"statistics" --- Mathematical statistics functions
**************************************************

バージョン 3.4 で追加.

**ソースコード:** Lib/statistics.py

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

このモジュールは、数値 ("Real" 型) データを数学的に統計計算するための
関数を提供します。

このモジュールは、 NumPy, SciPy のような third-party ライブラリや、
Minitab、SAS、Matlab のようなプロ統計家向けのフル装備なプロプライエタ
リ統計パッケージと競合することを意図していません。グラフ関数電卓のレベ
ルを対象としています。

特に明記しない限り、これらの関数は "int", "float", "Decimal" そして
"Fraction" をサポートします。他の型 (算術型及びそれ以外) は現在サポー
トされていません。型が混ざったコレクションも未定義で実装依存です。入力
データが複数の型からなる場合、"map()" を使用すると正しい結果が得られる
でしょう。 例: "map(float, input_data)"。

あるデータセットでは欠損値を表すために "NaN" (not a number) を使います
。 NaNは通常と異なる比較セマンティクスを持つため、ソートやカウントを行
う統計関数では、驚きや未定義の振る舞いを引き起こします。  影響を受ける
関数は "median()", "median_low()", "median_high()",
"median_grouped()", "mode()", "multimode()", そして "quantiles()" です
。  これらの関数を呼ぶ前に、 "NaN" 値を取り除く必要があります:

   >>> from statistics import median
   >>> from math import isnan
   >>> from itertools import filterfalse

   >>> data = [20.7, float('NaN'),19.2, 18.3, float('NaN'), 14.4]
   >>> sorted(data)  # This has surprising behavior
   [20.7, nan, 14.4, 18.3, 19.2, nan]
   >>> median(data)  # This result is unexpected
   16.35

   >>> sum(map(isnan, data))    # Number of missing values
   2
   >>> clean = list(filterfalse(isnan, data))  # Strip NaN values
   >>> clean
   [20.7, 19.2, 18.3, 14.4]
   >>> sorted(clean)  # Sorting now works as expected
   [14.4, 18.3, 19.2, 20.7]
   >>> median(clean)       # This result is now well defined
   18.75


平均及び中心位置の測度
======================

これらの関数は母集団または標本の平均値や標準値を計算します。

+-------------------------+-----------------------------------------------------------------+
| "mean()"                | データの算術平均 (いわゆる「平均」) 。                          |
+-------------------------+-----------------------------------------------------------------+
| "fmean()"               | Fast, floating point arithmetic mean, with optional weighting.  |
+-------------------------+-----------------------------------------------------------------+
| "geometric_mean()"      | データの幾何平均。                                              |
+-------------------------+-----------------------------------------------------------------+
| "harmonic_mean()"       | データの調和平均。                                              |
+-------------------------+-----------------------------------------------------------------+
| "median()"              | データのメジアン(中央値)。                                      |
+-------------------------+-----------------------------------------------------------------+
| "median_low()"          | データの low median。                                           |
+-------------------------+-----------------------------------------------------------------+
| "median_high()"         | データの high median。                                          |
+-------------------------+-----------------------------------------------------------------+
| "median_grouped()"      | Median, or 50th percentile, of grouped data.                    |
+-------------------------+-----------------------------------------------------------------+
| "mode()"                | 離散/名義尺度データの最頻値 (single mode) 。                    |
+-------------------------+-----------------------------------------------------------------+
| "multimode()"           | 離散/名義尺度データの最頻値 (list of modes) 。                  |
+-------------------------+-----------------------------------------------------------------+
| "quantiles()"           | データの等確率での分割。                                        |
+-------------------------+-----------------------------------------------------------------+


分散の測度
==========

これらの関数は母集団または標本が標準値や平均値からどれくらい離れている
かについて計算します。

+-------------------------+-----------------------------------------------+
| "pstdev()"              | データの母標準偏差。                          |
+-------------------------+-----------------------------------------------+
| "pvariance()"           | データの母分散。                              |
+-------------------------+-----------------------------------------------+
| "stdev()"               | データの標本標準偏差。                        |
+-------------------------+-----------------------------------------------+
| "variance()"            | データの標本標準分散。                        |
+-------------------------+-----------------------------------------------+


