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scipy.stats.

# gstd

scipy.stats.gstd(a, axis=0, ddof=1)

  Calculate the geometric standard deviation of an array.

  The geometric standard deviation describes the spread of a set of numbers where the geometric mean is preferred. It is a multiplicative factor, and so a dimensionless quantity.

  It is defined as the exponential of the standard deviation of the natural logarithms of the observations.

  Parameters:
    a array_like

      An array containing finite, strictly positive, real numbers.

      Deprecated since version 1.14.0: Support for masked array input was deprecated in SciPy 1.14.0 and will be removed in version 1.16.0.

    axis int, tuple or None, optional

      Axis along which to operate. Default is 0. If None, compute over the whole array a.

    ddof int, optional

      Degree of freedom correction in the calculation of the geometric standard deviation. Default is 1.

  Returns:
    gstd ndarray or float

      An array of the geometric standard deviation. If axis is None or a is a 1d array a float is returned.

  See also

  `gmean`

    Geometric mean

  `numpy.std`

    Standard deviation

  `gzscore`

    Geometric standard score

  Notes

  Mathematically, the sample geometric standard deviation \(s_G\) can be defined in terms of the natural logarithms of the observations \(y_i = \log(x_i)\):

  \[s_G = \exp(s), \quad s = \sqrt{\frac{1}{n - d} \sum_{i=1}^n (y_i - \bar y)^2}\]

  where \(n\) is the number of observations, \(d\) is the adjustment ddof to the degrees of freedom, and \(\bar y\) denotes the mean of the natural logarithms of the observations. Note that the default `ddof=1` is different from the default value used by similar functions, such as `numpy.std` and `numpy.var`.

  When an observation is infinite, the geometric standard deviation is NaN (undefined). Non-positive observations will also produce NaNs in the output because the natural logarithm (as opposed to the complex logarithm) is defined and finite only for positive reals. The geometric standard deviation is sometimes confused with the exponential of the standard deviation, `exp(std(a))`. Instead, the geometric standard deviation is `exp(std(log(a)))`.

  References

  [1]

  “Geometric standard deviation”, Wikipedia, https://en.wikipedia.org/wiki/Geometric_standard_deviation.

  [2]

  Kirkwood, T. B., “Geometric means and measures of dispersion”, Biometrics, vol. 35, pp. 908-909, 1979

  Examples

  Find the geometric standard deviation of a log-normally distributed sample. Note that the standard deviation of the distribution is one; on a log scale this evaluates to approximately `exp(1)`.

      >>> import numpy as np
      >>> from scipy.stats import gstd
      >>> rng = np.random.default_rng()
      >>> sample = rng.lognormal(mean=0, sigma=1, size=1000)
      >>> gstd(sample)
      2.810010162475324

  Compute the geometric standard deviation of a multidimensional array and of a given axis.

      >>> a = np.arange(1, 25).reshape(2, 3, 4)
      >>> gstd(a, axis=None)
      2.2944076136018947
      >>> gstd(a, axis=2)
      array([[1.82424757, 1.22436866, 1.13183117],
             [1.09348306, 1.07244798, 1.05914985]])
      >>> gstd(a, axis=(1,2))
      array([2.12939215, 1.22120169])