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

New in version 3.4.

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

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

This module provides functions for calculating mathematical statistics
of numeric ("Real"-valued) data.

The module is not intended to be a competitor to third-party libraries
such as NumPy, SciPy, or proprietary full-featured statistics packages
aimed at professional statisticians such as Minitab, SAS and Matlab.
It is aimed at the level of graphing and scientific calculators.

Unless explicitly noted, these functions support "int", "float",
"Decimal" and "Fraction". Behaviour with other types (whether in the
numeric tower or not) is currently unsupported.  Collections with a
mix of types are also undefined and implementation-dependent.  If your
input data consists of mixed types, you may be able to use "map()" to
ensure a consistent result, for example: "map(float, input_data)".

Some datasets use "NaN" (not a number) values to represent missing
data. Since NaNs have unusual comparison semantics, they cause
surprising or undefined behaviors in the statistics functions that
sort data or that count occurrences.  The functions affected are
"median()", "median_low()", "median_high()", "median_grouped()",
"mode()", "multimode()", and "quantiles()".  The "NaN" values should
be stripped before calling these functions:

   >>> 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


Averages and measures of central location
=========================================

These functions calculate an average or typical value from a
population or sample.

+-------------------------+-----------------------------------------------------------------+
| "mean()"                | Arithmetic mean ("average") of data.                            |
+-------------------------+-----------------------------------------------------------------+
| "fmean()"               | Fast, floating point arithmetic mean, with optional weighting.  |
+-------------------------+-----------------------------------------------------------------+
| "geometric_mean()"      | Geometric mean of data.                                         |
+-------------------------+-----------------------------------------------------------------+
| "harmonic_mean()"       | Harmonic mean of data.                                          |
+-------------------------+-----------------------------------------------------------------+
| "median()"              | Median (middle value) of data.                                  |
+-------------------------+-----------------------------------------------------------------+
| "median_low()"          | Low median of data.                                             |
+-------------------------+-----------------------------------------------------------------+
| "median_high()"         | High median of data.                                            |
+-------------------------+-----------------------------------------------------------------+
| "median_grouped()"      | Median, or 50th percentile, of grouped data.                    |
+-------------------------+-----------------------------------------------------------------+
| "mode()"                | Single mode (most common value) of discrete or nominal data.    |
+-------------------------+-----------------------------------------------------------------+
| "multimode()"           | List of modes (most common values) of discrete or nominal data. |
+-------------------------+-----------------------------------------------------------------+
| "quantiles()"           | Divide data into intervals with equal probability.              |
+-------------------------+-----------------------------------------------------------------+


Measures of spread
==================

These functions calculate a measure of how much the population or
sample tends to deviate from the typical or average values.

+-------------------------+-----------------------------------------------+
| "pstdev()"              | Population standard deviation of data.        |
+-------------------------+-----------------------------------------------+
| "pvariance()"           | Population variance of data.                  |
+-------------------------+-----------------------------------------------+
| "stdev()"               | Sample standard deviation of data.            |
+-------------------------+-----------------------------------------------+
| "variance()"            | Sample variance of data.                      |
+-------------------------+-----------------------------------------------+


Statistics for relations between two inputs
===========================================

These functions calculate statistics regarding relations between two
inputs.

+---------------------------+-------------------------------------------------------+
| "covariance()"            | Sample covariance for two variables.                  |
+---------------------------+-------------------------------------------------------+
| "correlation()"           | Pearson and Spearman's correlation coefficients.      |
+---------------------------+-------------------------------------------------------+
| "linear_regression()"     | Slope and intercept for simple linear regression.     |
+---------------------------+-------------------------------------------------------+


Function details
================

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')

   Note:

     The mean is strongly affected by outliers and is not necessarily
     a typical example of the data points. For a more robust, although
     less efficient, measure of central tendency, see "median()".The
     sample mean gives an unbiased estimate of the true population
     mean, so that when 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, weights=None)

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

   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.

   New in version 3.8.

   Changed in version 3.11: Added support for *weights*.

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

   New in version 3.8.

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.

   The harmonic 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)".  If one of the values is zero, the result will be zero.

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

   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.)

   New in version 3.6.

   Changed in version 3.10: Added support for *weights*.

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.

   See also:

     * "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'

   Changed in version 3.8: 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('')
      []

   New in version 3.8.

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)

   Note:

     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)

   Note:

     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]

   New in version 3.8.

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.

   Examples:

      >>> 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

   New in version 3.10.

statistics.correlation(x, y, /, *, method='linear')

   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 a linear
   relationship.

   If *method* is "ranked", computes Spearman's rank correlation
   coefficient for two inputs. The data is replaced by ranks.  Ties
   are averaged so that equal values receive the same rank.  The
   resulting coefficient measures the strength of a monotonic
   relationship.

   Spearman's correlation coefficient is appropriate for ordinal data
   or for continuous data that doesn't meet the linear proportion
   requirement for Pearson's correlation coefficient.

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

   Example with Kepler's laws of planetary motion:

      >>> # Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and  Neptune
      >>> orbital_period = [88, 225, 365, 687, 4331, 10_756, 30_687, 60_190]    # days
      >>> dist_from_sun = [58, 108, 150, 228, 778, 1_400, 2_900, 4_500] # million km

      >>> # Show that a perfect monotonic relationship exists
      >>> correlation(orbital_period, dist_from_sun, method='ranked')
      1.0

      >>> # Observe that a linear relationship is imperfect
      >>> round(correlation(orbital_period, dist_from_sun), 4)
      0.9882

      >>> # Demonstrate Kepler's third law: There is a linear correlation
      >>> # between the square of the orbital period and the cube of the
      >>> # distance from the sun.
      >>> period_squared = [p * p for p in orbital_period]
      >>> dist_cubed = [d * d * d for d in dist_from_sun]
      >>> round(correlation(period_squared, dist_cubed), 4)
      1.0

   New in version 3.10.

   Changed in version 3.12: Added support for Spearman's rank
   correlation coefficient.

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*

   New in version 3.10.

   Changed in version 3.11: Added support for *proportional*.


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.

   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".

      New in version 3.9.

   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

   New in version 3.8.


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


Classic probability problems
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

"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]


Monte Carlo inputs for simulations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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]


Approximating binomial distributions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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


Naive bayesian classifier
~~~~~~~~~~~~~~~~~~~~~~~~~

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'


Kernel density estimation
~~~~~~~~~~~~~~~~~~~~~~~~~

It is possible to estimate a continuous probability density function
from a fixed number of discrete samples.

The basic idea is to smooth the data using a kernel function such as a
normal distribution, triangular distribution, or uniform distribution.
The degree of smoothing is controlled by a single parameter, "h",
representing the variance of the kernel function.

   import math

   def kde_normal(sample, h):
       "Create a continuous probability density function from a sample."
       # Smooth the sample with a normal distribution of variance h.
       kernel_h = NormalDist(0.0, math.sqrt(h)).pdf
       n = len(sample)
       def pdf(x):
           return sum(kernel_h(x - x_i) for x_i in sample) / n
       return pdf

Wikipedia has an example where we can use the "kde_normal()" recipe to
generate and plot a probability density function estimated from a
small sample:

   >>> sample = [-2.1, -1.3, -0.4, 1.9, 5.1, 6.2]
   >>> f_hat = kde_normal(sample, h=2.25)
   >>> xarr = [i/100 for i in range(-750, 1100)]
   >>> yarr = [f_hat(x) for x in xarr]

The points in "xarr" and "yarr" can be used to make a PDF plot:

[image: Scatter plot of the estimated probability density
function.][image]
