diff options
| -rw-r--r-- | Doc/library/statistics.rst | 129 | ||||
| -rw-r--r-- | Lib/statistics.py | 38 | ||||
| -rw-r--r-- | Misc/ACKS | 1 |
3 files changed, 83 insertions, 85 deletions
diff --git a/Doc/library/statistics.rst b/Doc/library/statistics.rst index 04b731d..62d856b 100644 --- a/Doc/library/statistics.rst +++ b/Doc/library/statistics.rst @@ -26,10 +26,10 @@ numeric (:class:`Real`-valued) data. Unless explicitly noted otherwise, these functions support :class:`int`, :class:`float`, :class:`decimal.Decimal` and :class:`fractions.Fraction`. Behaviour with other types (whether in the numeric tower or not) is - currently unsupported. Mixed types are also undefined and - implementation-dependent. If your input data consists of mixed types, - you may be able to use :func:`map` to ensure a consistent result, e.g. - ``map(float, input_data)``. + currently unsupported. Collections with a mix of types are also undefined + and implementation-dependent. If your input data consists of mixed types, + you may be able to use :func:`map` to ensure a consistent result, for + example: ``map(float, input_data)``. Averages and measures of central location ----------------------------------------- @@ -102,11 +102,9 @@ However, for reading convenience, most of the examples show sorted sequences. .. note:: The mean is strongly affected by outliers and is not a robust estimator - for central location: the mean is not necessarily a typical example of the - data points. For more robust, although less efficient, measures of - central location, see :func:`median` and :func:`mode`. (In this case, - "efficient" refers to statistical efficiency rather than computational - efficiency.) + for central location: the mean is not necessarily a typical example of + the data points. For more robust measures of central location, see + :func:`median` and :func:`mode`. The sample mean gives an unbiased estimate of the true population mean, which means that, taken on average over all the possible samples, @@ -120,9 +118,8 @@ However, for reading convenience, most of the examples show sorted sequences. Convert *data* to floats and compute the arithmetic mean. This runs faster than the :func:`mean` function and it always returns a - :class:`float`. The result is highly accurate but not as perfect as - :func:`mean`. If the input dataset is empty, raises a - :exc:`StatisticsError`. + :class:`float`. The *data* may be a sequence or iterator. If the input + dataset is empty, raises a :exc:`StatisticsError`. .. doctest:: @@ -136,15 +133,20 @@ However, for reading convenience, most of the examples show sorted sequences. Convert *data* to floats and compute the geometric mean. + The geometric mean indicates the central tendency or typical value of the + *data* using the product of the values (as opposed to the arithmetic mean + which uses their sum). + Raises a :exc:`StatisticsError` if the input dataset is empty, if it contains a zero, or if it contains a negative value. + The *data* may be a sequence or iterator. No special efforts are made to achieve exact results. (However, this may change in the future.) .. doctest:: - >>> round(geometric_mean([54, 24, 36]), 9) + >>> round(geometric_mean([54, 24, 36]), 1) 36.0 .. versionadded:: 3.8 @@ -174,7 +176,7 @@ However, for reading convenience, most of the examples show sorted sequences. 3.6 Using the arithmetic mean would give an average of about 5.167, which - is too high. + is well over the aggregate P/E ratio. :exc:`StatisticsError` is raised if *data* is empty, or any element is less than zero. @@ -312,10 +314,10 @@ However, for reading convenience, most of the examples show sorted sequences. The mode (when it exists) is the most typical value and serves as a measure of central location. - If there are multiple modes, returns the first one encountered in the *data*. - If the smallest or largest of multiple modes is desired instead, use - ``min(multimode(data))`` or ``max(multimode(data))``. If the input *data* is - empty, :exc:`StatisticsError` is raised. + If there are multiple modes with the same frequency, returns the first one + encountered in the *data*. If the smallest or largest of those is + desired instead, use ``min(multimode(data))`` or ``max(multimode(data))``. + If the input *data* is empty, :exc:`StatisticsError` is raised. ``mode`` assumes discrete data, and returns a single value. This is the