# 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 positional-only parameters. Positional-only parameters have no externally-usable name. When a function accepting positional-only parameters is called, positional arguments are mapped to these parameters based solely on their order.

• Optional parameters must be keyword-only arguments.

• 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, floating-point operations must adhere to IEEE 754-2019.

## 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 implementation-defined. Specification-compliant libraries may choose to error, return a sentinel value (e.g., if  x  is a floating-point input array, return  NaN  ), or return the minimum possible value for the input array  x  data type (e.g., if  x  is a floating-point 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, if  False  , 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 zero-dimensional array containing the maximum value; otherwise, a non-zero-dimensional 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 floating-point input array  x  , let  N  equal the number of elements over which to compute the arithmetic mean and

• if  N  is  0  , the arithmetic mean is  NaN  .

#### Parameters ¶

• x : <array>

• input array. Should have a floating-point 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, if  False  , 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 zero-dimensional array containing the arithmetic mean; otherwise, a non-zero-dimensional 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 floating-point data type, specification-compliant array libraries may choose to accept input arrays having an integer data type. While mixed data type promotion is implementation-defined, if the input array  x  has an integer data type, the returned array must have the default floating-point 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 implementation-defined. Specification-compliant libraries may choose to error, return a sentinel value (e.g., if  x  is a floating-point input array, return  NaN  ), or return the maximum possible value for the input array  x  data type (e.g., if  x  is a floating-point 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, if  False  , 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 zero-dimensional array containing the minimum value; otherwise, a non-zero-dimensional 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  is  0  , the product is  1  (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 floating-point) of  x  has a smaller range of values than the data type of  x  (e.g.,  x  has data type  int64  and the default data type is  int32  , or  x  has data type  uint64  and the default data type is  int64  ), the returned array must have the same data type as  x  .

• if  x  has a floating-point data type, the returned array must have the default floating-point 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 is  int32  , the returned array must have a  uint32  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, if  False  , 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 zero-dimensional array containing the product; otherwise, a non-zero-dimensional 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 floating-point 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 to  0  , the standard deviation is  NaN  .

#### Parameters ¶

• x : <array>

• input array. Should have a floating-point 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 to  N-c  where  N  corresponds to the total number of elements over which the standard deviation is computed and  c  corresponds to the provided degrees of freedom adjustment. When computing the standard deviation of a population, setting this parameter to  0  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 to  1  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, if  False  , 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 zero-dimensional array containing the standard deviation; otherwise, a non-zero-dimensional 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 floating-point data type, specification-compliant array libraries may choose to accept input arrays having an integer data type. While mixed data type promotion is implementation-defined, if the input array  x  has an integer data type, the returned array must have the default floating-point 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  is  0  , the sum is  0  (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 floating-point) of  x  has a smaller range of values than the data type of  x  (e.g.,  x  has data type  int64  and the default data type is  int32  , or  x  has data type  uint64  and the default data type is  int64  ), the returned array must have the same data type as  x  .

• if  x  has a floating-point data type, the returned array must have the default floating-point 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 is  int32  , the returned array must have a  uint32  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, if  False  , 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 zero-dimensional 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 floating-point 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 to  0  , the variance is  NaN  .

#### Parameters ¶

• x : <array>

• input array. Should have a floating-point 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 to  N-c  where  N  corresponds to the total number of elements over which the variance is computed and  c  corresponds to the provided degrees of freedom adjustment. When computing the variance of a population, setting this parameter to  0  is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the unbiased sample variance, setting this parameter to  1  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, if  False  , 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 zero-dimensional array containing the variance; otherwise, a non-zero-dimensional 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 floating-point data type, specification-compliant array libraries may choose to accept input arrays having an integer data type. While mixed data type promotion is implementation-defined, if the input array  x  has an integer data type, the returned array must have the default floating-point data type.