"statistics" --- 數學統計函式
*****************************

3.4 版新加入.

**Source code:** Lib/statistics.py

======================================================================

這個模組提供計算數值 ("Real"-valued) 資料的數學統計函式。

備註: Unless explicitly noted otherwise, these functions support
  "int", "float", "decimal.Decimal" and "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 "map()" to ensure a consistent result,
  e.g. "map(float, input_data)".


平均值與中央位置量數
====================

這些函式計算來自一個母體或樣本的平均值或代表值。

+-------------------------+-----------------------------------------------------------------+
| "mean()"                | 數據的算術平均（平均值）。                                      |
+-------------------------+-----------------------------------------------------------------+
| "fmean()"               | Fast, floating point arithmetic mean.                           |
+-------------------------+-----------------------------------------------------------------+
| "geometric_mean()"      | Geometric mean of data.                                         |
+-------------------------+-----------------------------------------------------------------+
| "harmonic_mean()"       | Harmonic mean of data.                                          |
+-------------------------+-----------------------------------------------------------------+
| "median()"              | 數據的中位數（中間值）。                                        |
+-------------------------+-----------------------------------------------------------------+
| "median_low()"          | 數據中較小的中位數。                                            |
+-------------------------+-----------------------------------------------------------------+
| "median_high()"         | 數據中較大的中位數。                                            |
+-------------------------+-----------------------------------------------------------------+
| "median_grouped()"      | 分組數據的中位數或50%處。                                       |
+-------------------------+-----------------------------------------------------------------+
| "mode()"                | Single mode (most common value) of discrete or nominal data.    |
+-------------------------+-----------------------------------------------------------------+
| "multimode()"           | List of modes (most common values) of discrete or nomimal data. |
+-------------------------+-----------------------------------------------------------------+
| "quantiles()"           | Divide data into intervals with equal probability.              |
+-------------------------+-----------------------------------------------------------------+


離度 (spread) 的測量
====================

這些函式計算母體或樣本偏離平均值的程度。

+-------------------------+-----------------------------------------------+
| "pstdev()"              | 數據的母體標準差                              |
+-------------------------+-----------------------------------------------+
| "pvariance()"           | 數據的母體變異數                              |
+-------------------------+-----------------------------------------------+
| "stdev()"               | 數據的樣本標準差                              |
+-------------------------+-----------------------------------------------+
| "variance()"            | 數據的樣本變異數                              |
+-------------------------+-----------------------------------------------+


函式細節
========

Note: The functions do not require the data given to them to be
sorted. However, for reading convenience, most of the examples show
sorted sequences.

statistics.mean(data)

   Return the sample arithmetic mean of *data* which can be a sequence
   or iterator.

   The arithmetic mean is the sum of the data divided by the number of
   data points.  It is commonly called "the average", although it is
   only one of many different mathematical averages.  It is a measure
   of the central location of the data.

   If *data* is empty, "StatisticsError" will be raised.

   Some examples of use:

      >>> mean([1, 2, 3, 4, 4])
      2.8
      >>> mean([-1.0, 2.5, 3.25, 5.75])
      2.625

      >>> from fractions import Fraction as F
      >>> mean([F(3, 7), F(1, 21), F(5, 3), F(1, 3)])
      Fraction(13, 21)

      >>> from decimal import Decimal as D
      >>> mean([D("0.5"), D("0.75"), D("0.625"), D("0.375")])
      Decimal('0.5625')

   備註: 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 "median()" and "mode()".  (In this case, "efficient" refers
     to statistical efficiency rather than computational
     efficiency.)The sample mean gives an unbiased estimate of the
     true population mean, which means that, taken on average over all
     the possible samples, "mean(sample)" converges on the true mean
     of the entire population.  If *data* represents the entire
     population rather than a sample, then "mean(data)" is equivalent
     to calculating the true population mean μ.

statistics.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, raises a
   "StatisticsError".

      >>> fmean([3.5, 4.0, 5.25])
      4.25

   3.8 版新加入.

statistics.geometric_mean(data)

   Convert *data* to floats and compute the geometric mean.