standard treatment of the mode as commonly taught in schools: @@ -325,8 +327,8 @@ However, for reading convenience, most of the examples show sorted sequences. >>> mode([1, 1, 2, 3, 3, 3, 3, 4]) 3 - The mode is unique in that it is the only statistic which also applies - to nominal (non-numeric) data: + The mode is unique in that it is the only statistic in this package that + also applies to nominal (non-numeric) data: .. doctest:: @@ -368,15 +370,16 @@ However, for reading convenience, most of the examples show sorted sequences. .. function:: pvariance(data, mu=None) - Return the population variance of *data*, a non-empty iterable of real-valued - numbers. Variance, or second moment about the mean, is a measure of the - variability (spread or dispersion) of data. A large variance indicates that - the data is spread out; a small variance indicates it is clustered closely - around the mean. + Return the population variance of *data*, a non-empty sequence or iterator + of real-valued numbers. Variance, or second moment about the mean, is a + measure of the variability (spread or dispersion) of data. A large + variance indicates that the data is spread out; a small variance indicates + it is clustered closely around the mean. - If the optional second argument *mu* is given, it should be the mean of - *data*. If it is missing or ``None`` (the default), the mean is - automatically calculated. + If the optional second argument *mu* is given, it is typically the mean of + the *data*. It can also be used to compute the second moment around a + point that is not the mean. If it is missing or ``None`` (the default), + the arithmetic mean is automatically calculated. Use this function to calculate the variance from the entire population. To estimate the variance from a sample, the :func:`variance` function is usually @@ -401,10 +404,6 @@ However, for reading convenience, most of the examples show sorted sequences. >>> pvariance(data, mu) 1.25 - This function does not attempt to verify that you have passed the actual mean - as *mu*. Using arbitrary values for *mu* may lead to invalid or impossible - results. - Decimals and Fractions are supported: .. doctest:: @@ -423,11 +422,11 @@ However, for reading convenience, most of the examples show sorted sequences. σ². When called on a sample instead, this is the biased sample variance s², also known as variance with N degrees of freedom. - If you somehow know the true population mean μ, you may use this function - to calculate the variance of a sample, giving the known population mean as - the second argument. Provided the data points are representative - (e.g. independent and identically distributed), the result will be an - unbiased estimate of the population variance. + If you somehow know the true population mean μ, you may use this + function to calculate the variance of a sample, giving the known + population mean as the second argument. Provided the data points are a + random sample of the population, the result will be an unbiased estimate + of the population variance. .. function:: stdev(data, xbar=None) @@ -502,19 +501,19 @@ However, for reading convenience, most of the examples show sorted sequences. :func:`pvariance` function as the *mu* parameter to get the variance of a sample. -.. function:: quantiles(dist, *, n=4, method='exclusive') +.. function:: quantiles(data, *, n=4, method='exclusive') - Divide *dist* into *n* continuous intervals with equal probability. + Divide *data* into *n* continuous intervals with equal probability. Returns a list of ``n - 1`` cut points separating the intervals. Set *n* to 4 for quartiles (the default). Set *n* to 10 for deciles. Set *n* to 100 for percentiles which gives the 99 cuts points that separate - *dist* in to 100 equal sized groups. Raises :exc:`StatisticsError` if *n* + *data* in to 100 equal sized groups. Raises :exc:`StatisticsError` if *n* is not least 1. - The *dist* can be any iterable containing sample data or it can be an + The *data* can be any iterable containing sample data or it can be an instance of a class that defines an :meth:`~inv_cdf` method. For meaningful - results, the number of data points in *dist* should be larger than *n*. + results, the number of data points in *data* should be larger than *n*. Raises :exc:`StatisticsError` if there are not at least two data points. For sample data, the cut points are linearly