Source code for boltons.statsutils

# Copyright (c) 2013, Mahmoud Hashemi
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"""``statsutils`` provides tools aimed primarily at descriptive
statistics for data analysis, such as :func:`mean` (average),
:func:`median`, :func:`variance`, and many others,

The :class:`Stats` type provides all the main functionality of the
``statsutils`` module. A :class:`Stats` object wraps a given dataset,
providing all statistical measures as property attributes. These
attributes cache their results, which allows efficient computation of
multiple measures, as many measures rely on other measures. For
example, relative standard deviation (:attr:`Stats.rel_std_dev`)
relies on both the mean and standard deviation. The Stats object
caches those results so no rework is done.

The :class:`Stats` type's attributes have module-level counterparts for
convenience when the computation reuse advantages do not apply.

>>> stats = Stats(range(42))
>>> stats.mean
>>> mean(range(42))

Statistics is a large field, and ``statsutils`` is focused on a few
basic techniques that are useful in software. The following is a brief
introduction to those techniques. For a more in-depth introduction,
`Statistics for Software
an article I wrote on the topic. It introduces key terminology vital
to effective usage of statistics.

Statistical moments

Python programmers are probably familiar with the concept of the
*mean* or *average*, which gives a rough quantitiative middle value by
which a sample can be can be generalized. However, the mean is just
the first of four `moment`_-based measures by which a sample or
distribution can be measured.

The four `Standardized moments`_ are:

  1. `Mean`_ - :func:`mean` - theoretical middle value
  2. `Variance`_ - :func:`variance` - width of value dispersion
  3. `Skewness`_ - :func:`skewness` - symmetry of distribution
  4. `Kurtosis`_ - :func:`kurtosis` - "peakiness" or "long-tailed"-ness

For more information check out `the Moment article on Wikipedia`_.

.. _moment:
.. _Standardized moments:
.. _Mean:
.. _Variance:
.. _Skewness:
.. _Kurtosis:
.. _the Moment article on Wikipedia:

Keep in mind that while these moments can give a bit more insight into
the shape and distribution of data, they do not guarantee a complete
picture. Wildly different datasets can have the same values for all
four moments, so generalize wisely.

Robust statistics

Moment-based statistics are notorious for being easily skewed by
outliers. The whole field of robust statistics aims to mitigate this
dilemma. ``statsutils`` also includes several robust statistical methods:

  * `Median`_ - The middle value of a sorted dataset
  * `Trimean`_ - Another robust measure of the data's central tendency
  * `Median Absolute Deviation`_ (MAD) - A robust measure of
    variability, a natural counterpart to :func:`variance`.
  * `Trimming`_ - Reducing a dataset to only the middle majority of
    data is a simple way of making other estimators more robust.

.. _Median:
.. _Trimean:
.. _Median Absolute Deviation:
.. _Trimming:

Online and Offline Statistics

Unrelated to computer networking, `online`_ statistics involve
calculating statistics in a `streaming`_ fashion, without all the data
being available. The :class:`Stats` type is meant for the more
traditional offline statistics when all the data is available. For
pure-Python online statistics accumulators, look at the `Lithoxyl`_
system instrumentation package.

.. _Online:
.. _streaming:
.. _Lithoxyl:


import bisect
from math import floor, ceil
from collections import Counter

class _StatsProperty:
    def __init__(self, name, func): = name
        self.func = func
        self.internal_name = '_' + name

        doc = func.__doc__ or ''
        pre_doctest_doc, _, _ = doc.partition('>>>')
        self.__doc__ = pre_doctest_doc

    def __get__(self, obj, objtype=None):
        if obj is None:
            return self
        if not
            return obj.default
            return getattr(obj, self.internal_name)
        except AttributeError:
            setattr(obj, self.internal_name, self.func(obj))
            return getattr(obj, self.internal_name)

