# Statistical

ivy.einsum(equation, *operands, out=None)[source]

Sums the product of the elements of the input operands along dimensions specified using a notation based on the Einstein summation convention.

Parameters
• equation (`str`) – A str describing the contraction, in the same format as numpy.einsum.

• operands (`Union`[`Array`, `NativeArray`]) – seq of arrays, the inputs to contract (each one an ivy.Array), whose shapes should be consistent with equation.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. (default: `None`)

Return type

`Array`

Returns

ret – The array with sums computed.

Functional Examples

With :code: ‘ivy.Array’ input:

```>>> x = ivy.array([[0, 1, 2], [3, 4, 5], [6, 7, 8]])
>>> y = ivy.einsum('ii', x)
>>> print(y)
ivy.array(12)
```
```>>> x = ivy.array([[0, 1, 2], [3, 4, 5], [6, 7, 8]])
>>> z = ivy.einsum('ij -> j', x)
>>> print(z)
ivy.array([ 9, 12, 15])
```
```>>> A = ivy.array([0, 1, 2])
>>> B = ivy.array([[ 0,  1,  2,  3],                       [ 4,  5,  6,  7],                       [ 8,  9, 10, 11]])
>>> C = ivy.einsum('i,ij->i', A, B)
>>> print(C)
ivy.array([ 0, 22, 76])
```
```>>> A = ivy.array([[1, 1, 1],                       [2, 2, 2],                       [5, 5, 5]])
>>> B = ivy.array([[0, 1, 0],                       [1, 1, 0],                       [1, 1, 1]])
>>> C = ivy.einsum('ij,jk->ik', A, B)
>>> print(C)
ivy.array([[ 2,  3,  1],
[ 4,  6,  2],
[10, 15,  5]])
```
```>>> A = ivy.arange(10)
>>> B = ivy.arange(5, 15)
>>> C = ivy.einsum('i->', A)
>>> print(C)
ivy.array(45)
```
```>>> A = ivy.arange(10)
>>> B = ivy.arange(5, 15)
>>> C = ivy.einsum('i,i->i', A, B)
>>> print(C)
ivy.array([  0,   6,  14,  24,  36,  50,  66,  84, 104, 126])
```
```>>> A = ivy.arange(10)
>>> B = ivy.arange(5, 15)
>>> C = ivy.einsum('i,i->', A, B) # or just use 'i,i'
>>> print(C)
ivy.array(510)
```
```>>> A = ivy.arange(10)
>>> B = ivy.arange(5, 15)
>>> C = ivy.einsum('i,j->ij', A, B)
>>> print(C)
ivy.array([[  0,   0,   0,   0,   0,   0,   0,   0,   0,   0],
[  5,   6,   7,   8,   9,  10,  11,  12,  13,  14],
[ 10,  12,  14,  16,  18,  20,  22,  24,  26,  28],
[ 15,  18,  21,  24,  27,  30,  33,  36,  39,  42],
[ 20,  24,  28,  32,  36,  40,  44,  48,  52,  56],
[ 25,  30,  35,  40,  45,  50,  55,  60,  65,  70],
[ 30,  36,  42,  48,  54,  60,  66,  72,  78,  84],
[ 35,  42,  49,  56,  63,  70,  77,  84,  91,  98],
[ 40,  48,  56,  64,  72,  80,  88,  96, 104, 112],
[ 45,  54,  63,  72,  81,  90,  99, 108, 117, 126]])
```

With :code:’ivy.NativeArray’ input:

```>>> x = ivy.native_array([[0, 1, 2], [3, 4, 5], [6, 7, 8]])
>>> y = ivy.einsum('ii', x)
>>> print(y)
ivy.array(12)
```

With a mix of code: ‘ivy.Array’ and code: ‘ivy.NativeArray’ inputs:

```>>> A = ivy.array([0, 1, 2])
>>> B = ivy.native_array([[ 0, 1, 2, 3],                              [ 4, 5, 6, 7],                              [ 8, 9, 10, 11]])
>>> C = ivy.einsum('i,ij->i', A, B)
>>> print(C)
ivy.array([ 0, 22, 76])
```

