Averages and measures of central location¶

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

Measures of spread¶

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

Statistics for relations between two inputs¶

These functions calculate statistics regarding relations between two inputs.

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 ofdata which can be a sequence or iterable.

The arithmetic mean is the sum of the data divided by the number of datapoints. It is commonly called «the average», although it is only one of manydifferent mathematical averages. It is a measure of the central location ofthe data.

Ifdata 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>>>fromfractionsimportFractionasF>>>mean([F(3,7),F(1,21),F(5,3),F(1,3)])Fraction(13, 21)>>>fromdecimalimportDecimalasD>>>mean([D("0.5"),D("0.75"),D("0.625"),D("0.375")])Decimal('0.5625')

Σημείωση

The mean is strongly affected byoutliers and is not necessarily atypical example of the data points. For a more robust, although lessefficient, measure ofcentral tendency, seemedian().

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. Ifdata represents the entire population rather than a sample, thenmean(data) is equivalent to calculating the true population mean μ.

statistics.fmean(data,weights=None)¶

Convertdata to floats and compute the arithmetic mean.

This runs faster than themean() function and it always returns afloat. Thedata may be a sequence or iterable. If the inputdataset is empty, raises aStatisticsError.

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

Optional weighting is supported. For example, a professor assigns agrade for a course by weighting quizzes at 20%, homework at 20%, amidterm 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

Ifweights is supplied, it must be the same length as thedata oraValueError will be raised.

Added in version 3.8.

Άλλαξε στην έκδοση 3.11:Added support forweights.

statistics.geometric_mean(data)¶

Convertdata to floats and compute the geometric mean.

The geometric mean indicates the central tendency or typical value of thedata using the product of the values (as opposed to the arithmetic meanwhich uses their sum).

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

Added in version 3.8.

statistics.harmonic_mean(data,weights=None)¶

Return the harmonic mean ofdata, a sequence or iterable ofreal-valued numbers. Ifweights is omitted orNone, thenequal weighting is assumed.

The harmonic mean is the reciprocal of the arithmeticmean() of thereciprocals of the data. For example, the harmonic mean of three valuesa,b andc will be equivalent to3/(1/a+1/b+1/c). If one of thevalues is zero, the result will be zero.

The harmonic mean is a type of average, a measure of the centrallocation of the data. It is often appropriate when averagingratios 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. Whatis the average speed?

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

StatisticsError is raised ifdata is empty, any elementis less than zero, or if the weighted sum isn’t positive.

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

Added in version 3.6.

Άλλαξε στην έκδοση 3.10:Added support forweights.

statistics.median(data)¶

Return the median (middle value) of numeric data, using the common «mean ofmiddle two» method. Ifdata 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 bythe presence of outliers. When the number of data points is odd, themiddle data point is returned:

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

When the number of data points is even, the median is interpolated by takingthe 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 themedian may not be an actual data point.

If the data is ordinal (supports order operations) but not numeric (doesn’tsupport addition), consider usingmedian_low() ormedian_high()instead.

statistics.median_low(data)¶

Return the low median of numeric data. Ifdata 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 datapoints is odd, the middle value is returned. When it is even, the smaller ofthe 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 tobe an actual data point rather than interpolated.

statistics.median_high(data)¶

Return the high median of data. Ifdata is empty,StatisticsErroris raised.data can be a sequence or iterable.

The high median is always a member of the data set. When the number of datapoints is odd, the middle value is returned. When it is even, the larger ofthe 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 tobe an actual data point rather than interpolated.

statistics.median_grouped(data,interval=1.0)¶

Estimates the median for numeric data that has beengrouped or binned around the midpointsof consecutive, fixed-width intervals.

Thedata can be any iterable of numeric data with each value beingexactly the midpoint of a bin. At least one value must be present.

Theinterval is the width of each bin.

For example, demographic information may have been summarized intoconsecutive ten-year age groups with each group being representedby the 5-year midpoints of the intervals:

>>>fromcollectionsimportCounter>>>demographics=Counter({...25:172,# 20 to 30 years old...35:484,# 30 to 40 years old...45:387,# 40 to 50 years old...55:22,# 50 to 60 years old...65:6,# 60 to 70 years old...})...

