Skip to main content
`Ctrl`+`K`
scipy.stats.

# bartlett

scipy.stats.bartlett(*samples, axis=0, nan_policy='propagate', keepdims=False)

  Perform Bartlett’s test for equal variances.

  Bartlett’s test tests the null hypothesis that all input samples are from populations with equal variances. For samples from significantly non-normal populations, Levene’s test `levene` is more robust.

  Parameters:
    sample1, sample2, …array_like

      arrays of sample data. Only 1d arrays are accepted, they may have different lengths.

    axis int or None, default: 0

      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:
    statistic float

      The test statistic.

    pvalue float

      The p-value of the test.

  See also

  `fligner`

    A non-parametric test for the equality of k variances

  `levene`

    A robust parametric test for equality of k variances

  Bartlett’s test for equal variances

    Extended example

  Notes

  Conover et al. (1981) examine many of the existing parametric and nonparametric tests by extensive simulations and they conclude that the tests proposed by Fligner and Killeen (1976) and Levene (1960) appear to be superior in terms of robustness of departures from normality and power ([3]).

  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`.

  Array API Standard Support

  `bartlett` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ✅ | ✅
  Dask | ✅ | n/a

  > See Support for the array API standard for more information.

  References

  [1]

  https://www.itl.nist.gov/div898/handbook/eda/section3/eda357.htm

  [2]

  Snedecor, George W. and Cochran, William G. (1989), Statistical Methods, Eighth Edition, Iowa State University Press.

  [3]

  Park, C. and Lindsay, B. G. (1999). Robust Scale Estimation and Hypothesis Testing based on Quadratic Inference Function. Technical Report #99-03, Center for Likelihood Studies, Pennsylvania State University.

  [4]

  Bartlett, M. S. (1937). Properties of Sufficiency and Statistical Tests. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 160, No.901, pp. 268-282.

  Examples

  Test whether the lists a, b and c come from populations with equal variances.

      >>> import numpy as np
      >>> from scipy import stats
      >>> a = [8.88, 9.12, 9.04, 8.98, 9.00, 9.08, 9.01, 8.85, 9.06, 8.99]
      >>> b = [8.88, 8.95, 9.29, 9.44, 9.15, 9.58, 8.36, 9.18, 8.67, 9.05]
      >>> c = [8.95, 9.12, 8.95, 8.85, 9.03, 8.84, 9.07, 8.98, 8.86, 8.98]
      >>> stat, p = stats.bartlett(a, b, c)
      >>> p
      1.1254782518834628e-05

  The very small p-value suggests that the populations do not have equal variances.

  This is not surprising, given that the sample variance of b is much larger than that of a and c:

      >>> [np.var(x, ddof=1) for x in [a, b, c]]
      [0.007054444444444413, 0.13073888888888888, 0.008890000000000002]

  For a more detailed example, see Bartlett’s test for equal variances.