With a mix of `ivy.Array` and `ivy.Container` inputs:

```>>> x = ivy.array([0, 1, 2])
>>> y = ivy.Container(a=ivy.array([[ 0,  1,  2,  3],                                       [ 4,  5,  6,  7],                                       [ 8,  9, 10, 11]]),                          b=ivy.array([[ 0,  1,  2],                                       [ 4,  5,  6],                                       [ 8,  9, 10]]))
>>> z = ivy.einsum('i,ij->i', x, y)
>>> print(z)
{
a: ivy.array([0, 22, 76]),
b: ivy.array([0, 15, 54])
}
```

With :code: ‘ivy.Container’ input:

```>>> x = ivy.Container(a=ivy.array([[0, 1, 0],[1, 1, 0],[1, 1, 1]]),                          b=ivy.array([[0, 1, 2],[4, 5, 6],[8, 9, 10]]))
>>> y = ivy.einsum('ii', x)
>>> print(y)
{
a: ivy.array(2),
b: ivy.array(15)
}
```

Instance Method Examples

Using :code: ‘ivy.Array’ instance method:

```>>> x = ivy.array([[0, 1, 2], [3, 4, 5], [6, 7, 8]])
>>> y = x.einsum('ii')
>>> print(y)
ivy.array(12)
```

Using :code: ‘ivy.Container’ instance method:

```>>> x = ivy.Container(a=ivy.array([[0, 1, 0],[1, 1, 0],[1, 1, 1]]),                          b=ivy.array([[0, 1, 2],[4, 5, 6],[8, 9, 10]]))
>>> y = x.einsum('ii')
>>> print(y)
{
a: ivy.array(2),
b: ivy.array(15)
}
```
ivy.max(x, axis=None, keepdims=False, *, out=None)[source]

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 raise an 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`).

Special Cases

For floating-point operands,

• If `x_i` is `NaN`, the maximum value is `NaN` (i.e., `NaN` values propagate).

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array. Should have a numeric data type.

• axis (`Optional`[`Union`[`int`, `Sequence`[`int`]]]) – axis or axes along which maximum values must be computed. By default, the (default: `None`) 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 (`Optional`[`bool`]) – if `True`, the reduced axes (dimensions) must be included in the result as (default: `False`) 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`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. (default: `None`)

Return type

`Array`

Returns

• ret – 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`.

• This method conforms to the `Array API Standard

• <https (//data-apis.org/array-api/latest/>`_. This docstring is an extension of the)

• `docstring <https (//data-apis.org/array-api/latest/API_specification/generated/signatures.elementwise_functions.max.html>`_ )

• in the standard.

• Both the description and the type hints above assumes an array input for simplicity,

• but this function is nestable, and therefore also accepts `ivy.Container`

• instances in place of any of the arguments.

Examples

With `ivy.Array` input:

```>>> x = ivy.array([1, 2, 3])
>>> z = x.max()
>>> print(z)
ivy.array(3)
```
```>>> x = ivy.array([0, 1, 2])
>>> z = ivy.array([0,0,0])
>>> y = ivy.max(x, out=z)
>>> print(z)
ivy.array(2)
```
```>>> x = ivy.array([[0, 1, 2], [4, 6, 10]])
>>> y = ivy.max(x, 0, True)
>>> print(y)
ivy.array([[4, 6, 10]])
```
```>>> x = ivy.native_array([[0, 1, 2], [4, 6, 10]])
>>> y = ivy.max(x)
>>> print(y)
ivy.array(10)
```

With `ivy.Container` input:

```>>> x = ivy.Container(a=ivy.array([0., 1., 2.]), b=ivy.array([3., 4., 5.]))
>>> y = ivy.max(x)
>>> print(y)
{
a: ivy.array(2.),
b: ivy.array(5.)
}
```
```>>> x = ivy.Container(a=ivy.array([1, 2, 3]),                          b=ivy.array([2, 3, 4]))
>>> z = x.max()
>>> print(z)
{
a: ivy.array(3),
b: ivy.array(4)
}
```
ivy.mean(x, axis=None, keepdims=False, *, out=None)[source]

Calculates the arithmetic mean of the input array `x`.

Special Cases

Let `N` equal the number of elements over which to compute the arithmetic mean. - If `N` is `0`, the arithmetic mean is `NaN`. - If `x_i` is `NaN`, the arithmetic mean is `NaN` (i.e., `NaN` values

propagate).

Parameters
• x (`Union`[`Array`, `NativeArray`]) – 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 (default: `None`) must be computed over the entire array. If a tuple of integers, arithmetic means must be computed over multiple axes. Default: `None`.