The 50th percentile (median) is the 536th person out of the 1071member cohort. That person is in the 30 to 40 year old age group.

The regularmedian() function would assume that everyone in thetricenarian age group was exactly 35 years old. A more tenableassumption is that the 484 members of that age group are evenlydistributed between 30 and 40. For that, we usemedian_grouped():

>>>data=list(demographics.elements())>>>median(data)35>>>round(median_grouped(data,interval=10),1)37.5

The caller is responsible for making sure the data points are separatedby exact multiples ofinterval. This is essential for getting acorrect result. The function does not check this precondition.

Inputs may be any numeric type that can be coerced to a float duringthe interpolation step.

statistics.mode(data)¶

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

If there are multiple modes with the same frequency, returns the first oneencountered in thedata. If the smallest or largest of those isdesired instead, usemin(multimode(data)) ormax(multimode(data)).If the inputdata is empty,StatisticsError is raised.

mode assumes discrete data and returns a single value. This is thestandard 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 thatalso applies to nominal (non-numeric) data:

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

Only hashable inputs are supported. To handle typeset,consider casting tofrozenset. To handle typelist,consider casting totuple. For mixed or nested inputs, considerusing this slower quadratic algorithm that only depends on equality tests:max(data,key=data.count).

Άλλαξε στην έκδοση 3.8:Now handles multimodal datasets by returning the first mode encountered.Formerly, it raisedStatisticsError when more than one mode wasfound.

statistics.multimode(data)¶

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

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

Added in version 3.8.

statistics.pstdev(data,mu=None)¶

Return the population standard deviation (the square root of the populationvariance). Seepvariance() 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 ofdata, a non-empty sequence or iterableof real-valued numbers. Variance, or second moment about the mean, is ameasure of the variability (spread or dispersion) of data. A largevariance indicates that the data is spread out; a small variance indicatesit is clustered closely around the mean.

If the optional second argumentmu is given, it should be thepopulationmean of thedata. It can also be used to compute the second moment arounda point that is not the mean. If it is missing orNone (the default),the arithmetic mean is automatically calculated.

Use this function to calculate the variance from the entire population. Toestimate the variance from a sample, thevariance() function is usuallya better choice.

RaisesStatisticsError ifdata 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 theoptional second argumentmu to avoid recalculation:

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

Decimals and Fractions are supported:

>>>fromdecimalimportDecimalasD>>>pvariance([D("27.5"),D("30.25"),D("30.25"),D("34.5"),D("41.75")])Decimal('24.815')>>>fromfractionsimportFractionasF>>>pvariance([F(1,4),F(5,4),F(1,2)])Fraction(13, 72)

Σημείωση

When called with the entire population, this gives the population varianceσ². When called on a sample instead, this is the biased sample variances², also known as variance with N degrees of freedom.

If you somehow know the true population mean μ, you may use thisfunction to calculate the variance of a sample, giving the knownpopulation mean as the second argument. Provided the data points are arandom sample of the population, the result will be an unbiased estimateof the population variance.

statistics.stdev(data,xbar=None)¶

Return the sample standard deviation (the square root of the samplevariance). Seevariance() 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 ofdata, an iterable of at least two real-valuednumbers. Variance, or second moment about the mean, is a measure of thevariability (spread or dispersion) of data. A large variance indicates thatthe data is spread out; a small variance indicates it is clustered closelyaround the mean.

If the optional second argumentxbar is given, it should be thesamplemean ofdata. If it is missing orNone (the default), the mean isautomatically calculated.

Use this function when your data is a sample from a population. To calculatethe variance from the entire population, seepvariance().

RaisesStatisticsError ifdata 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 sample mean of your data, you can pass itas the optional second argumentxbar to avoid recalculation:

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

This function does not attempt to verify that you have passed the actual meanasxbar. Using arbitrary values forxbar can lead to invalid orimpossible results.

