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

3.4 版新加入.

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

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

该模块提供了用于计算数字 ("Real"-valued) 数据的数理统计量的函数。

此模块并不是诸如 NumPy ， SciPy 等第三方库或者诸如 Minitab ， SAS ，
Matlab 等针对专业统计学家的专有全功能统计软件包的竟品。此模块针对图形
和科学计算器的水平。

除非明确注释，这些函数支持 "int" ， "float" ， "Decimal" 和 "Fraction"
。当前不支持同其他类型（是否在数字塔中）的行为。混合类型的集合也是未定
义的，并且依赖于实现。如果你输入的数据由混合类型组成，你应该能够使用
"map()" 来确保一个一致的结果，比如： "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()"               | 快速的，浮点算数平均数。                                        |
+-------------------------+-----------------------------------------------------------------+
| "geometric_mean()"      | 数据的几何平均数                                                |
+-------------------------+-----------------------------------------------------------------+
| "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()"                | 离散的或标称的数据的单模（最常见的值）。                        |
+-------------------------+-----------------------------------------------------------------+
| "multimode()"           | 离散的或标称的数据的模式列表（最常见的值）。                    |
+-------------------------+-----------------------------------------------------------------+
| "quantiles()"           | 将数据以相等的概率分为多个间隔。                                |
+-------------------------+-----------------------------------------------------------------+


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 iterable.

   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: 均值非常受异常值的影响并且这不是中心位置的可靠估计：均值不
     一定是 数据点的典型示例。如需要更可靠的的估计，请参考 "median()"
     和 "mode()" 。样本均值给出了一个无偏向的真实总体均值的估计，因此
     当平 均抽取所有可能的样本， "mean(sample)" 收敛于整个总体的真实均
     值。 如果 *data* 代表整个总体而不是样本，那么 "mean(data)" 等同于
     计算 真实整体均值 μ 。

statistics.fmean(data)

   将浮点数转换成 *data* 并且计算算术平均数。

   This runs faster than the "mean()" function and it always returns a
   "float".  The *data* may be a sequence or iterable.  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)

   将浮点数转换成 *data* 并且计算几何平均数。

   几何平均值使用值的乘积表示 *数据* 的中心趋势或典型值（与使用它们的
   总和的算术平均值相反）。

   Raises a "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 iterable.

   无需做出特殊努力即可获得准确的结果。（但是，将来或许会修改。）

      >>> round(geometric_mean([54, 24, 36]), 1)
      36.0

   3.8 版新加入.

statistics.harmonic_mean(data)

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

   调和均值,也叫次相反均值，所有数据的倒数的算术平均数 "mean()" 的倒数
   。比如说，数据 *a* ， *b* ， *c* 的调和均值等于 "3/(1/a + 1/b +
   1/c)" 。如果其中一个值为零，结果为零。

   调和均值是一种均值类型，是数据中心位置的度量。它通常适合于求比率和
   比例的平均值，比如速率。

   假设一辆车在 40 km/hr 的速度下行驶了 10 km ，然后又以 60 km/hr 的速
   度行驶了 10 km 。车辆的平均速率是多少？

      >>> harmonic_mean([40, 60])
      48.0

   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

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

   当前算法在输入中遇到零时会提前退出。这意味着不会测试后续输入的有效
   性。（此行为将来可能会更改。）

   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 iterable.

   The median is a robust measure of central location and is less
   affected by the presence of outliers.  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 the data is ordinal (supports order operations) but not numeric
   (doesn't support addition), consider using "median_low()" or
   "median_high()" instead.

statistics.median_low(data)

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

   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 iterable.

   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 iterable.

      >>> 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 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,
   "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 in this package
   that 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 sequence or
   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 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
   "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

   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 a random sample of the population,
     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(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 *data* into 100 equal sized groups.  Raises
   "StatisticsError" if *n* is not least 1.

   The *data* can be any iterable containing sample data.  For
   meaningful results, the number of data points in *data* should be
   larger than *n*. Raises "StatisticsError" if there are not at least
   two data points.

   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* 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 *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%.

      # 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]

   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 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.

   median

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

   mode

      A read-only property for the mode 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 with *mu* and *sigma*
      parameters estimated from the *data* using "fmean()" and
      "stdev()".

      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 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 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)

      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.

   quantiles(n=4)

      Divide the normal distribution 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 the normal distribution into 100 equal
      sized groups.

   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:

   >>> list(map(round, sat.quantiles()))
   [928, 1060, 1192]
   >>> list(map(round, sat.quantiles(n=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
   >>> 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))       
   [1.4591308524824727, 1.8035946855390597, 2.175091447274739]

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'