• keepdims (`bool`) – bool, if `True`, the reduced axes (dimensions) must be included in the result (default: `False`) 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`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. (default: `None`)

Return type

`Array`

Returns

ret – 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.
```

ivy.min(x, axis=None, keepdims=False, *, out=None)[source]

Calculates the minimum value of the input array x.

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 raise an 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).

Special Cases

For floating-point operands,

If x_i is NaN, the minimum value is NaN (i.e., NaN values propagate).

Parameters
• x (`Union`[`Array`, `NativeArray`]) – Input array containing elements to min.

• axis (`Optional`[`Union`[`int`, `Tuple`[`int`]]]) – axis or axes along which minimum values must be computed. By default, the (default: `None`) 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`) – optional boolean, if True, the reduced axes (dimensions) must be included in the (default: `False`) 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.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. (default: `None`)

Return type

`Array`

Returns

ret – 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.

ivy.prod(x, *, axis=None, dtype=None, keepdims=False, out=None)[source]

Calculates the product of input array x elements.

x

input array. Should have a numeric data type.

axis

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.

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.

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.

out

optional output array, for writing the result to.

Return type

`Array`

Returns

• ret – 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.

• >>> x = ivy.array([1, 2, 3])

• >>> z = ivy.prod(x)

• >>> print(z)

• ivy.array(6)

• >>> x = ivy.array([1, 0, 3])

• >>> z = ivy.prod(x)

• >>> print(z)

• ivy.array(0)

ivy.std(x, axis=None, correction=0.0, keepdims=False, *, out=None)[source]

Calculates the standard deviation of the input array `x`.

Special Cases

Let `N` equal the number of elements over which to compute the standard deviation.

• If `N` is `0`, the standard deviation is `0` (i.e., the empty standard deviation).

• If `x_i` is `NaN`, the standard deviation is `NaN` (i.e., `NaN` values propagate).

Parameters
• x (`Union`[`Array`, `NativeArray`]) – 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 (default: `None`) 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 (default: `0.0`) 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 (default: `False`) 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`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. (default: `None`)

Return type

`Array`

Returns

ret – if the sum was computed over the entire array, a zero-dimensional array containing the standard deviation; otherwise, an array containing the standard deviations. The returned array must have a data type as described by the `dtype` parameter above.

Examples

```>>> x = ivy.array([-1., 0., 1.])
>>> y = ivy.std(x)
>>> print(y)
ivy.array(0.8164966)
```
ivy.sum(x, *, axis=None, dtype=None, keepdims=False, out=None)[source]

Calculates the sum of the input array `x`.

Special Cases

Let `N` equal the number of elements over which to compute the sum. - If `N` is `0`, the sum is `0` (i.e., the empty sum).

For floating-point operands, - If `x_i` is `NaN`, the sum is `NaN` (i.e., `NaN` values propagate).

Parameters
• x (`Union`[`Array`, `NativeArray`]) – 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 (default: `None`) computed over the entire array. If a tuple of integers, sums must be computed over multiple axes. Default: `None`.

• dtype (`Optional`[`Union`[`Dtype`, `NativeDtype`]]) –

Data type of the returned array. If `None`, (default: `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

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 (default: `False`) 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`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. (default: `None`)

Return type

`Array`

Returns

ret – 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.

Examples

```>>> x = ivy.array([0.41, 0.89])
>>> y = ivy.sum(x)
>>> print(y)
ivy.array(1.3)
```
ivy.var(x, axis=None, correction=0.0, keepdims=False, *, out=None)[source]

Calculates the variance of the input array x.

Special Cases

Let N equal the number of elements over which to compute the variance.

If N - correction is less than or equal to 0, the variance is NaN.

If x_i is NaN, the variance is NaN (i.e., NaN values propagate).

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array. Should have a floating-point data type.

• axis (`Optional`[`Union`[`int`, `Sequence`[`int`]]]) – axis or axes along which variances must be computed. By default, the variance (default: `None`) 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 (default: `0.0`) 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 (`Optional`[`bool`]) – if True, the reduced axes (dimensions) must be included in the result as (default: `False`) 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.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. (default: `None`)

Return type

`Array`

Returns

ret – 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.

Examples

With `ivy.Array` input:

```>>> x = ivy.array([0.1, 0.2, 0.3, 0.3, 0.9, 0.10])
>>> y = ivy.var(x)
>>> print(y)
ivy.array(0.07472222)
```
```>>> x = ivy.array([0.1, 0.2, 0.3, 0.3, 0.9, 0.10])
>>> y = ivy.zeros(6)
>>> ivy.var(x, out=y)
>>> print(y)
ivy.array(0.07472222)
```
```>>> x = ivy.array([0.1, 0.2, 0.3, 0.3, 0.9, 0.10])
>>> ivy.var(x, out=x)
>>> print(x)
ivy.array(0.07472222)
```

With `ivy.native_array` input:

```>>> x = ivy.native_array([0.1, 0.2, 0.3, 0.3, 0.9, 0.10])
>>> y = ivy.var(x)
>>> print(y)
ivy.array(0.07472222)
```

With `ivy.Container` input:

```>>> x = ivy.Container(a=ivy.array([0.1, 0.2, 0.9]),                           b=ivy.array([0.7, 0.1, 0.9]))
>>> y = ivy.var(x)
>>> print(y)
{
a: ivy.array(0.12666667),
b: ivy.array(0.11555555)
}
```

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Functional Examples

With `ivy.Array` input:

```>>> x = ivy.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0])
>>> y = ivy.var(x)
>>> print(y)
ivy.array(6.6666665)
```
```>>> x = ivy.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0])
>>> y = ivy.array(0.0)
>>> ivy.var(x, out=y)
>>> print(y)
ivy.array(6.6666665)
```
```>>> x = ivy.array([[0.0, 1.0, 2.0], [3.0, 4.0, 5.0], [6.0, 7.0, 8.0]])
>>> y = ivy.array(0.0)
>>> ivy.var(x, out=y)
>>> print(y)
ivy.array(6.6666665)
```
```>>> x = ivy.array([[0.0, 1.0, 2.0], [3.0, 4.0, 5.0],[6.0, 7.0, 8.0]])
>>> y = ivy.zeros(3)
>>> ivy.var(x, axis=1, out=y)
>>> print(y)
ivy.array([0.667, 0.667, 0.667])
```
```>>> x = ivy.array([[0.0, 1.0, 2.0], [3.0, 4.0, 5.0], [6.0, 7.0, 8.0]])
>>> y = ivy.zeros(3)
>>> ivy.var(x, axis=0, out=y)
>>> print(y)
ivy.array([6., 6., 6.])
```

With `ivy.NativeArray` input:

```>>> x = ivy.native_array([1.0, 2.0, 2.0, 3.0])
>>> y = ivy.var(x)
>>> print(y)
ivy.array(0.5)
```
```>>> x = ivy.native_array([1.0, 2.0, 2.0, 3.0])
>>> y = ivy.array(0.0)
>>> ivy.var(x, out=y)
>>> print(y)
ivy.array(0.5)
```

With `ivy.Container` input:

```>>> x = ivy.Container(a=ivy.array([0.0, 1.0, 2.0]), b=ivy.array([3.0, 4.0, 5.0]))
>>> y = ivy.var(x)
>>> print(y)
{
a: ivy.array(0.6666667),
b: ivy.array(0.6666667)
}
```
```>>> x = ivy.Container(a=ivy.array([0.0, 1.0, 2.0]), b=ivy.array([3.0, 4.0, 5.0]))
>>> y = ivy.Container.static_var(x)
>>> print(y)
{
a: ivy.array(0.6666667),
b: ivy.array(0.6666667)
}
```

Instance Method Examples

Using `ivy.Array` instance method:

```>>> x = ivy.array([[0.0, 1.0, 2.0], [3.0, 4.0, 5.0],[6.0, 7.0, 8.0]])
>>> y = x.var()
>>> print(y)
ivy.array(6.6666665)
```
```>>> x = ivy.array([[0.0, 1.0, 2.0], [3.0, 4.0, 5.0],[6.0, 7.0, 8.0]])
>>> y = x.var(axis=0)
>>> print(y)
ivy.array([6., 6., 6.])
```

Using `ivy.Container` instance method:

```>>> x = ivy.Container(a=ivy.array([0.0, 1.0, 2.0]), b=ivy.array([3.0, 4.0, 5.0]))
>>> y = x.var()
>>> print(y)
{
a: ivy.array(0.6666667),
b: ivy.array(0.6666667)
}
```