[docs] class Stats: """The ``Stats`` type is used to represent a group of unordered statistical datapoints for calculations such as mean, median, and variance. Args: data (list): List or other iterable containing numeric values. default (float): A value to be returned when a given statistical measure is not defined. 0.0 by default, but ``float('nan')`` is appropriate for stricter applications. use_copy (bool): By default Stats objects copy the initial data into a new list to avoid issues with modifications. Pass ``False`` to disable this behavior. is_sorted (bool): Presorted data can skip an extra sorting step for a little speed boost. Defaults to False. """ def __init__(self, data, default=0.0, use_copy=True, is_sorted=False): self._use_copy = use_copy self._is_sorted = is_sorted if use_copy: = list(data) else: = data self.default = default cls = self.__class__ self._prop_attr_names = [a for a in dir(self) if isinstance(getattr(cls, a, None), _StatsProperty)] self._pearson_precision = 0 def __len__(self): return len( def __iter__(self): return iter( def _get_sorted_data(self): """When using a copy of the data, it's better to have that copy be sorted, but we do it lazily using this method, in case no sorted measures are used. I.e., if median is never called, sorting would be a waste. When not using a copy, it's presumed that all optimizations are on the user. """ if not self._use_copy: return sorted( elif not self._is_sorted: return
[docs] def clear_cache(self): """``Stats`` objects automatically cache intermediary calculations that can be reused. For instance, accessing the ``std_dev`` attribute after the ``variance`` attribute will be significantly faster for medium-to-large datasets. If you modify the object by adding additional data points, call this function to have the cached statistics recomputed. """ for attr_name in self._prop_attr_names: attr_name = getattr(self.__class__, attr_name).internal_name if not hasattr(self, attr_name): continue delattr(self, attr_name) return
def _calc_count(self): """The number of items in this Stats object. Returns the same as :func:`len` on a Stats object, but provided for pandas terminology parallelism. >>> Stats(range(20)).count 20 """ return len( count = _StatsProperty('count', _calc_count) def _calc_mean(self): """ The arithmetic mean, or "average". Sum of the values divided by the number of values. >>> mean(range(20)) 9.5 >>> mean(list(range(19)) + [949]) # 949 is an arbitrary outlier 56.0 """ return sum(, 0.0) / len( mean = _StatsProperty('mean', _calc_mean) def _calc_max(self): """ The maximum value present in the data. >>> Stats([2, 1, 3]).max 3 """ if self._is_sorted: return[-1] return max( max = _StatsProperty('max', _calc_max) def _calc_min(self): """ The minimum value present in the data. >>> Stats([2, 1, 3]).min 1 """ if self._is_sorted: return[0] return min( min = _StatsProperty('min', _calc_min) def _calc_median(self): """ The median is either the middle value or the average of the two middle values of a sample. Compared to the mean, it's generally more resilient to the presence of outliers in the sample. >>> median([2, 1, 3]) 2 >>> median(range(97)) 48 >>> median(list(range(96)) + [1066]) # 1066 is an arbitrary outlier 48 """ return self._get_quantile(self._get_sorted_data(), 0.5) median = _StatsProperty('median', _calc_median) def _calc_iqr(self): """Inter-quartile range (IQR) is the difference between the 75th percentile and 25th percentile. IQR is a robust measure of dispersion, like standard deviation, but safer to compare between datasets, as it is less influenced by outliers. >>> iqr([1, 2, 3, 4, 5]) 2 >>> iqr(range(1001)) 500 """ return self.get_quantile(0.75) - self.get_quantile(0.25) iqr = _StatsProperty('iqr', _calc_iqr) def _calc_trimean(self): """The trimean is a robust measure of central tendency, like the median, that takes the weighted average of the median and the upper and lower quartiles. >>> trimean([2, 1, 3]) 2.0 >>> trimean(range(97)) 48.0 >>> trimean(list(range(96)) + [1066]) # 1066 is an arbitrary outlier 48.0 """ sorted_data = self._get_sorted_data() gq = lambda q: self._get_quantile(sorted_data, q) return (gq(0.25) + (2 * gq(0.5)) + gq(0.75)) / 4.0 trimean = _StatsProperty('trimean', _calc_trimean) def _calc_variance(self): """\ Variance is the average of the squares of the difference between each value and the mean. >>> variance(range(97)) 784.0 """ global mean # defined