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# scipy.stats.circstd

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

  Compute the circular standard deviation for samples assumed to be in the range [low to high].

  Parameters:
    samples array_like

      Input array.

    high float or int, optional

      High boundary for the sample range. Default is `2*pi`.

    low float or int, optional

      Low boundary for the sample range. Default is 0.

    normalize boolean, optional

      If True, the returned value is equal to `sqrt(-2*log(R))` and does not depend on the variable units. If False (default), the returned value is scaled by `((high-low)/(2*pi))`.

    axis int 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.

    keepdims bool, 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.

  Returns:
    circstd float

      Circular standard deviation.

  See also

  `circmean`

    Circular mean.

  `circvar`

    Circular variance.

  Notes

  This uses a definition of circular standard deviation from [1]. Essentially, the calculation is as follows.

      import numpy as np
      C = np.cos(samples).mean()
      S = np.sin(samples).mean()
      R = np.sqrt(C**2 + S**2)
      l = 2*np.pi / (high-low)
      circstd = np.sqrt(-2*np.log(R)) / l

  In the limit of small angles, it returns a number close to the ‘linear’ standard deviation.

  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`.

  References

  [1]

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

  Examples

      >>> 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}")
      >>> plt.show()