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def var(a, axis=None, dtype=None, out=None, ddof=0, keepdims=<no value>, *, where=<no value>, mean=<no value>, correction=<no value>)
Compute the variance along the specified axis.
Returns the variance of the array elements, a measure of the spread of a
distribution. The variance is computed for the flattened array by
default, otherwise over the specified axis.
Parameters
----------
a : array_like
Array containing numbers whose variance is desired. If `a` is not an
array, a conversion is attempted.
axis : None or int or tuple of ints, optional
Axis or axes along which the variance is computed. The default is to
compute the variance of the flattened array.
If this is a tuple of ints, a variance is performed over multiple axes,
instead of a single axis or all the axes as before.
dtype : data-type, optional
Type to use in computing the variance. For arrays of integer type
the default is `float64`; for arrays of float types it is the same as
the array type.
out : ndarray, optional
Alternate output array in which to place the result. It must have
the same shape as the expected output, but the type is cast if
necessary.
ddof : {int, float}, optional
"Delta Degrees of Freedom": the divisor used in the calculation is
``N - ddof``, where ``N`` represents the number of elements. By
default `ddof` is zero. See notes for details about use of `ddof`.
keepdims : bool, optional
If this is set to True, the axes which are reduced are left
in the result as dimensions with size one. With this option,
the result will broadcast correctly against the input array.
If the default value is passed, then `keepdims` will not be
passed through to the `var` method of sub-classes of
`ndarray`, however any non-default value will be. If the
sub-class' method does not implement `keepdims` any
exceptions will be raised.
where : array_like of bool, optional
Elements to include in the variance. See `~numpy.ufunc.reduce` for
details.
.. versionadded:: 1.20.0
mean : array like, optional
Provide the mean to prevent its recalculation. The mean should have
a shape as if it was calculated with ``keepdims=True``.
The axis for the calculation of the mean should be the same as used in
the call to this var function.
.. versionadded:: 2.0.0
correction : {int, float}, optional
Array API compatible name for the ``ddof`` parameter. Only one of them
can be provided at the same time.
.. versionadded:: 2.0.0
Returns
-------
variance : ndarray, see dtype parameter above
If ``out=None``, returns a new array containing the variance;
otherwise, a reference to the output array is returned.
See Also
--------
std, mean, nanmean, nanstd, nanvar
:ref:`ufuncs-output-type`
Notes
-----
There are several common variants of the array variance calculation.
Assuming the input `a` is a one-dimensional NumPy array and ``mean`` is
either provided as an argument or computed as ``a.mean()``, NumPy
computes the variance of an array as::
N = len(a)
d2 = abs(a - mean)**2 # abs is for complex `a`
var = d2.sum() / (N - ddof) # note use of `ddof`
Different values of the argument `ddof` are useful in different
contexts. NumPy's default ``ddof=0`` corresponds with the expression:
.. math::
\frac{\sum_i{|a_i - \bar{a}|^2 }}{N}
which is sometimes called the "population variance" in the field of
statistics because it applies the definition of variance to `a` as if `a`
were a complete population of possible observations.
Many other libraries define the variance of an array differently, e.g.:
.. math::
\frac{\sum_i{|a_i - \bar{a}|^2}}{N - 1}
In statistics, the resulting quantity is sometimes called the "sample
variance" because if `a` is a random sample from a larger population,
this calculation provides an unbiased estimate of the variance of the
population. The use of :math:`N-1` in the denominator is often called
"Bessel's correction" because it corrects for bias (toward lower values)
in the variance estimate introduced when the sample mean of `a` is used
in place of the true mean of the population. For this quantity, use
``ddof=1``.
Note that for complex numbers, the absolute value is taken before
squaring, so that the result is always real and nonnegative.
For floating-point input, the variance is computed using the same
precision the input has. Depending on the input data, this can cause
the results to be inaccurate, especially for `float32` (see example
below). Specifying a higher-accuracy accumulator using the ``dtype``
keyword can alleviate this issue.
Examples
--------
>>> import numpy as np
>>> a = np.array([[1, 2], [3, 4]])
>>> np.var(a)
1.25
>>> np.var(a, axis=0)
array([1., 1.])
>>> np.var(a, axis=1)
array([0.25, 0.25])
In single precision, var() can be inaccurate:
>>> a = np.zeros((2, 512*512), dtype=np.float32)
>>> a[0, :] = 1.0
>>> a[1, :] = 0.1
>>> np.var(a)
np.float32(0.20250003)
Computing the variance in float64 is more accurate:
>>> np.var(a, dtype=np.float64)
0.20249999932944759 # may vary
>>> ((1-0.55)**2 + (0.1-0.55)**2)/2
0.2025
Specifying a where argument:
>>> a = np.array([[14, 8, 11, 10], [7, 9, 10, 11], [10, 15, 5, 10]])
>>> np.var(a)
6.833333333333333 # may vary
>>> np.var(a, where=[[True], [True], [False]])
4.0
Using the mean keyword to save computation time:
>>> import numpy as np
>>> from timeit import timeit
>>>
>>> a = np.array([[14, 8, 11, 10], [7, 9, 10, 11], [10, 15, 5, 10]])
>>> mean = np.mean(a, axis=1, keepdims=True)
>>>
>>> g = globals()
>>> n = 10000
>>> t1 = timeit("var = np.var(a, axis=1, mean=mean)", globals=g, number=n)
>>> t2 = timeit("var = np.var(a, axis=1)", globals=g, number=n)
>>> print(f'Percentage execution time saved {100*(t2-t1)/t2:.0f}%')
#doctest: +SKIP
Percentage execution time saved 32%
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