interpolated from the @@ -523,7 +522,7 @@ However, for reading convenience, most of the examples show sorted sequences. cut-point will evaluate to ``104``. The *method* for computing quantiles can be varied depending on - whether the data in *dist* includes or excludes the lowest and + whether the data in *data* includes or excludes the lowest and highest possible values from the population. The default *method* is "exclusive" and is used for data sampled from @@ -535,14 +534,14 @@ However, for reading convenience, most of the examples show sorted sequences. Setting the *method* to "inclusive" is used for describing population data or for samples that are known to include the most extreme values - from the population. The minimum value in *dist* is treated as the 0th + from the population. The minimum value in *data* is treated as the 0th percentile and the maximum value is treated as the 100th percentile. The portion of the population falling below the *i-th* of *m* sorted data points is computed as ``(i - 1) / (m - 1)``. Given 11 sample values, the method sorts them and assigns the following percentiles: 0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 100%. - If *dist* is an instance of a class that defines an + If *data* is an instance of a class that defines an :meth:`~inv_cdf` method, setting *method* has no effect. .. doctest:: @@ -580,7 +579,7 @@ A single exception is defined: :class:`NormalDist` is a tool for creating and manipulating normal distributions of a `random variable <http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm>`_. It is a -composite class that treats the mean and standard deviation of data +class that treats the mean and standard deviation of data measurements as a single entity. Normal distributions arise from the `Central Limit Theorem @@ -616,13 +615,14 @@ of applications in statistics. .. classmethod:: NormalDist.from_samples(data) - Makes a normal distribution instance computed from sample data. The - *data* can be any :term:`iterable` and should consist of values that - can be converted to type :class:`float`. + Makes a normal distribution instance with *mu* and *sigma* parameters + estimated from the *data* using :func:`fmean` and :func:`stdev`. - If *data* does not contain at least two elements, raises - :exc:`StatisticsError` because it takes at least one point to estimate - a central value and at least two points to estimate dispersion. + The *data* can be any :term:`iterable` and should consist of values + that can be converted to type :class:`float`. If *data* does not + contain at least two elements, raises :exc:`StatisticsError` because it + takes at least one point to estimate a central value and at least two + points to estimate dispersion. .. method:: NormalDist.samples(n, *, seed=None) @@ -636,10 +636,10 @@ of applications in statistics. .. method:: NormalDist.pdf(x) Using a `probability density function (pdf) - <https://en.wikipedia.org/wiki/Probability_density_function>`_, - compute the relative likelihood that a random variable *X* will be near - the given value *x*. Mathematically, it is the ratio ``P(x <= X < - x+dx) / dx``. + <https://en.wikipedia.org/wiki/Probability_density_function>`_, compute + the relative likelihood that a random variable *X* will be near the + given value *x*. Mathematically, it is the limit of the ratio ``P(x <= + X < x+dx) / dx`` as *dx* approaches zero. The relative likelihood is computed as the probability of a sample occurring in a narrow range divided by the width of the range (hence @@ -667,8 +667,10 @@ of applications in statistics. .. method:: NormalDist.overlap(other) - Returns a value between 0.0 and 1.0 giving the overlapping area for - the two probability density functions. + Measures the agreement between two normal probability distributions. + Returns a value between 0.0 and 1.0 giving `the overlapping area for + the two probability density functions + <https://www.rasch.org/rmt/rmt101r.htm>`_. Instances of :class:`NormalDist` support addition, subtraction, multiplication and division by a constant. These operations @@ -740,12 +742,11 @@ Carlo simulation <https://en.wikipedia.org/wiki/Monte_Carlo_method>`_: ... return (3*x + 7*x*y - 5*y) / (11 * z) ... >>> n = 100_000 - >>> seed = 86753099035768 - >>> X = NormalDist(10, 2.5).samples(n, seed=seed) - >>> Y = NormalDist(15, 1.75).samples(n, seed=seed) - >>> Z = NormalDist(50, 