2入力間の関係の統計
===================

これらの関数は、2つの入力間の関係について統計量を計算します。

+---------------------------+-------------------------------------------------------+
| "covariance()"            | 2変数の標本共分散。                                   |
+---------------------------+-------------------------------------------------------+
| "correlation()"           | Pearson's correlation coefficient for two variables.  |
+---------------------------+-------------------------------------------------------+
| "linear_regression()"     | 単回帰の傾きと切片。                                  |
+---------------------------+-------------------------------------------------------+


関数の詳細
==========

註釈: 関数の引数となるデータをソートしておく必要はありません。例の多く
がソートされているのは見やすさのためです。

statistics.mean(data)

   シーケンス型またはイテラブルになり得る *data* の標本算術平均を返し
   ます。

   算術平均はデータの総和をデータ数で除したものです。単に「平均」と呼
   ばれることも多いですが、数学における平均の一種に過ぎません。データ
   の中心位置の測度の一つです。

   *data* が空の場合 "StatisticsError" を送出します。

   使用例:

      >>> 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')

   注釈:

     mean は 外れ値 (outliers) の影響を強く受け、必ずしもデータ点の典
     型例とはなりません。効率が悪いものの、より堅牢な、 中心的な傾向
     (central tendency) の尺度については、 "median()" を参照してくださ
     い。標本平均は真の母平均の不偏推定量のため、出来る限り多くの標本
     から平均を求めると、"mean(sample)" は真の母平均に収束します (訳注
     : 大数の法則)。*data* が標本ではなく母集団全体の場合、
     "mean(data)" は真の母平均 μ を計算することと等価です。

statistics.fmean(data, weights=None)

   *data* を float に変換し、算術平均を計算します。

   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

   Optional weighting is supported.  For example, a professor assigns
   a grade for a course by weighting quizzes at 20%, homework at 20%,
   a midterm exam at 30%, and a final exam at 30%:

      >>> grades = [85, 92, 83, 91]
      >>> weights = [0.20, 0.20, 0.30, 0.30]
      >>> fmean(grades, weights)
      87.6

   If *weights* is supplied, it must be the same length as the *data*
   or a "ValueError" will be raised.

   バージョン 3.8 で追加.

   バージョン 3.11 で変更: *weights* のサポートが追加されました。

statistics.geometric_mean(data)

   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 "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.

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

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

   バージョン 3.8 で追加.

statistics.harmonic_mean(data, weights=None)

   Return the harmonic mean of *data*, a sequence or iterable of real-
   valued numbers.  If *weights* is omitted or *None*, then equal
   weighting is assumed.

   調和平均（harmonic mean）は、データの逆数の算術平均 "mean()" の逆数
   です。例えば、3つの値 *a*, *b*, *c* の調和平均は``3/(1/a + 1/b +
   1/c)`` になります。いずれかの値が0の場合、結果は0になります。

   調和平均は平均の一種で、データの中心位置の測度です。速度のような比
   (ratios)や率(rates)を平均するときにしばしば適切です。

   Suppose a car travels 10 km at 40 km/hr, then another 10 km at 60
   km/hr. What is the average speed?

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

   Suppose a car travels 40 km/hr for 5 km, and when traffic clears,
   speeds-up to 60 km/hr for the remaining 30 km of the journey. What
   is the average speed?

      >>> harmonic_mean([40, 60], weights=[5, 30])
      56.0

   "StatisticsError" is raised if *data* is empty, any element is less
   than zero, or if the weighted sum isn't positive.

   The current algorithm has an early-out when it encounters a zero in
   the input.  This means that the subsequent inputs are not tested
   for validity.  (This behavior may change in the future.)

   バージョン 3.6 で追加.

   バージョン 3.10 で変更: *weights* のサポートが追加されました。

statistics.median(data)

   一般的な「中央2つの平均をとる」方法を使用して、数値データの中央値（
   中間値）を返します。もし *data* が空の場合、例外 "StatisticsError"
   が送出されます。*data* はシーケンス型またはイテラブルにもなれます。

   中央値は外れ値の影響を受けにくいため、中心位置のロバストな測度です
   。データ数が奇数の場合は中央の値を返します:

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

   データ数が偶数の場合は、中央値は中央に最も近い2値の算術平均で補間さ
   れます:

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

   データが離散的で、中央値がデータ点にない値でも構わない場合に適して
   います。

   もしあなたのデータが（注文操作をサポートする）序数で、（追加操作を
   サポートしない）数値でないならば、代わりに "median_low()" または
   "median_high()" の使用を検討してください。

statistics.median_low(data)

   数値データの小さい方の中央値(low median)を返します。もし *data* が
   空の場合、"StatisticsError" が送出されます。*data* はシーケンス型ま
   たはイテラブルにもなれます。

   low median は必ずデータに含まれています。データ数が奇数の場合は中央
   の値を返し、偶数の場合は中央の2値の小さい方を返します。

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

   データが離散的で、中央値が補間値よりもデータ点にある値の方が良い場
   合に用いてください。

statistics.median_high(data)

   データの大きい方の中央値(high median)を返します。もし *data* が空の
   場合、"StatisticsError" が送出されます。 *data* はシーケンス型やイ
   テラブルにもなれます。

   high median は必ずデータに含まれています。データ数が奇数の場合は中
   央の値を返し、偶数の場合は中央の2値の大きい方を返します。

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

   データが離散的で、中央値が補間値よりもデータ点にある値の方が良い場
   合に用いてください。

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 実装の詳細:** Under some circumstances,
   "median_grouped()" may coerce data points to floats.  This
   behaviour is likely to change in the future.

   参考:

     * "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" は離散データであることを想定していて、1つの値を返します。こ
   れは学校で教わるような最頻値の標準的な取扱いです:

      >>> 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'

   バージョン 3.8 で変更: 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('')
      []

   バージョン 3.8 で追加.

statistics.pstdev(data, mu=None)

   母標準偏差 (母分散の平方根) を返します。引数や詳細は "pvariance()"
   を参照してください。

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

statistics.pvariance(data, mu=None)

   *data* の母分散を返します。*data* は実数の空でないシーケンスまたは
   イテラブルです。分散、すなわち2次の中心化モーメントはデータの散らば
   り具合の測度です。分散が大きいデータはばらつきが大きく、分散が小さ
   いデータは平均値のまわりに固まっています。

   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.

   母集団全体から分散を計算する場合に用いてください。標本から分散を推
   定する場合は "variance()" を使いましょう。

   *data* が空の場合 "StatisticsError" を送出します。

   例:

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

   既にデータの平均値を計算している場合、それを第2引数 *mu* に渡して再
   計算を避けることが出来ます:

      >>> mu = mean(data)
      >>> pvariance(data, mu)
      1.25

   Decimal と Fraction がサポートされています:

      >>> 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)

   注釈:

     母集団全体で呼んだ場合は母分散 σ² を返します。代わりに標本で呼ん
     だ場合は biased variance s²、すなわち自由度 N の分散を返します。
     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)

   標本標準偏差 (標本分散の平方根) を返します。引数や詳細は
   "variance()" を参照してください。

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

statistics.variance(data, xbar=None)

   *data* の標本分散を返します。*data* は少なくとも2つの実数の
   iterable です。分散、すなわち2次の中心化モーメントはデータの散らば
   り具合の測度です。分散が大きいデータはばらつきが大きく、分散が小さ
   いデータは平均値のまわりに固まっています。

   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.

   データが母集団の標本であるときに用いてください。母集団全体から分散
   を計算するには "pvariance()" を参照してください。

   *data* の値が2より少ない場合 "StatisticsError" を送出します。

   例:

      >>> 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

   この関数は引数として渡した *xbar* が実際の平均値かどうかチェックし
   ません。任意の値を *xbar* に渡すと無効な結果やありえない結果が返る
   ことがあります。

   Decimal と Fraction がサポートされています:

      >>> 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)

   注釈:

     Bessel 補正済みの標本分散 s²、すなわち自由度 N-1 の分散です。与え
     られたデータ点が代表的 (たとえば独立で均等に分布) な場合、戻り値
     は母分散の不偏推定量になります。何らかの方法で真の母平均 μ を知っ
     ている場合、それを  "pvariance()"  の引数 *mu* に渡して標本の分散
     を計算することが出来ます。

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]

   バージョン 3.8 で追加.

statistics.covariance(x, y, /)

   Return the sample covariance of two inputs *x* and *y*. Covariance
   is a measure of the joint variability of two inputs.

   Both inputs must be of the same length (no less than two),
   otherwise "StatisticsError" is raised.

