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Help on module statistics: NAME statistics - Basic statistics module. MODULE REFERENCE https://docs.python.org/3.10/library/statistics.html The following documentation is automatically generated from the Python source files. It may be incomplete, incorrect or include features that are considered implementation detail and may vary between Python implementations. When in doubt, consult the module reference at the location listed above. DESCRIPTION This module provides functions for calculating statistics of data, including averages, variance, and standard deviation. Calculating averages -------------------- ================== ================================================== Function Description ================== ================================================== mean Arithmetic mean (average) of data. fmean Fast, floating point arithmetic mean. 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 Mode (most common value) of data. multimode List of modes (most common values of data). quantiles Divide data into intervals with equal probability. ================== ================================================== Calculate the arithmetic mean ("the average") of data: >>> mean([-1.0, 2.5, 3.25, 5.75]) 2.625 Calculate the standard median of discrete data: >>> median([2, 3, 4, 5]) 3.5 Calculate the median, or 50th percentile, of data grouped into class intervals centred on the data values provided. E.g. if your data points are rounded to the nearest whole number: >>> median_grouped([2, 2, 3, 3, 3, 4]) #doctest: +ELLIPSIS 2.8333333333... This should be interpreted in this way: you have two data points in the class interval 1.5-2.5, three data points in the class interval 2.5-3.5, and one in the class interval 3.5-4.5. The median of these data points is 2.8333... Calculating variability or spread --------------------------------- ================== ============================================= Function Description ================== ============================================= pvariance Population variance of data. variance Sample variance of data. pstdev Population standard deviation of data. stdev Sample standard deviation of data. ================== ============================================= Calculate the standard deviation of sample data: >>> stdev([2.5, 3.25, 5.5, 11.25, 11.75]) #doctest: +ELLIPSIS 4.38961843444... If you have previously calculated the mean, you can pass it as the optional second argument to the four "spread" functions to avoid recalculating it: >>> data = [1, 2, 2, 4, 4, 4, 5, 6] >>> mu = mean(data) >>> pvariance(data, mu) 2.5 Statistics for relations between two inputs ------------------------------------------- ================== ==================================================== Function Description ================== ==================================================== covariance Sample covariance for two variables. correlation Pearson's correlation coefficient for two variables. linear_regression Intercept and slope for simple linear regression. ================== ==================================================== Calculate covariance, Pearson's correlation, and simple linear regression for two inputs: >>> 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 >>> correlation(x, y) #doctest: +ELLIPSIS 0.31622776601... >>> linear_regression(x, y) #doctest: LinearRegression(slope=0.1, intercept=1.5) Exceptions ---------- A single exception is defined: StatisticsError is a subclass of ValueError. CLASSES builtins.ValueError(builtins.Exception) StatisticsError builtins.object NormalDist class NormalDist(builtins.object) | NormalDist(mu=0.0, sigma=1.0) | | Normal distribution of a random variable | | Methods defined here: | | __add__(x1, x2) | Add a constant or another NormalDist instance. | | If *other* is a constant, translate mu by the constant, | leaving sigma unchanged. | | If *other* is a NormalDist, add both the means and the variances. | Mathematically, this works only if the two distributions are | independent or if they are jointly normally distributed. | | __eq__(x1, x2) | Two NormalDist objects are equal if their mu and sigma are both equal. | | __getstate__(self) | | __hash__(self) | NormalDist objects hash equal if their mu and sigma are both equal. | | __init__(self, mu=0.0, sigma=1.0) | NormalDist where mu is the mean and sigma is the standard deviation. | | __mul__(x1, x2) | Multiply both mu and sigma by a constant. | | Used for rescaling, perhaps to change measurement units. | Sigma is scaled with the absolute value of the constant. | | __neg__(x1) | Negates mu while keeping sigma the same. | | __pos__(x1) | Return a copy of the instance. | | __radd__ = __add__(x1, x2) | | __repr__(self) | Return repr(self). | | __rmul__ = __mul__(x1, x2) | | __rsub__(x1, x2) | Subtract a NormalDist from a constant or another NormalDist. | | __setstate__(self, state) | | __sub__(x1, x2) | Subtract