scipy.stats.circstd(samples, high=6.283185307179586, low=0, axis=None, nan_policy='propagate', *, normalize=False, keepdims=False)[source]#

Compute the circular standard deviation of a sample of angle observations.

Given \(n\) angle observations \(x_1, \cdots, x_n\) measured in radians, their circular standard deviation is defined by ([2], Eq. 2.3.11)

\[\sqrt{ -2 \log \left| \frac{1}{n} \sum_{k=1}^n e^{i x_k} \right| }\]

where \(i\) is the imaginary unit and \(|z|\) gives the length of the complex number \(z\). \(|z|\) in the above expression is known as the mean resultant length.


Input array of angle observations. The value of a full angle is equal to (high - low).

highfloat, optional

Upper boundary of the principal value of an angle. Default is 2*pi.

lowfloat, optional

Lower boundary of the principal value of an angle. Default is 0.

normalizeboolean, optional

If False (the default), the return value is computed from the above formula with the input scaled by (2*pi)/(high-low) and the output scaled (back) by (high-low)/(2*pi). If True, the output is not scaled and is returned directly.

axisint or None, default: None

If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If None, the input will be raveled before computing the statistic.

nan_policy{‘propagate’, ‘omit’, ‘raise’}

Defines how to handle input NaNs.

  • propagate: if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN.

  • omit: NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN.

  • raise: if a NaN is present, a ValueError will be raised.

keepdimsbool, default: False

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.


Circular standard deviation, optionally normalized.

If the input array is empty, np.nan is returned.

See also


Circular mean.


Circular variance.


In the limit of small angles, the circular standard deviation is close to the ‘linear’ standard deviation if normalize is False.

Beginning in SciPy 1.9, np.matrix inputs (not recommended for new code) are converted to np.ndarray before the calculation is performed. In this case, the output will be a scalar or np.ndarray of appropriate shape rather than a 2D np.matrix. Similarly, while masked elements of masked arrays are ignored, the output will be a scalar or np.ndarray rather than a masked array with mask=False.



Mardia, K. V. (1972). 2. In Statistics of Directional Data (pp. 18-24). Academic Press. DOI:10.1016/C2013-0-07425-7.


Mardia, K. V. and Jupp, P. E. Directional Statistics. John Wiley & Sons, 1999.


>>> import numpy as np
>>> from scipy.stats import circstd
>>> import matplotlib.pyplot as plt
>>> samples_1 = np.array([0.072, -0.158, 0.077, 0.108, 0.286,
...                       0.133, -0.473, -0.001, -0.348, 0.131])
>>> samples_2 = np.array([0.111, -0.879, 0.078, 0.733, 0.421,
...                       0.104, -0.136, -0.867,  0.012,  0.105])
>>> circstd_1 = circstd(samples_1)
>>> circstd_2 = circstd(samples_2)

Plot the samples.

>>> fig, (left, right) = plt.subplots(ncols=2)
>>> for image in (left, right):
...     image.plot(np.cos(np.linspace(0, 2*np.pi, 500)),
...                np.sin(np.linspace(0, 2*np.pi, 500)),
...                c='k')
...     image.axis('equal')
...     image.axis('off')
>>> left.scatter(np.cos(samples_1), np.sin(samples_1), c='k', s=15)
>>> left.set_title(f"circular std: {np.round(circstd_1, 2)!r}")
>>> right.plot(np.cos(np.linspace(0, 2*np.pi, 500)),
...            np.sin(np.linspace(0, 2*np.pi, 500)),
...            c='k')
>>> right.scatter(np.cos(samples_2), np.sin(samples_2), c='k', s=15)
>>> right.set_title(f"circular std: {np.round(circstd_2, 2)!r}")