elsewhere in this file return mean(self._get_pow_diffs(2)) variance = _StatsProperty('variance', _calc_variance) def _calc_std_dev(self): """\ Standard deviation. Square root of the variance. >>> std_dev(range(97)) 28.0 """ return self.variance ** 0.5 std_dev = _StatsProperty('std_dev', _calc_std_dev) def _calc_median_abs_dev(self): """\ Median Absolute Deviation is a robust measure of statistical dispersion: >>> median_abs_dev(range(97)) 24.0 """ global median # defined elsewhere in this file sorted_vals = sorted( x = float(median(sorted_vals)) return median([abs(x - v) for v in sorted_vals]) median_abs_dev = _StatsProperty('median_abs_dev', _calc_median_abs_dev) mad = median_abs_dev # convenience def _calc_rel_std_dev(self): """\ Standard deviation divided by the absolute value of the average. >>> print('%1.3f' % rel_std_dev(range(97))) 0.583 """ abs_mean = abs(self.mean) if abs_mean: return self.std_dev / abs_mean else: return self.default rel_std_dev = _StatsProperty('rel_std_dev', _calc_rel_std_dev) def _calc_skewness(self): """\ Indicates the asymmetry of a curve. Positive values mean the bulk of the values are on the left side of the average and vice versa. See the module docstring for more about statistical moments. >>> skewness(range(97)) # symmetrical around 48.0 0.0 >>> left_skewed = skewness(list(range(97)) + list(range(10))) >>> right_skewed = skewness(list(range(97)) + list(range(87, 97))) >>> round(left_skewed, 3), round(right_skewed, 3) (0.114, -0.114) """ data, s_dev =, self.std_dev if len(data) > 1 and s_dev > 0: return (sum(self._get_pow_diffs(3)) / float((len(data) - 1) * (s_dev ** 3))) else: return self.default skewness = _StatsProperty('skewness', _calc_skewness) def _calc_kurtosis(self): """\ Indicates how much data is in the tails of the distribution. The result is always positive, with the normal "bell-curve" distribution having a kurtosis of 3. See the module docstring for more about statistical moments. >>> kurtosis(range(9)) 1.99125 With a kurtosis of 1.99125, [0, 1, 2, 3, 4, 5, 6, 7, 8] is more centrally distributed than the normal curve. """ data, s_dev =, self.std_dev if len(data) > 1 and s_dev > 0: return (sum(self._get_pow_diffs(4)) / float((len(data) - 1) * (s_dev ** 4))) else: return 0.0 kurtosis = _StatsProperty('kurtosis', _calc_kurtosis) def _calc_pearson_type(self): precision = self._pearson_precision skewness = self.skewness kurtosis = self.kurtosis beta1 = skewness ** 2.0 beta2 = kurtosis * 1.0 # TODO: range checks? c0 = (4 * beta2) - (3 * beta1) c1 = skewness * (beta2 + 3) c2 = (2 * beta2) - (3 * beta1) - 6 if round(c1, precision) == 0: if round(beta2, precision) == 3: return 0 # Normal else: if beta2 < 3: return 2 # Symmetric Beta elif beta2 > 3: return 7 elif round(c2, precision) == 0: return 3 # Gamma else: k = c1 ** 2 / (4 * c0 * c2) if k < 0: return 1 # Beta raise RuntimeError('missed a spot') pearson_type = _StatsProperty('pearson_type', _calc_pearson_type) @staticmethod def _get_quantile(sorted_data, q): data, n = sorted_data, len(sorted_data) idx = q / 1.0 * (n - 1) idx_f, idx_c = int(floor(idx)), int(ceil(idx)) if idx_f == idx_c: return data[idx_f] return (data[idx_f] * (idx_c - idx)) + (data[idx_c] * (idx - idx_f))
[docs] def get_quantile(self, q): """Get a quantile from the dataset. Quantiles are floating point values between ``0.0`` and ``1.0``, with ``0.0`` representing the minimum value in the dataset and ``1.0`` representing the maximum. ``0.5`` represents the median: >>> Stats(range(100)).get_quantile(0.5) 49.5 """ q = float(q) if not 0.0 <= q <= 1.0: raise ValueError('expected q between 0.0 and 1.0, not %r' % q) elif not return self.default return self._get_quantile(self._get_sorted_data(), q)
[docs] def get_zscore(self, value): """Get the z-score for *value* in the group. If the standard deviation is 0, 0 inf or -inf will be returned to indicate whether the value is equal to, greater than or below the group's mean. """ mean = self.mean if self.std_dev == 0: if value == mean: return 0 if value > mean: return float('inf') if value < mean: return float('-inf') return (float(value) - mean) / self.std_dev