a constant or another NormalDist instance. | | If *other* is a constant, translate by the constant mu, | leaving sigma unchanged. | | If *other* is a NormalDist, subtract the means and add the variances. | Mathematically, this works only if the two distributions are | independent or if they are jointly normally distributed. | | __truediv__(x1, x2) | Divide both mu and sigma by a constant. | | Used for rescaling, perhaps to change measurement units. | Sigma is scaled with the absolute value of the constant. | | cdf(self, x) | Cumulative distribution function. P(X <= x) | | inv_cdf(self, p) | Inverse cumulative distribution function. x : P(X <= x) = p | | Finds the value of the random variable such that the probability of | the variable being less than or equal to that value equals the given | probability. | | This function is also called the percent point function or quantile | function. | | overlap(self, other) | Compute the overlapping coefficient (OVL) between two normal distributions. | | Measures the agreement between two normal probability distributions. | Returns a value between 0.0 and 1.0 giving the overlapping area in | the two underlying probability density functions. | | >>> N1 = NormalDist(2.4, 1.6) | >>> N2 = NormalDist(3.2, 2.0) | >>> N1.overlap(N2) | 0.8035050657330205 | | pdf(self, x) | Probability density function. P(x <= X < x+dx) / dx | | quantiles(self, n=4) | Divide 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 in to 100 equal sized groups. | | samples(self, n, *, seed=None) | Generate *n* samples for a given mean and standard deviation. | | zscore(self, x) | Compute the Standard Score. (x - mean) / stdev | | Describes *x* in terms of the number of standard deviations | above or below the mean of the normal distribution. | | ---------------------------------------------------------------------- | Class methods defined here: | | from_samples(data) from builtins.type | Make a normal distribution instance from sample data. | | ---------------------------------------------------------------------- | Readonly properties defined here: | | mean | Arithmetic mean of the normal distribution. | | median | Return the median of the normal distribution | | mode | Return the mode of the normal distribution | | The mode is the value x where which the probability density | function (pdf) takes its maximum value. | | stdev | Standard deviation of the normal distribution. | | variance | Square of the standard deviation. class StatisticsError(builtins.ValueError) | Method resolution order: | StatisticsError | builtins.ValueError | builtins.Exception | builtins.BaseException | builtins.object | | Data descriptors defined here: | | __weakref__ | list of weak references to the object (if defined) | | ---------------------------------------------------------------------- | Methods inherited from builtins.ValueError: | | __init__(self, /, *args, **kwargs) | Initialize self. See help(type(self)) for accurate signature. | | ---------------------------------------------------------------------- | Static methods inherited from builtins.ValueError: | | __new__(*args, **kwargs) from builtins.type | Create and return a new object. See help(type) for accurate signature. | | ---------------------------------------------------------------------- | Methods inherited from builtins.BaseException: | | __delattr__(self, name, /) | Implement delattr(self, name). | | __getattribute__(self, name, /) | Return getattr(self, name). | | __reduce__(...) | Helper for pickle. | | __repr__(self, /) | Return repr(self). | | __setattr__(self, name, value, /) | Implement setattr(self, name, value). | | __setstate__(...) | | __str__(self, /) | Return str(self). | | with_traceback(...) | Exception.with_traceback(tb) -- | set self.__traceback__ to tb and return self. | | ---------------------------------------------------------------------- | Data descriptors inherited from builtins.BaseException: | | __cause__ | exception cause | | __context__ | exception context | | __dict__ | | __suppress_context__ | | __traceback__ | | args FUNCTIONS correlation(x, y, /) Pearson's correlation coefficient 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. >>> 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 covariance(x, y, /) Covariance Return the sample covariance of two inputs *x* and *y*. Covariance is a measure of the joint variability of two inputs. >>> 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 fmean(data) Convert data to floats and compute the arithmetic mean. This runs faster than the mean() function and it always returns a float. If the input dataset is empty, it raises a StatisticsError. >>> fmean([3.5, 4.0, 5.25]) 4.25 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 harmonic_mean(data, weights=None) Return the harmonic mean of data. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the data. It can be used for averaging ratios or rates, for example speeds. Suppose a car travels 40 km/hr for 5 km and then speeds-up to 60 km/hr for another 5 km. 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 If ``data`` is empty, or any element is less than zero, ``harmonic_mean`` will raise ``StatisticsError``. linear_regression(x, y, /) Slope and intercept for simple linear regression. Return the slope and intercept of simple linear regression parameters estimated using ordinary least squares. Simple linear regression describes relationship between an independent variable *x* and a dependent variable *y* in terms of 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). The parameters are returned as a named tuple. >>> x = [1, 2, 3, 4, 5] >>> noise = NormalDist().samples(5, seed=42) >>> y = [3 * x[i] + 2 + noise[i] for i in range(5)] >>> linear_regression(x, y) #doctest: +ELLIPSIS LinearRegression(slope=3.09078914170..., intercept=1.75684970486...) mean(data) Return the sample arithmetic mean of data. >>> mean([1, 2, 3, 4, 4]) 2.8 >>> 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') If ``data`` is empty, StatisticsError will be raised. median(data) Return the median (middle value) of numeric data. When the number of data points is odd, return the middle data point. 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]) 3 >>> median([1, 3, 5, 7]) 4.0 median_grouped(data, interval=1) Return the 50th percentile (median) of grouped continuous data. >>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5]) 3.7 >>> median_grouped([52, 52, 53, 54]) 52.5 This calculates the median as the 50th percentile, and should be used when your data is continuous and grouped. In the above example, the values 1, 2, 3, etc. actually represent the midpoint of classes 0.5-1.5, 1.5-2.5, 2.5-3.5, etc. The middle value falls somewhere in class 3.5-4.5, and interpolation is used to estimate it. Optional argument ``interval`` represents the class interval, and defaults to 1. Changing the class interval naturally will change the interpolated 50th percentile value: >>> 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. median_high(data) Return the high median of data. 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 median_low(data) Return the low median of numeric data. 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 mode(data) Return the most common data point from discrete or nominal data. ``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 This also works with nominal (non-numeric) data: >>> mode(["red", "blue", "blue", "red", "green", "red", "red"]) 'red' If there are multiple modes with same frequency, return the first one encountered: >>> mode(['red', 'red', 'green', 'blue', 'blue']) 'red' If *data* is empty, ``mode``, raises StatisticsError. multimode(data) Return a list of the most frequently occurring values. Will return more than one result if there are multiple modes or an empty list if *data* is empty. >>> multimode('aabbbbbbbbcc') ['b'] >>> multimode('aabbbbccddddeeffffgg') ['b', 'd', 'f'] >>> multimode('') [] pstdev(data, mu=None) Return 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 pvariance(data, mu=None) Return the population variance of ``data``. data should be a sequence or iterable of Real-valued numbers, with at least one value. The optional argument mu, if given, should be the mean of the data. If it is missing or None, 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. 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 the data, you can pass it as the optional second argument to avoid recalculating it: >>> 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) 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* in to 100 equal sized groups. The *data* can be any iterable containing sample. The cut points are linearly interpolated between data points. If *method* is set to *inclusive*, *data* is treated as population data. The minimum value is treated as the 0th percentile and the maximum value is treated as the 100th percentile. stdev(data, xbar=None) Return 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 variance(data, xbar=None) Return the sample variance of data. data should be an iterable of Real-valued numbers, with at least two values. The optional argument xbar, if given, should be the mean of the data. If it is missing or None, 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``. 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 recalculating it: >>> m = mean(data) >>> variance(data, m) 1.3720238095238095 This function does not check that ``xbar`` is actually the mean of ``data``. Giving arbitrary values for ``xbar`` may lead to invalid or impossible results. Decimals and Fractions 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) DATA __all__ = ['NormalDist', 'StatisticsError', 'correlation', 'covariance... FILE /usr/lib/python3.10/statistics.py
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