   Raises a "StatisticsError" if the input dataset is empty, if it
   contains a zero, or if it contains a negative value.

   No special efforts are made to achieve exact results. (However,
   this may change in the future.)

      >>> round(geometric_mean([54, 24, 36]), 9)
      36.0

   3.8 版新加入.

statistics.harmonic_mean(data)

   Return the harmonic mean of *data*, a sequence or iterator of real-
   valued numbers.

   The harmonic mean, sometimes called the subcontrary mean, is the
   reciprocal of the arithmetic "mean()" of the reciprocals of the
   data. For example, the harmonic mean of three values *a*, *b* and
   *c* will be equivalent to "3/(1/a + 1/b + 1/c)".

   The harmonic mean is a type of average, a measure of the central
   location of the data.  It is often appropriate when averaging
   quantities which are rates or ratios, for example speeds. For
   example:

   Suppose an investor purchases an equal value of shares in each of
   three companies, with P/E (price/earning) ratios of 2.5, 3 and 10.
   What is the average P/E ratio for the investor's portfolio?

      >>> harmonic_mean([2.5, 3, 10])  # For an equal investment portfolio.
      3.6

   Using the arithmetic mean would give an average of about 5.167,
   which is too high.

   "StatisticsError" is raised if *data* is empty, or any element is
   less than zero.

   3.6 版新加入.

statistics.median(data)

   Return the median (middle value) of numeric data, using the common
   "mean of middle two" method.  If *data* is empty, "StatisticsError"
   is raised. *data* can be a sequence or iterator.

   The median is a robust measure of central location, and is less
   affected by the presence of outliers in your data.  When the number
   of data points is odd, the middle data point is returned:

      >>> median([1, 3, 5])
      3

   When the number of data points is even, the median is interpolated
   by taking the average of the two middle values:

      >>> median([1, 3, 5, 7])
      4.0

   This is suited for when your data is discrete, and you don't mind
   that the median may not be an actual data point.

   If your data is ordinal (supports order operations) but not numeric
   (doesn't support addition), you should use "median_low()" or
   "median_high()" instead.

   也參考: "median_low()", "median_high()", "median_grouped()"

statistics.median_low(data)

   Return the low median of numeric data.  If *data* is empty,
   "StatisticsError" is raised.  *data* can be a sequence or iterator.

   The low median is always a member of the data set.  When the number
   of data points is odd, the middle value is returned.  When it is
   even, the smaller of the two middle values is returned.

      >>> median_low([1, 3, 5])
      3
      >>> median_low([1, 3, 5, 7])
      3

   Use the low median when your data are discrete and you prefer the
   median to be an actual data point rather than interpolated.

statistics.median_high(data)

   Return the high median of data.  If *data* is empty,
   "StatisticsError" is raised.  *data* can be a sequence or iterator.

   The high median is always a member of the data set.  When the
   number of data points is odd, the middle value is returned.  When
   it is even, the larger of the two middle values is returned.

      >>> median_high([1, 3, 5])
      3
      >>> median_high([1, 3, 5, 7])
      5

   Use the high median when your data are discrete and you prefer the
   median to be an actual data point rather than interpolated.

statistics.median_grouped(data, interval=1)

   Return the median of grouped continuous data, calculated as the
   50th percentile, using interpolation.  If *data* is empty,
   "StatisticsError" is raised.  *data* can be a sequence or iterator.

      >>> median_grouped([52, 52, 53, 54])
      52.5

   In the following example, the data are rounded, so that each value
   represents the midpoint of data classes, e.g. 1 is the midpoint of
   the class 0.5--1.5, 2 is the midpoint of 1.5--2.5, 3 is the
   midpoint of 2.5--3.5, etc.  With the data given, the middle value
   falls somewhere in the class 3.5--4.5, and interpolation is used to
   estimate it:

      >>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5])
      3.7

   Optional argument *interval* represents the class interval, and
   defaults to 1.  Changing the class interval naturally will change
   the interpolation:

      >>> median_grouped([1, 3, 3, 5, 7], interval=1)
      3.25
      >>> median_grouped([1, 3, 3, 5, 7], interval=2)
      3.5

   This function does not check whether the data points are at least
   *interval* apart.

