scipy.stats.

quantile#

scipy.stats.quantile(x, p, *, method='linear', axis=0, nan_policy='propagate', keepdims=None)[source]#

Compute the p-th quantile of the data along the specified axis.

Parameters:
xarray_like of real numbers

Data array.

parray_like of float

Probability or sequence of probabilities of the quantiles to compute. Values must be between 0 and 1 (inclusive). Must have length 1 along axis unless keepdims=True.

methodstr, default: ‘linear’

The method to use for estimating the quantile. The available options, numbered as they appear in [1], are:

  1. ‘inverted_cdf’

  2. ‘averaged_inverted_cdf’

  3. ‘closest_observation’

  4. ‘interpolated_inverted_cdf’

  5. ‘hazen’

  6. ‘weibull’

  7. ‘linear’ (default)

  8. ‘median_unbiased’

  9. ‘normal_unbiased’

‘harrell-davis’ is also available to compute the quantile estimate according to [2]. See Notes for details.

axisint or None, default: 0

Axis along which the quantiles are computed. None ravels both x and p before performing the calculation, without checking whether the original shapes were compatible.

nan_policystr, default: ‘propagate’

Defines how to handle NaNs in the input data x.

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

  • 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 slice of the output will contain NaN(s).

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

If NaNs are present in p, a ValueError will be raised.

keepdimsbool, optional

Consider the case in which x is 1-D and p is a scalar: the quantile is a reducing statistic, and the default behavior is to return a scalar. If keepdims is set to True, the axis will not be reduced away, and the result will be a 1-D array with one element.

The general case is more subtle, since multiple quantiles may be requested for each axis-slice of x. For instance, if both x and p are 1-D and p.size > 1, no axis can be reduced away; there must be an axis to contain the number of quantiles given by p.size. Therefore:

  • By default, the axis will be reduced away if possible (i.e. if there is exactly one element of q per axis-slice of x).

  • If keepdims is set to True, the axis will not be reduced away.

  • If keepdims is set to False, the axis will be reduced away if possible, and an error will be raised otherwise.

Returns:
quantilescalar or ndarray

The resulting quantile(s). The dtype is the result dtype of x and p.

Notes

Given a sample x from an underlying distribution, quantile provides a nonparametric estimate of the inverse cumulative distribution function.

By default, this is done by interpolating between adjacent elements in y, a sorted copy of x:

(1-g)*y[j] + g*y[j+1]

where the index j and coefficient g are the integral and fractional components of p * (n-1), and n is the number of elements in the sample.

This is a special case of Equation 1 of H&F [1]. More generally,

  • j = (p*n + m - 1) // 1, and

  • g = (p*n + m - 1) % 1,

where m may be defined according to several different conventions. The preferred convention may be selected using the method parameter:

method

number in H&F

m

interpolated_inverted_cdf

4

0

hazen

5

1/2

weibull

6

p

linear (default)

7

1 - p

median_unbiased

8

p/3 + 1/3

normal_unbiased

9

p/4 + 3/8

Note that indices j and j + 1 are clipped to the range 0 to n - 1 when the results of the formula would be outside the allowed range of non-negative indices. When j is clipped to zero, g is set to zero as well. The -1 in the formulas for j and g accounts for Python’s 0-based indexing.

The table above includes only the estimators from [1] that are continuous functions of probability p (estimators 4-9). SciPy also provides the three discontinuous estimators from [1] (estimators 1-3), where j is defined as above, m is defined as follows, and g is 0 when index = p*n + m - 1 is less than 0 and otherwise is defined below.

  1. inverted_cdf: m = 0 and g = int(index - j > 0)

  2. averaged_inverted_cdf: m = 0 and g = (1 + int(index - j > 0)) / 2

  3. closest_observation: m = -1/2 and g = 1 - int((index == j) & (j%2 == 1))

A different strategy for computing quantiles from [2], method='harrell-davis', uses a weighted combination of all elements. The weights are computed as:

\[w_{n, i} = I_{i/n}(a, b) - I_{(i - 1)/n}(a, b)\]

where \(n\) is the number of elements in the sample, \(i\) are the indices \(1, 2, ..., n-1, n\) of the sorted elements, \(a = p (n + 1)\), \(b = (1 - p)(n + 1)\), \(p\) is the probability of the quantile, and \(I\) is the regularized, lower incomplete beta function (scipy.special.betainc).

References

[1] (1,2,3,4)

R. J. Hyndman and Y. Fan, “Sample quantiles in statistical packages,” The American Statistician, 50(4), pp. 361-365, 1996

[2] (1,2)

Harrell, Frank E., and C. E. Davis. “A new distribution-free quantile estimator.” Biometrika 69.3 (1982): 635-640.

Examples

>>> import numpy as np
>>> from scipy import stats
>>> x = np.asarray([[10, 8, 7, 5, 4],
...                 [0, 1, 2, 3, 5]])

Take the median along the last axis.

>>> stats.quantile(x, 0.5, axis=-1)
array([7.,  2.])

Take a different quantile along each axis.

>>> stats.quantile(x, [[0.25], [0.75]], axis=-1, keepdims=True)
array([[5.],
       [3.]])

Take multiple quantiles along each axis.

>>> stats.quantile(x, [0.25, 0.75], axis=-1)
array([[5., 8.],
       [1., 3.]])
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