# `statistics` --- 數學統計函式

3.4 版新加入.

Source code: Lib/statistics.py

這個模組提供計算數值 (`Real`-valued) 資料的數學統計函式。

備註

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

## 平均值與中央位置量數

這些函式計算來自一個母體或樣本的平均值或代表值。

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

## 離度 (spread) 的測量

這些函式計算母體或樣本偏離平均值的程度。

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

## 函式細節

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

`statistics.``mean`(data)

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

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

  If data is empty, `StatisticsError` will be raised.

  Some examples of use:

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

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

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

  備註

  The mean is strongly affected by outliers and is not a robust estimator for central location: the mean is not necessarily a typical example of the data points. For more robust, although less efficient, measures of central location, see `median()` and `mode()`. (In this case, "efficient" refers to statistical efficiency rather than computational efficiency.)

  The sample mean gives an unbiased estimate of the true population mean, which means that, taken on average over all the possible samples, `mean(sample)` converges on the true mean of the entire population. If data represents the entire population rather than a sample, then `mean(data)` is equivalent to calculating the true population mean μ.

`statistics.``fmean`(data)

  Convert data to floats and compute the arithmetic mean.

  This runs faster than the `mean()` function and it always returns a `float`. The result is highly accurate but not as perfect as `mean()`. If the input dataset is empty, raises a `StatisticsError`.

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

  3.8 版新加入.

`statistics.``geometric_mean`(data)

  Convert data to floats and compute the geometric mean.

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

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

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

  3.8 版新加入.

`statistics.``harmonic_mean`(data)

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

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

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

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

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

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

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

  3.6 版新加入.

`statistics.``median`(data)

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

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

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

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

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

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

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

  也參考

  `median_low()`, `median_high()`, `median_grouped()`

`statistics.``median_low`(data)

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

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

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

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

`statistics.``median_high`(data)

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

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

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

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

`statistics.``median_grouped`(data, interval=1)

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

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

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

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

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

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

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

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

  也參考

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

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

`statistics.``mode`(data)

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

  If there are multiple modes, returns the first one encountered in the data. If data is empty, `StatisticsError` is raised.

  `mode` assumes discrete data, and returns a single value. This is the standard treatment of the mode as commonly taught in schools:

      >>> mode([1, 1, 2, 3, 3, 3, 3, 4])
      3

  The mode is unique in that it is the only statistic which also applies to nominal (non-numeric) data:

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

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

`statistics.``multimode`(data)

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

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

  3.8 版新加入.

`statistics.``pstdev`(data, mu=None)

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

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

`statistics.``pvariance`(data, mu=None)

  Return the population variance of data, a non-empty iterable of real-valued numbers. Variance, or second moment about the mean, is a measure of the variability (spread or dispersion) of data. A large variance indicates that the data is spread out; a small variance indicates it is clustered closely around the mean.

  If the optional second argument mu is given, it should be the mean of data. If it is missing or `None` (the default), the mean is automatically calculated.

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

  Raises `StatisticsError` if data is empty.

  Examples:

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

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

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

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

  Decimals and Fractions are supported:

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

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

  備註

  When called with the entire population, this gives the population variance σ². When called on a sample instead, this is the biased sample variance s², also known as variance with N degrees of freedom.

  If you somehow know the true population mean μ, you may use this function to calculate the variance of a sample, giving the known population mean as the second argument. Provided the data points are representative (e.g. independent and identically distributed), the result will be an unbiased estimate of the population variance.

`statistics.``stdev`(data, xbar=None)

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

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

`statistics.``variance`(data, xbar=None)

  Return the sample variance of data, an iterable of at least two real-valued numbers. Variance, or second moment about the mean, is a measure of the variability (spread or dispersion) of data. A large variance indicates that the data is spread out; a small variance indicates it is clustered closely around the mean.

  If the optional second argument xbar is given, it should be the mean of data. If it is missing or `None` (the default), the mean is automatically calculated.

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

  Raises `StatisticsError` if data has fewer than two values.

  Examples:

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

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

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

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

  Decimal and Fraction values are supported:

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

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

  備註

  This is the sample variance s² with Bessel's correction, also known as variance with N-1 degrees of freedom. Provided that the data points are representative (e.g. independent and identically distributed), the result should be an unbiased estimate of the true population variance.

  If you somehow know the actual population mean μ you should pass it to the `pvariance()` function as the mu parameter to get the variance of a sample.

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

  Divide dist into n continuous intervals with equal probability. Returns a list of `n - 1` cut points separating the intervals.

  Set n to 4 for quartiles (the default). Set n to 10 for deciles. Set n to 100 for percentiles which gives the 99 cuts points that separate dist in to 100 equal sized groups. Raises `StatisticsError` if n is not least 1.

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

  For sample data, the cut points are linearly interpolated from the two nearest data points. For example, if a cut point falls one-third of the distance between two sample values, `100` and `112`, the cut-point will evaluate to `104`.

  The method for computing quantiles can be varied depending on whether the data in dist includes or excludes the lowest and highest possible values from the population.

  The default method is "exclusive" and is used for data sampled from a population that can have more extreme values than found in the samples. The portion of the population falling below the i-th of m data points is computed as `i / (m + 1)`.

  Setting the method to "inclusive" is used for describing population data or for samples that include the extreme points. The minimum value in dist is treated as the 0th percentile and the maximum value is treated as the 100th percentile. The portion of the population falling below the i-th of m data points is computed as `(i - 1) / (m - 1)`.

  If dist is an instance of a class that defines an `inv_cdf()` method, setting method has no effect.

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

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

  3.8 版新加入.

## Exceptions

A single exception is defined:

exception `statistics.``StatisticsError`

  Subclass of `ValueError` for statistics-related exceptions.

## `NormalDist` objects

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

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

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

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

  If sigma is negative, raises `StatisticsError`.

  `mean`

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

  `stdev`

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

  `variance`

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

  classmethod `from_samples`(data)

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

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

  `samples`(n, *, seed=None)

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

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

  `pdf`(x)

    Using a probability density function (pdf), compute the relative likelihood that a random variable X will be near the given value x. Mathematically, it is the ratio `P(x <= X < x+dx) / dx`.

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

  `cdf`(x)

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

  `inv_cdf`(p)

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

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

  `overlap`(other)

    Compute the overlapping coefficient (OVL) between two normal distributions, giving a measure of agreement. Returns a value between 0.0 and 1.0 giving the overlapping area for the two probability density functions.

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

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

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

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

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

  3.8 版新加入.

### `NormalDist` Examples and Recipes

`NormalDist` readily solves classic probability problems.

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

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

Find the quartiles and deciles for the SAT scores:

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

What percentage of men and women will have the same height in two normally distributed populations with known means and standard deviations?

    >>> men = NormalDist(70, 4)
    >>> women = NormalDist(65, 3.5)
    >>> ovl = men.overlap(women)
    >>> round(ovl * 100.0, 1)
    50.3

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)
    >>> Y = NormalDist(15, 1.75).samples(n)
    >>> Z = NormalDist(5, 1.25).samples(n)
    >>> NormalDist.from_samples(map(model, X, Y, Z))
    NormalDist(mu=19.640137307085507, sigma=47.03273142191088)

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'