
, the returned array must have auint32
data type).
If the data type (either specified or resolved) differs from the data type of
x
, the input array should be cast to the specified data type before computing the sum. Default:None
.Note
This keyword argument is intended to help prevent data type overflows.



keepdims : bool

If
True
, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see Broadcasting ). Otherwise, ifFalse
, the reduced axes (dimensions) must not be included in the result. Default:False
.

Returns ¶

out : <array>

if the sum was computed over the entire array, a zerodimensional array containing the sum; otherwise, an array containing the sums. The returned array must have a data type as described by the
dtype
parameter above.

var(x, /, *, axis=None, correction=0.0, keepdims=False) ¶
Calculates the variance of the input array
x
.
Special Cases ¶
For a floatingpoint input array
x
, let
N
equal the number of elements over which to compute the variance and

if
N  correction
is less than or equal to0
, the variance isNaN
.
Parameters ¶

x : <array>

input array. Should have a floatingpoint data type.


axis : Optional[ Union[ int, Tuple[ int, … ] ] ]

axis or axes along which variances must be computed. By default, the variance must be computed over the entire array. If a tuple of integers, variances must be computed over multiple axes. Default:
None
.


correction : Union[ int, float ]

degrees of freedom adjustment. Setting this parameter to a value other than
0
has the effect of adjusting the divisor during the calculation of the variance according toNc
whereN
corresponds to the total number of elements over which the variance is computed andc
corresponds to the provided degrees of freedom adjustment. When computing the variance of a population, setting this parameter to0
is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the unbiased sample variance, setting this parameter to1
is the standard choice (i.e., the provided array contains data sampled from a larger population; this is commonly referred to as Bessel’s correction). Default:0
.


keepdims : bool

If
True
, the reduced axes (dimensions) must be included in the result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see Broadcasting ). Otherwise, ifFalse
, the reduced axes (dimensions) must not be included in the result. Default:False
.

Returns ¶

out : <array>

if the variance was computed over the entire array, a zerodimensional array containing the variance; otherwise, a nonzerodimensional array containing the variances. The returned array must have the same data type as
x
.Note
While this specification recommends that this function only accept input arrays having a floatingpoint data type, specificationcompliant array libraries may choose to accept input arrays having an integer data type. While mixed data type promotion is implementationdefined, if the input array
x
has an integer data type, the returned array must have the default floatingpoint data type.