Decimal and Fraction values are supported:

>>>fromdecimalimportDecimalasD>>>variance([D("27.5"),D("30.25"),D("30.25"),D("34.5"),D("41.75")])Decimal('31.01875')>>>fromfractionsimportFractionasF>>>variance([F(1,6),F(1,2),F(5,3)])Fraction(67, 108)

Σημείωση

This is the sample variance s² with Bessel’s correction, also known asvariance with N-1 degrees of freedom. Provided that the data points arerepresentative (e.g. independent and identically distributed), the resultshould be an unbiased estimate of the true population variance.

If you somehow know the actual population mean μ you should pass it to thepvariance() function as themu parameter to get the variance of asample.

statistics.quantiles(data,*,n=4,method='exclusive')¶

Dividedata inton continuous intervals with equal probability.Returns a list ofn-1 cut points separating the intervals.

Setn to 4 for quartiles (the default). Setn to 10 for deciles. Setn to 100 for percentiles which gives the 99 cuts points that separatedata into 100 equal sized groups. RaisesStatisticsError ifnis not least 1.

Thedata can be any iterable containing sample data. For meaningfulresults, the number of data points indata should be larger thann.RaisesStatisticsError if there are not at least two data points.

The cut points are linearly interpolated from thetwo nearest data points. For example, if a cut point falls one-thirdof the distance between two sample values,100 and112, thecut-point will evaluate to104.

Themethod for computing quantiles can be varied depending onwhether thedata includes or excludes the lowest andhighest possible values from the population.

The defaultmethod is «exclusive» and is used for data sampled froma population that can have more extreme values than found in thesamples. The portion of the population falling below thei-th ofm sorted data points is computed asi/(m+1). Given ninesample values, the method sorts them and assigns the followingpercentiles: 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%.

Setting themethod to «inclusive» is used for describing populationdata or for samples that are known to include the most extreme valuesfrom the population. The minimum value indata is treated as the 0thpercentile and the maximum value is treated as the 100th percentile.The portion of the population falling below thei-th ofm sorteddata points is computed as(i-1)/(m-1). Given 11 samplevalues, 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)forqinquantiles(data,n=10)][81.0, 86.2, 89.0, 99.4, 102.5, 103.6, 106.0, 109.8, 111.0]

Added in version 3.8.

statistics.covariance(x,y,/)¶

Return the sample covariance of two inputsx andy. Covarianceis a measure of the joint variability of two inputs.

Both inputs must be of the same length (no less than two), otherwiseStatisticsError 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

Added in version 3.10.

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

Return thePearson’s correlation coefficientfor two inputs. Pearson’s correlation coefficientr takes valuesbetween -1 and +1. It measures the strength and direction of a linearrelationship.

Ifmethod is «ranked», computesSpearman’s rank correlation coefficientfor two inputs. The data is replaced by ranks. Ties are averaged so thatequal values receive the same rank. The resulting coefficient measures thestrength of a monotonic relationship.

Spearman’s correlation coefficient is appropriate for ordinal data or forcontinuous data that doesn’t meet the linear proportion requirement forPearson’s correlation coefficient.

Both inputs must be of the same length (no less than two), and neednot to be constant, otherwiseStatisticsError is raised.

Example withKepler’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*pforpinorbital_period]>>>dist_cubed=[d*d*dfordindist_from_sun]>>>round(correlation(period_squared,dist_cubed),4)1.0

Added in version 3.10.

Άλλαξε στην έκδοση 3.12:Added support for Spearman’s rank correlation coefficient.

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

Return the slope and intercept ofsimple linear regressionparameters estimated using ordinary least squares. Simple linearregression describes the relationship between an independent variablex anda dependent variabley in terms of this linear function:

y = slope * x + intercept + noise

whereslope andintercept are the regression parameters that areestimated, andnoise represents thevariability of the data that was not explained by the linear regression(it is equal to the difference between predicted and actual valuesof the dependent variable).

Both inputs must be of the same length (no less than two), andthe independent variablex cannot be constant;otherwise aStatisticsError is raised.