   **CPython implementation detail:** Under some circumstances,
   "median_grouped()" may coerce data points to floats.  This
   behaviour is likely to change in the future.

   也參考:

     * "Statistics for the Behavioral Sciences", Frederick J
       Gravetter and Larry B Wallnau (8th Edition).

     * The SSMEDIAN function in the Gnome Gnumeric spreadsheet,
       including this discussion.

statistics.mode(data)

   Return the single most common data point from discrete or nominal
   *data*. 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, "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:

      >>> 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:

      >>> mode(["red", "blue", "blue", "red", "green", "red", "red"])
      'red'

   3.8 版更變: Now handles multimodal datasets by returning the first
   mode encountered. Formerly, it raised "StatisticsError" when more
   than one mode was found.

statistics.multimode(data)

   Return a list of the most frequently occurring values in the order
   they were first encountered in the *data*.  Will return more than
   one result if there are multiple modes or an empty list if the
   *data* is empty:

      >>> multimode('aabbbbccddddeeffffgg')
      ['b', 'd', 'f']
      >>> multimode('')
      []

   3.8 版新加入.

statistics.pstdev(data, mu=None)

   Return the population standard deviation (the square root of the
   population variance).  See "pvariance()" for arguments and other
   details.

      >>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
      0.986893273527251

statistics.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.

   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.

   Use this function to calculate the variance from the entire
   population.  To estimate the variance from a sample, the
   "variance()" function is usually a better choice.

   Raises "StatisticsError" if *data* is empty.

   Examples:

      >>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25]
      >>> pvariance(data)
      1.25

   If you have already calculated the mean of your data, you can pass
   it as the optional second argument *mu* to avoid recalculation:

      >>> mu = mean(data)
      >>> 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:

      >>> from decimal import Decimal as D
      >>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
      Decimal('24.815')

      >>> from fractions import Fraction as F
      >>> pvariance([F(1, 4), F(5, 4), F(1, 2)])
      Fraction(13, 72)

   備註: When called with the entire population, this gives the
     population variance 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.

statistics.stdev(data, xbar=None)

   Return the sample standard deviation (the square root of the sample
   variance).  See "variance()" for arguments and other details.

      >>> stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
      1.0810874155219827

statistics.variance(data, xbar=None)

   Return the sample variance of *data*, an iterable of at least two
   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 *xbar* is given, it should be the
   mean of *data*.  If it is missing or "None" (the default), the mean
   is automatically calculated.

   Use this function when your data is a sample from a population. To
   calculate the variance from the entire population, see
   "pvariance()".

   Raises "StatisticsError" if *data* has fewer than two values.

   Examples:

      >>> data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5]
      >>> variance(data)
      1.3720238095238095

   If you have already calculated the mean of your data, you can pass
   it as the optional second argument *xbar* to avoid recalculation:

      >>> m = mean(data)
      >>> variance(data, m)
      1.3720238095238095

   This function does not attempt to verify that you have passed the
   actual mean as *xbar*.  Using arbitrary values for *xbar* can lead
   to invalid or impossible results.

   Decimal and Fraction values are supported:

      >>> from decimal import Decimal as D
      >>> variance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
      Decimal('31.01875')

      >>> from fractions import Fraction as F
      >>> variance([F(1, 6), F(1, 2), F(5, 3)])
      Fraction(67, 108)

   備註: This is the sample variance s² with Bessel's correction,
     also known as variance with N-1 degrees of freedom.  Provided
     that the data points are representative (e.g. independent and
     identically distributed), the result should be an unbiased
     estimate of the true population variance.If you somehow know the
     actual population mean μ you should pass it to the "pvariance()"
     function as the *mu* parameter to get the variance of a sample.

statistics.quantiles(dist, *, n=4, method='exclusive')

   Divide *dist* 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
   "StatisticsError" if *n* is not least 1.