[docs] def trim_relative(self, amount=0.15): """A utility function used to cut a proportion of values off each end of a list of values. This has the effect of limiting the effect of outliers. Args: amount (float): A value between 0.0 and 0.5 to trim off of each side of the data. .. note: This operation modifies the data in-place. It does not make or return a copy. """ trim = float(amount) if not 0.0 <= trim < 0.5: raise ValueError('expected amount between 0.0 and 0.5, not %r' % trim) size = len( size_diff = int(size * trim) if size_diff == 0.0: return = self._get_sorted_data()[size_diff:-size_diff] self.clear_cache()
def _get_pow_diffs(self, power): """ A utility function used for calculating statistical moments. """ m = self.mean return [(v - m) ** power for v in] def _get_bin_bounds(self, count=None, with_max=False): if not return [0.0] # TODO: raise? data = len_data, min_data, max_data = len(data), min(data), max(data) if len_data < 4: if not count: count = len_data dx = (max_data - min_data) / float(count) bins = [min_data + (dx * i) for i in range(count)] elif count is None: # freedman algorithm for fixed-width bin selection q25, q75 = self.get_quantile(0.25), self.get_quantile(0.75) dx = 2 * (q75 - q25) / (len_data ** (1 / 3.0)) bin_count = max(1, int(ceil((max_data - min_data) / dx))) bins = [min_data + (dx * i) for i in range(bin_count + 1)] bins = [b for b in bins if b < max_data] else: dx = (max_data - min_data) / float(count) bins = [min_data + (dx * i) for i in range(count)] if with_max: bins.append(float(max_data)) return bins
[docs] def get_histogram_counts(self, bins=None, **kw): """Produces a list of ``(bin, count)`` pairs comprising a histogram of the Stats object's data, using fixed-width bins. See :meth:`Stats.format_histogram` for more details. Args: bins (int): maximum number of bins, or list of floating-point bin boundaries. Defaults to the output of Freedman's algorithm. bin_digits (int): Number of digits used to round down the bin boundaries. Defaults to 1. The output of this method can be stored and/or modified, and then passed to :func:`statsutils.format_histogram_counts` to achieve the same text formatting as the :meth:`~Stats.format_histogram` method. This can be useful for snapshotting over time. """ bin_digits = int(kw.pop('bin_digits', 1)) if kw: raise TypeError('unexpected keyword arguments: %r' % kw.keys()) if not bins: bins = self._get_bin_bounds() else: try: bin_count = int(bins) except TypeError: try: bins = [float(x) for x in bins] except Exception: raise ValueError('bins expected integer bin count or list' ' of float bin boundaries, not %r' % bins) if self.min < bins[0]: bins = [self.min] + bins else: bins = self._get_bin_bounds(bin_count) # floor and ceil really should have taken ndigits, like round() round_factor = 10.0 ** bin_digits bins = [floor(b * round_factor) / round_factor for b in bins] bins = sorted(set(bins)) idxs = [bisect.bisect(bins, d) - 1 for d in] count_map = Counter(idxs) bin_counts = [(b, count_map.get(i, 0)) for i, b in enumerate(bins)] return bin_counts
[docs] def format_histogram(self, bins=None, **kw): """Produces a textual histogram of the data, using fixed-width bins, allowing for simple visualization, even in console environments. >>> data = list(range(20)) + list(range(5, 15)) + [10] >>> print(Stats(data).format_histogram(width=30)) 0.0: 5 ######### 4.4: 8 ############### 8.9: 11 #################### 13.3: 5 ######### 17.8: 2 #### In this histogram, five values are between 0.0 and 4.4, eight are between 4.4 and 8.9, and two values lie between 17.8 and the max. You can specify the number of bins, or provide a list of bin boundaries themselves. If no bins are provided, as in the example above, `Freedman's algorithm`_ for bin selection is used. Args: bins (int): Maximum number of bins for the histogram. Also accepts a list of floating-point bin boundaries. If the minimum boundary is still greater than the minimum value in the data, that boundary will be implicitly added. Defaults to the bin boundaries returned by `Freedman's algorithm`_. bin_digits (int): Number of digits to round each bin to. Note that bins are always rounded down to avoid clipping any data. Defaults to 1. width (int): integer number of columns in the longest line in the histogram. Defaults to console width on Python 3.3+, or 80 if that is not available. format_bin (callable): Called on each bin to create a label for the final output. Use this function to add units, such as "ms" for milliseconds. Should you want something more programmatically reusable, see the :meth:`~Stats.get_histogram_counts` method, the output of is used by format_histogram. The :meth:`~Stats.describe` method is another useful summarization method, albeit less visual. .. _Freedman's algorithm: """ width = kw.pop('width', None) format_bin = kw.pop('format_bin', None) bin_counts = self.get_histogram_counts(bins=bins, **kw) return format_histogram_counts(bin_counts, width=width, format_bin=format_bin)
