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# scipy.stats.median_abs_deviation

scipy.stats.median_abs_deviation(x, axis=0, center=<function median>, scale=1.0, nan_policy='propagate')

  Compute the median absolute deviation of the data along the given axis.

  The median absolute deviation (MAD, [1]) computes the median over the absolute deviations from the median. It is a measure of dispersion similar to the standard deviation but more robust to outliers [2].

  The MAD of an empty array is `np.nan`.

  New in version 1.5.0.

  Parameters:
    x array_like

      Input array or object that can be converted to an array.

    axis int or None, optional

      Axis along which the range is computed. Default is 0. If None, compute the MAD over the entire array.

    center callable, optional

      A function that will return the central value. The default is to use np.median. Any user defined function used will need to have the function signature `func(arr, axis)`.

    scale scalar or str, optional

      The numerical value of scale will be divided out of the final result. The default is 1.0. The string “normal” is also accepted, and results in scale being the inverse of the standard normal quantile function at 0.75, which is approximately 0.67449. Array-like scale is also allowed, as long as it broadcasts correctly to the output such that `out / scale` is a valid operation. The output dimensions depend on the input array, x, and the axis argument.

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

      Defines how to handle when input contains nan. The following options are available (default is ‘propagate’):

      - ‘propagate’: returns nan

      - ‘raise’: throws an error

      - ‘omit’: performs the calculations ignoring nan values

  Returns:
    mad scalar or ndarray

      If `axis=None`, a scalar is returned. If the input contains integers or floats of smaller precision than `np.float64`, then the output data-type is `np.float64`. Otherwise, the output data-type is the same as that of the input.

  See also

  `numpy.std`, `numpy.var`, `numpy.median`, `scipy.stats.iqr`, `scipy.stats.tmean`
  `scipy.stats.tstd`, `scipy.stats.tvar`

  Notes

  The center argument only affects the calculation of the central value around which the MAD is calculated. That is, passing in `center=np.mean` will calculate the MAD around the mean - it will not calculate the mean absolute deviation.

  The input array may contain inf, but if center returns inf, the corresponding MAD for that data will be nan.

  References

  [1]

  “Median absolute deviation”, https://en.wikipedia.org/wiki/Median_absolute_deviation

  [2]

  “Robust measures of scale”, https://en.wikipedia.org/wiki/Robust_measures_of_scale

  Examples

  When comparing the behavior of `median_abs_deviation` with `np.std`, the latter is affected when we change a single value of an array to have an outlier value while the MAD hardly changes:

      >>> import numpy as np
      >>> from scipy import stats
      >>> x = stats.norm.rvs(size=100, scale=1, random_state=123456)
      >>> x.std()
      0.9973906394005013
      >>> stats.median_abs_deviation(x)
      0.82832610097857
      >>> x[0] = 345.6
      >>> x.std()
      34.42304872314415
      >>> stats.median_abs_deviation(x)
      0.8323442311590675

  Axis handling example:

      >>> x = np.array([[10, 7, 4], [3, 2, 1]])
      >>> x
      array([[10,  7,  4],
             [ 3,  2,  1]])
      >>> stats.median_abs_deviation(x)
      array([3.5, 2.5, 1.5])
      >>> stats.median_abs_deviation(x, axis=None)
      2.0

  Scale normal example:

      >>> x = stats.norm.rvs(size=1000000, scale=2, random_state=123456)
      >>> stats.median_abs_deviation(x)
      1.3487398527041636
      >>> stats.median_abs_deviation(x, scale='normal')
      1.9996446978061115