   The *dist* can be any iterable containing sample data or it can be
   an instance of a class that defines an "inv_cdf()" method.  For
   meaningful results, the number of data points in *dist* should be
   larger than *n*. Raises "StatisticsError" if there are not at least
   two data points.

   For sample data, the cut points are linearly interpolated from the
   two nearest data points.  For example, if a cut point falls one-
   third of the distance between two sample values, "100" and "112",
   the 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
   highest possible values from the population.

   The default *method* is "exclusive" and is used for data sampled
   from a population that can have more extreme values than found in
   the samples.  The portion of the population falling below the
   *i-th* of *m* sorted data points is computed as "i / (m + 1)".
   Given nine sample values, the method sorts them and assigns the
   following percentiles: 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%.

   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 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 "inv_cdf()"
   method, setting *method* has no effect.

      # Decile cut points for empirically sampled data
      >>> data = [105, 129, 87, 86, 111, 111, 89, 81, 108, 92, 110,
      ...         100, 75, 105, 103, 109, 76, 119, 99, 91, 103, 129,
      ...         106, 101, 84, 111, 74, 87, 86, 103, 103, 106, 86,
      ...         111, 75, 87, 102, 121, 111, 88, 89, 101, 106, 95,
      ...         103, 107, 101, 81, 109, 104]
      >>> [round(q, 1) for q in quantiles(data, n=10)]
      [81.0, 86.2, 89.0, 99.4, 102.5, 103.6, 106.0, 109.8, 111.0]

      >>> # Quartile cut points for the standard normal distibution
      >>> Z = NormalDist()
      >>> [round(q, 4) for q in quantiles(Z, n=4)]
      [-0.6745, 0.0, 0.6745]

   3.8 版新加入.


Exceptions
==========

A single exception is defined:

exception statistics.StatisticsError

   Subclass of "ValueError" for statistics-related exceptions.


"NormalDist" objects
====================

"NormalDist" is a tool for creating and manipulating normal
distributions of a random variable.  It is a composite class that
treats the mean and standard deviation of data measurements as a
single entity.

Normal distributions arise from the Central Limit Theorem and have a
wide range of applications in statistics.

class statistics.NormalDist(mu=0.0, sigma=1.0)

   Returns a new *NormalDist* object where *mu* represents the
   arithmetic mean and *sigma* represents the standard deviation.

   If *sigma* is negative, raises "StatisticsError".

   mean

      A read-only property for the arithmetic mean of a normal
      distribution.

   stdev

      A read-only property for the standard deviation of a normal
      distribution.

   variance

      A read-only property for the variance of a normal distribution.
      Equal to the square of the standard deviation.

   classmethod from_samples(data)

      Makes a normal distribution instance computed from sample data.
      The *data* can be any *iterable* and should consist of values
      that can be converted to type "float".

      If *data* does not contain at least two elements, raises
      "StatisticsError" because it takes at least one point to
      estimate a central value and at least two points to estimate
      dispersion.

   samples(n, *, seed=None)

      Generates *n* random samples for a given mean and standard
      deviation. Returns a "list" of "float" values.

      If *seed* is given, creates a new instance of the underlying
      random number generator.  This is useful for creating
      reproducible results, even in a multi-threading context.

   pdf(x)

      Using a probability density function (pdf), 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".

      The relative likelihood is computed as the probability of a
      sample occurring in a narrow range divided by the width of the
      range (hence the word "density").  Since the likelihood is
      relative to other points, its value can be greater than *1.0*.

   cdf(x)

      Using a cumulative distribution function (cdf), compute the
      probability that a random variable *X* will be less than or
      equal to *x*.  Mathematically, it is written "P(X <= x)".

   inv_cdf(p)

      Compute the inverse cumulative distribution function, also known
      as the quantile function or the percent-point function.
      Mathematically, it is written "x : P(X <= x) = p".