[docs] def describe(self, quantiles=None, format=None): """Provides standard summary statistics for the data in the Stats object, in one of several convenient formats. Args: quantiles (list): A list of numeric values to use as quantiles in the resulting summary. All values must be 0.0-1.0, with 0.5 representing the median. Defaults to ``[0.25, 0.5, 0.75]``, representing the standard quartiles. format (str): Controls the return type of the function, with one of three valid values: ``"dict"`` gives back a :class:`dict` with the appropriate keys and values. ``"list"`` is a list of key-value pairs in an order suitable to pass to an OrderedDict or HTML table. ``"text"`` converts the values to text suitable for printing, as seen below. Here is the information returned by a default ``describe``, as presented in the ``"text"`` format: >>> stats = Stats(range(1, 8)) >>> print(stats.describe(format='text')) count: 7 mean: 4.0 std_dev: 2.0 mad: 2.0 min: 1 0.25: 2.5 0.5: 4 0.75: 5.5 max: 7 For more advanced descriptive statistics, check out my blog post on the topic `Statistics for Software <>`_. """ if format is None: format = 'dict' elif format not in ('dict', 'list', 'text'): raise ValueError('invalid format for describe,' ' expected one of "dict"/"list"/"text", not %r' % format) quantiles = quantiles or [0.25, 0.5, 0.75] q_items = [] for q in quantiles: q_val = self.get_quantile(q) q_items.append((str(q), q_val)) items = [('count', self.count), ('mean', self.mean), ('std_dev', self.std_dev), ('mad', self.mad), ('min', self.min)] items.extend(q_items) items.append(('max', self.max)) if format == 'dict': ret = dict(items) elif format == 'list': ret = items elif format == 'text': ret = '\n'.join(['{}{}'.format((label + ':').ljust(10), val) for label, val in items]) return ret
[docs] def describe(data, quantiles=None, format=None): """A convenience function to get standard summary statistics useful for describing most data. See :meth:`Stats.describe` for more details. >>> print(describe(range(7), format='text')) count: 7 mean: 3.0 std_dev: 2.0 mad: 2.0 min: 0 0.25: 1.5 0.5: 3 0.75: 4.5 max: 6 See :meth:`Stats.format_histogram` for another very useful summarization that uses textual visualization. """ return Stats(data).describe(quantiles=quantiles, format=format)
def _get_conv_func(attr_name): def stats_helper(data, default=0.0): return getattr(Stats(data, default=default, use_copy=False), attr_name) return stats_helper for attr_name, attr in list(Stats.__dict__.items()): if isinstance(attr, _StatsProperty): if attr_name in ('max', 'min', 'count'): # don't shadow builtins continue if attr_name in ('mad',): # convenience aliases continue func = _get_conv_func(attr_name) func.__doc__ = attr.func.__doc__ globals()[attr_name] = func delattr(Stats, '_calc_' + attr_name) # cleanup del attr del attr_name del func
[docs] def format_histogram_counts(bin_counts, width=None, format_bin=None): """The formatting logic behind :meth:`Stats.format_histogram`, which takes the output of :meth:`Stats.get_histogram_counts`, and passes them to this function. Args: bin_counts (list): A list of bin values to counts. width (int): Number of character columns in the text output, defaults to 80 or console width in Python 3.3+. format_bin (callable): Used to convert bin values into string labels. """ lines = [] if not format_bin: format_bin = lambda v: v if not width: try: import shutil # python 3 convenience width = shutil.get_terminal_size()[0] except Exception: width = 80 bins = [b for b, _ in bin_counts] count_max = max([count for _, count in bin_counts]) count_cols = len(str(count_max)) labels = ['%s' % format_bin(b) for b in bins] label_cols = max([len(l) for l in labels]) tmp_line = '{}: {} #'.format('x' * label_cols, count_max) bar_cols = max(width - len(tmp_line), 3) line_k = float(bar_cols) / count_max tmpl = "{label:>{label_cols}}: {count:>{count_cols}} {bar}" for label, (bin_val, count) in zip(labels, bin_counts): bar_len = int(round(count * line_k)) bar = ('#' * bar_len) or '|' line = tmpl.format(label=label, label_cols=label_cols, count=count, count_cols=count_cols, bar=bar) lines.append(line) return '\n'.join(lines)