9.7. "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)".


9.7.1. 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）。                    |
+-------------------------+-----------------------------------------------+
| "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()"                | 離散資料中的眾數（出現次數最多次）。          |
+-------------------------+-----------------------------------------------+


9.7.2. 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                              |
+-------------------------+-----------------------------------------------+


9.7.3. 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.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).

     * Calculating the median.

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

statistics.mode(data)

   Return the most common data point from discrete or nominal *data*.
   The mode (when it exists) is the most typical value, and is a
   robust measure of central location.

   If *data* is empty, or if there is not exactly one most common
   value, "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'

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.


9.7.4. Exceptions
=================

A single exception is defined:

exception statistics.StatisticsError

   Subclass of "ValueError" for statistics-related exceptions.