   例:

      >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
      >>> y = [1, 2, 3, 1, 2, 3, 1, 2, 3]
      >>> covariance(x, y)
      0.75
      >>> z = [9, 8, 7, 6, 5, 4, 3, 2, 1]
      >>> covariance(x, z)
      -7.5
      >>> covariance(z, x)
      -7.5

   バージョン 3.10 で追加.

statistics.correlation(x, y, /)

   Return the Pearson's correlation coefficient for two inputs.
   Pearson's correlation coefficient *r* takes values between -1 and
   +1. It measures the strength and direction of the linear
   relationship, where +1 means very strong, positive linear
   relationship, -1 very strong, negative linear relationship, and 0
   no linear relationship.

   Both inputs must be of the same length (no less than two), and need
   not to be constant, otherwise "StatisticsError" is raised.

   例:

      >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
      >>> y = [9, 8, 7, 6, 5, 4, 3, 2, 1]
      >>> correlation(x, x)
      1.0
      >>> correlation(x, y)
      -1.0

   バージョン 3.10 で追加.

statistics.linear_regression(x, y, /, *, proportional=False)

   Return the slope and intercept of simple linear regression
   parameters estimated using ordinary least squares. Simple linear
   regression describes the relationship between an independent
   variable *x* and a dependent variable *y* in terms of this linear
   function:

      *y = slope * x + intercept + noise*

   where "slope" and "intercept" are the regression parameters that
   are estimated, and "noise" represents the variability of the data
   that was not explained by the linear regression (it is equal to the
   difference between predicted and actual values of the dependent
   variable).

   Both inputs must be of the same length (no less than two), and the
   independent variable *x* cannot be constant; otherwise a
   "StatisticsError" is raised.

   For example, we can use the release dates of the Monty Python films
   to predict the cumulative number of Monty Python films that would
   have been produced by 2019 assuming that they had kept the pace.

      >>> year = [1971, 1975, 1979, 1982, 1983]
      >>> films_total = [1, 2, 3, 4, 5]
      >>> slope, intercept = linear_regression(year, films_total)
      >>> round(slope * 2019 + intercept)
      16

   If *proportional* is true, the independent variable *x* and the
   dependent variable *y* are assumed to be directly proportional. The
   data is fit to a line passing through the origin. Since the
   *intercept* will always be 0.0, the underlying linear function
   simplifies to:

      *y = slope * x + noise*

   バージョン 3.10 で追加.

   バージョン 3.11 で変更: Added support for *proportional*.


例外
====

例外が1つ定義されています:

exception statistics.StatisticsError

   統計関係の例外。"ValueError" の派生クラス。


"NormalDist" オブジェクト
=========================

"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.

   *sigma* が負の数の場合 "StatisticsError" を送出します。

   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.

   zscore(x)

      Compute the Standard Score describing *x* in terms of the number
      of standard deviations above or below the mean of the normal
      distribution: "(x - mean) / stdev".

      バージョン 3.9 で追加.

   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

   バージョン 3.8 で追加.


"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 195, 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 can be used to approximate Binomial distributions
when the sample size is large and when the probability of a successful
trial is near 50%.

For example, an open source conference has 750 attendees and two rooms
with a 500 person capacity.  There is a talk about Python and another
about Ruby. In previous conferences, 65% of the attendees preferred to
listen to Python talks.  Assuming the population preferences haven't
changed, what is the probability that the Python room will stay within
its capacity limits?

   >>> n = 750             # Sample size
   >>> p = 0.65            # Preference for Python
   >>> q = 1.0 - p         # Preference for Ruby
   >>> k = 500             # Room capacity

   >>> # Approximation using the cumulative normal distribution
   >>> from math import sqrt
   >>> round(NormalDist(mu=n*p, sigma=sqrt(n*p*q)).cdf(k + 0.5), 4)
   0.8402

   >>> # Solution using the cumulative binomial distribution
   >>> from math import comb, fsum
   >>> round(fsum(comb(n, r) * p**r * q**(n-r) for r in range(k+1)), 4)
   0.8402

   >>> # Approximation using a simulation
   >>> from random import seed, choices
   >>> seed(8675309)
   >>> def trial():
   ...     return choices(('Python', 'Ruby'), (p, q), k=n).count('Python')
   >>> mean(trial() <= k for i in range(10_000))
   0.8398

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'