      Finds the value *x* of the random variable *X* such that the
      probability of the variable being less than or equal to that
      value equals the given probability *p*.

   overlap(other)

      Returns a value between 0.0 and 1.0 giving the overlapping area
      for the two probability density functions.

   Instances of "NormalDist" support addition, subtraction,
   multiplication and division by a constant.  These operations are
   used for translation and scaling.  For example:

      >>> temperature_february = NormalDist(5, 2.5)             # Celsius
      >>> temperature_february * (9/5) + 32                     # Fahrenheit
      NormalDist(mu=41.0, sigma=4.5)

   Dividing a constant by an instance of "NormalDist" is not supported
   because the result wouldn't be normally distributed.

   Since normal distributions arise from additive effects of
   independent variables, it is possible to add and subtract two
   independent normally distributed random variables represented as
   instances of "NormalDist".  For example:

      >>> birth_weights = NormalDist.from_samples([2.5, 3.1, 2.1, 2.4, 2.7, 3.5])
      >>> drug_effects = NormalDist(0.4, 0.15)
      >>> combined = birth_weights + drug_effects
      >>> round(combined.mean, 1)
      3.1
      >>> round(combined.stdev, 1)
      0.5

   3.8 版新加入.


"NormalDist" Examples and Recipes
---------------------------------

"NormalDist" readily solves classic probability problems.

For example, given historical data for SAT exams showing that scores
are normally distributed with a mean of 1060 and a standard deviation
of 192, determine the percentage of students with test scores between
1100 and 1200, after rounding to the nearest whole number:

   >>> sat = NormalDist(1060, 195)
   >>> fraction = sat.cdf(1200 + 0.5) - sat.cdf(1100 - 0.5)
   >>> round(fraction * 100.0, 1)
   18.4

Find the quartiles and deciles for the SAT scores:

   >>> [round(sat.inv_cdf(p)) for p in (0.25, 0.50, 0.75)]
   [928, 1060, 1192]
   >>> [round(sat.inv_cdf(p / 10)) for p in range(1, 10)]
   [810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]

To estimate the distribution for a model than isn't easy to solve
analytically, "NormalDist" can generate input samples for a Monte
Carlo simulation:

   >>> def model(x, y, z):
   ...     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))     
   NormalDist(mu=1.8661894803304777, sigma=0.65238717376862)

Normal distributions commonly arise in machine learning problems.

Wikipedia has a nice example of a Naive Bayesian Classifier. The
challenge is to predict a person's gender from measurements of
normally distributed features including height, weight, and foot size.

We're given a training dataset with measurements for eight people.
The measurements are assumed to be normally distributed, so we
summarize the data with "NormalDist":

   >>> height_male = NormalDist.from_samples([6, 5.92, 5.58, 5.92])
   >>> height_female = NormalDist.from_samples([5, 5.5, 5.42, 5.75])
   >>> weight_male = NormalDist.from_samples([180, 190, 170, 165])
   >>> weight_female = NormalDist.from_samples([100, 150, 130, 150])
   >>> foot_size_male = NormalDist.from_samples([12, 11, 12, 10])
   >>> foot_size_female = NormalDist.from_samples([6, 8, 7, 9])

Next, we encounter a new person whose feature measurements are known
but whose gender is unknown:

   >>> ht = 6.0        # height
   >>> wt = 130        # weight
   >>> fs = 8          # foot size

Starting with a 50% prior probability of being male or female, we
compute the posterior as the prior times the product of likelihoods
for the feature measurements given the gender:

   >>> prior_male = 0.5
   >>> prior_female = 0.5
   >>> posterior_male = (prior_male * height_male.pdf(ht) *
   ...                   weight_male.pdf(wt) * foot_size_male.pdf(fs))

   >>> posterior_female = (prior_female * height_female.pdf(ht) *
   ...                     weight_female.pdf(wt) * foot_size_female.pdf(fs))

The final prediction goes to the largest posterior. This is known as
the maximum a posteriori or MAP:

   >>> 'male' if posterior_male > posterior_female else 'female'
   'female'