1.25).samples(n, seed=seed) - >>> NormalDist.from_samples(map(model, X, Y, Z)) # doctest: +SKIP - NormalDist(mu=1.8661894803304777, sigma=0.65238717376862) + >>> X = NormalDist(10, 2.5).samples(n, seed=3652260728) + >>> Y = NormalDist(15, 1.75).samples(n, seed=4582495471) + >>> Z = NormalDist(50, 1.25).samples(n, seed=6582483453) + >>> quantiles(map(model, X, Y, Z)) # doctest: +SKIP + [1.4591308524824727, 1.8035946855390597, 2.175091447274739] Normal distributions commonly arise in machine learning problems. diff --git a/Lib/statistics.py b/Lib/statistics.py index c7d6568..4b17266 100644 --- a/Lib/statistics.py +++ b/Lib/statistics.py @@ -322,7 +322,6 @@ def fmean(data): """Convert data to floats and compute the arithmetic mean. This runs faster than the mean() function and it always returns a float. - The result is highly accurate but not as perfect as mean(). If the input dataset is empty, it raises a StatisticsError. >>> fmean([3.5, 4.0, 5.25]) @@ -538,15 +537,16 @@ def mode(data): ``mode`` assumes discrete data, and returns a single value. This is the standard treatment of the mode as commonly taught in schools: - >>> mode([1, 1, 2, 3, 3, 3, 3, 4]) - 3 + >>> mode([1, 1, 2, 3, 3, 3, 3, 4]) + 3 This also works with nominal (non-numeric) data: - >>> mode(["red", "blue", "blue", "red", "green", "red", "red"]) - 'red' + >>> mode(["red", "blue", "blue", "red", "green", "red", "red"]) + 'red' - If there are multiple modes, return the first one encountered. + If there are multiple modes with same frequency, return the first one + encountered: >>> mode(['red', 'red', 'green', 'blue', 'blue']) 'red' @@ -615,28 +615,28 @@ def multimode(data): # position is that fewer options make for easier choices and that # external packages can be used for anything more advanced. -def quantiles(dist, /, *, n=4, method='exclusive'): - """Divide *dist* into *n* continuous intervals with equal probability. +def quantiles(data, /, *, n=4, method='exclusive'): + """Divide *data* into *n* continuous intervals with equal probability. Returns a list of (n - 1) cut points separating the intervals. Set *n* to 4 for quartiles (the default). Set *n* to 10 for deciles. Set *n* to 100 for percentiles which gives the 99 cuts points that - separate *dist* in to 100 equal sized groups. + separate *data* in to 100 equal sized groups. - The *dist* can be any iterable containing sample data or it can be + The *data* can be any iterable containing sample data or it can be an instance of a class that defines an inv_cdf() method. For sample data, the cut points are linearly interpolated between data points. - If *method* is set to *inclusive*, *dist* is treated as population + If *method* is set to *inclusive*, *data* is treated as population data. The minimum value is treated as the 0th percentile and the maximum value is treated as the 100th percentile. """ if n < 1: raise StatisticsError('n must be at least 1') - if hasattr(dist, 'inv_cdf'): - return [dist.inv_cdf(i / n) for i in range(1, n)] - data = sorted(dist) + if hasattr(data, 'inv_cdf'): + return [data.inv_cdf(i / n) for i in range(1, n)] + data = sorted(data) ld = len(data) if ld < 2: raise StatisticsError('must have at least two data points') @@ -745,7 +745,7 @@ def variance(data, xbar=None): def pvariance(data, mu=None): """Return the population variance of ``data``. - data should be an iterable of Real-valued numbers, with at least one + data should be a sequence or iterator of Real-valued numbers, with at least one value. The optional argument mu, if given, should be the mean of the data. If it is missing or None, the mean is automatically calculated. @@ -766,10 +766,6 @@ def pvariance(data, mu=None): >>> pvariance(data, mu) 1.25 - This function does not check that ``mu`` is actually the mean of ``data``. - Giving arbitrary values for ``mu`` may lead to invalid or impossible - results. - Decimals and Fractions are supported: >>> from decimal import Decimal as D @@ -913,8 +909,8 @@ class NormalDist: "NormalDist where mu is the mean and sigma is the standard deviation." if sigma < 0.0: raise StatisticsError('sigma must be non-negative') - self._mu = mu - self._sigma = sigma + self._mu = float(mu) + self._sigma = float(sigma) @classmethod def from_samples(cls, data): @@ -416,6 +416,7 @@ Dima Dorfman Yves Dorfsman Michael Dorman Steve Dower +Allen Downey Cesar Douady Dean Draayer Fred L. Drake, Jr. |