numpy.var#

numpy.var(a,axis=None,dtype=None,out=None,ddof=0,keepdims=<novalue>,*,where=<novalue>,mean=<novalue>,correction=<novalue>)[source]#

Compute the variance along the specified axis.

Returns the variance of the array elements, a measure of the spread of adistribution. The variance is computed for the flattened array bydefault, otherwise over the specified axis.

Parameters:
aarray_like

Array containing numbers whose variance is desired. Ifa is not anarray, a conversion is attempted.

axisNone or int or tuple of ints, optional

Axis or axes along which the variance is computed. The default is tocompute 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.

dtypedata-type, optional

Type to use in computing the variance. For arrays of integer typethe default isfloat64; for arrays of float types it is the same asthe array type.

outndarray, optional

Alternate output array in which to place the result. It must havethe same shape as the expected output, but the type is cast ifnecessary.

ddof{int, float}, optional

“Delta Degrees of Freedom”: the divisor used in the calculation isN-ddof, whereN represents the number of elements. Bydefaultddof is zero. See notes for details about use ofddof.

keepdimsbool, optional

If this is set to True, the axes which are reduced are leftin 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, thenkeepdims will not bepassed through to thevar method of sub-classes ofndarray, however any non-default value will be. If thesub-class’ method does not implementkeepdims anyexceptions will be raised.

wherearray_like of bool, optional

Elements to include in the variance. Seereduce fordetails.

New in version 1.20.0.

meanarray like, optional

Provide the mean to prevent its recalculation. The mean should havea shape as if it was calculated withkeepdims=True.The axis for the calculation of the mean should be the same as used inthe call to this var function.

New in version 2.0.0.

correction{int, float}, optional

Array API compatible name for theddof parameter. Only one of themcan be provided at the same time.

New in version 2.0.0.

Returns:
variancendarray, see dtype parameter above

Ifout=None, returns a new array containing the variance;otherwise, a reference to the output array is returned.

Notes

There are several common variants of the array variance calculation.Assuming the inputa is a one-dimensional NumPy array andmean iseither provided as an argument or computed asa.mean(), NumPycomputes 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 argumentddof are useful in differentcontexts. NumPy’s defaultddof=0 corresponds with the expression:

\[\frac{\sum_i{|a_i - \bar{a}|^2 }}{N}\]

which is sometimes called the “population variance” in the field ofstatistics because it applies the definition of variance toa as ifawere a complete population of possible observations.

Many other libraries define the variance of an array differently, e.g.:

\[\frac{\sum_i{|a_i - \bar{a}|^2}}{N - 1}\]

In statistics, the resulting quantity is sometimes called the “samplevariance” because ifa is a random sample from a larger population,this calculation provides an unbiased estimate of the variance of thepopulation. The use of\(N-1\) 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 ofa is usedin place of the true mean of the population. For this quantity, useddof=1.

Note that for complex numbers, the absolute value is taken beforesquaring, so that the result is always real and nonnegative.

For floating-point input, the variance is computed using the sameprecision the input has. Depending on the input data, this can causethe results to be inaccurate, especially forfloat32 (see examplebelow). Specifying a higher-accuracy accumulator using thedtypekeyword can alleviate this issue.

Examples

>>>importnumpyasnp>>>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)/20.2025

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:

>>>importnumpyasnp>>>fromtimeitimporttimeit>>>>>>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}%')Percentage execution time saved 32%
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