numpy.std

numpy.std(a, axis=None, dtype=None, out=None, ddof=0, keepdims=<no value>, *, where=<no value>, mean=<no value>, correction=<no value>)

Compute the standard deviation along the specified axis.

Returns the standard deviation, a measure of the spread of a distribution, of the array elements. The standard deviation is computed for the flattened array by default, otherwise over the specified axis.

Parameters:

aarray_like

Calculate the standard deviation of these values.

axisNone or int or tuple of ints, optional

Axis or axes along which the standard deviation is computed. The default is to compute the standard deviation of the flattened array.

New in version 1.7.0.

If this is a tuple of ints, a standard deviation is performed over multiple axes, instead of a single axis or all the axes as before.

dtypedtype, optional

Type to use in computing the standard deviation. For arrays of integer type the default is float64, for arrays of float types it is the same as the array type.

outndarray, optional

Alternative output array in which to place the result. It must have the same shape as the expected output but the type (of the calculated values) will be cast if necessary. See Output type determination for more details.

ddof{int, float}, optional

Means Delta Degrees of Freedom. The divisor used in calculations is N - ddof, where N represents the number of elements. By default ddof is zero. See Notes for details about use of ddof.

keepdimsbool, 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 std 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.

wherearray_like of bool, optional

Elements to include in the standard deviation. See reduce for details.

New in version 1.20.0.

meanarray_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 std function.

New in version 1.26.0.

correction{int, float}, optional

Array API compatible name for the ddof parameter. Only one of them can be provided at the same time.

New in version 2.0.0.

Returns:

standard_deviationndarray, see dtype parameter above.

If out is None, return a new array containing the standard deviation, otherwise return a reference to the output array.

See also

var, mean, nanmean, nanstd, nanvar

Output type determination

Notes

There are several common variants of the array standard deviation 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 standard deviation 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`
std = var**0.5

Different values of the argument ddof are useful in different contexts. NumPy’s default ddof=0 corresponds with the expression:

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

which is sometimes called the “population standard deviation” in the field of statistics because it applies the definition of standard deviation to a as if a were a complete population of possible observations.

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

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

In statistics, the resulting quantity is sometimed called the “sample standard deviation” because if a is a random sample from a larger population, this calculation provides the square root of an unbiased estimate of the variance of the population. 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 of a is used in place of the true mean of the population. The resulting estimate of the standard deviation is still biased, but less than it would have been without the correction. For this quantity, use ddof=1.

Note that, for complex numbers, std takes the absolute value before squaring, so that the result is always real and nonnegative.

For floating-point input, the standard deviation 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.std(a)
1.1180339887498949 # may vary
>>> np.std(a, axis=0)
array([1.,  1.])
>>> np.std(a, axis=1)
array([0.5,  0.5])

In single precision, std() can be inaccurate:

>>> a = np.zeros((2, 512*512), dtype=np.float32)
>>> a[0, :] = 1.0
>>> a[1, :] = 0.1
>>> np.std(a)
0.45000005

Computing the standard deviation in float64 is more accurate:

>>> np.std(a, dtype=np.float64)
0.44999999925494177 # may vary

Specifying a where argument:

>>> a = np.array([[14, 8, 11, 10], [7, 9, 10, 11], [10, 15, 5, 10]])
>>> np.std(a)
2.614064523559687 # may vary
>>> np.std(a, where=[[True], [True], [False]])
2.0

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("std = np.std(a, axis=1, mean=mean)", globals=g, number=n)
>>> t2 = timeit("std = np.std(a, axis=1)", globals=g, number=n)
>>> print(f'Percentage execution time saved {100*(t2-t1)/t2:.0f}%')

Percentage execution time saved 30%