For example, we can use therelease dates of the MontyPython filmsto predict the cumulative number of Monty Python filmsthat would have been produced by 2019assuming 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

Ifproportional is true, the independent variablex and thedependent variabley are assumed to be directly proportional.The data is fit to a line passing through the origin.Since theintercept will always be 0.0, the underlying linearfunction simplifies to:

y = slope * x + noise

Continuing the example fromcorrelation(), we look to seehow well a model based on major planets can predict the orbitaldistances for dwarf planets:

>>>model=linear_regression(period_squared,dist_cubed,proportional=True)>>>slope=model.slope>>># Dwarf planets:   Pluto,  Eris,    Makemake, Haumea, Ceres>>>orbital_periods=[90_560,204_199,111_845,103_410,1_680]# days>>>predicted_dist=[math.cbrt(slope*(p*p))forpinorbital_periods]>>>list(map(round,predicted_dist))[5912, 10166, 6806, 6459, 414]>>>[5_906,10_152,6_796,6_450,414]# actual distance in million km[5906, 10152, 6796, 6450, 414]

Added in version 3.10.

Άλλαξε στην έκδοση 3.11:Added support forproportional.

Exceptions¶

A single exception is defined:

exceptionstatistics.StatisticsError¶

Subclass ofValueError for statistics-related exceptions.

NormalDist objects¶

NormalDist is a tool for creating and manipulating normaldistributions of arandom variable. It is aclass that treats the mean and standard deviation of datameasurements as a single entity.

Normal distributions arise from theCentral Limit Theorem and have a wide rangeof applications in statistics.

classstatistics.NormalDist(mu=0.0,sigma=1.0)¶

Returns a newNormalDist object wheremu represents thearithmeticmean andsigmarepresents thestandard deviation.

Ifsigma is negative, raisesStatisticsError.

mean¶

A read-only property for thearithmetic mean of a normaldistribution.

median¶

A read-only property for themedian of a normaldistribution.

mode¶

A read-only property for themode of a normaldistribution.

stdev¶

A read-only property for thestandard deviation of a normaldistribution.

variance¶

A read-only property for thevariance of a normaldistribution. Equal to the square of the standard deviation.

classmethodfrom_samples(data)¶

Makes a normal distribution instance withmu andsigma parametersestimated from thedata usingfmean() andstdev().

Thedata can be anyiterable and should consist of valuesthat can be converted to typefloat. Ifdata does notcontain at least two elements, raisesStatisticsError because ittakes at least one point to estimate a central value and at least twopoints to estimate dispersion.

samples(n,*,seed=None)¶

Generatesn random samples for a given mean and standard deviation.Returns alist offloat values.

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

pdf(x)¶

Using aprobability density function (pdf), computethe relative likelihood that a random variableX will be near thegiven valuex. Mathematically, it is the limit of the ratioP(x<=X<x+dx)/dx asdx approaches zero.

The relative likelihood is computed as the probability of a sampleoccurring in a narrow range divided by the width of the range (hencethe word «density»). Since the likelihood is relative to other points,its value can be greater than1.0.

cdf(x)¶

Using acumulative distribution function (cdf),compute the probability that a random variableX will be less than orequal tox. Mathematically, it is writtenP(X<=x).

inv_cdf(p)¶

Compute the inverse cumulative distribution function, also known as thequantile functionor thepercent-pointfunction. Mathematically, it is writtenx:P(X<=x)=p.

Finds the valuex of the random variableX such that theprobability of the variable being less than or equal to that valueequals the given probabilityp.

overlap(other)¶

Measures the agreement between two normal probability distributions.Returns a value between 0.0 and 1.0 givingthe overlapping area forthe two probability density functions.

quantiles(n=4)¶

Divide the normal distribution inton continuous intervals withequal probability. Returns a list of (n - 1) cut points separatingthe intervals.

Setn to 4 for quartiles (the default). Setn to 10 for deciles.Setn to 100 for percentiles which gives the 99 cuts points thatseparate the normal distribution into 100 equal sized groups.

zscore(x)¶

Compute theStandard Scoredescribingx in terms of the number of standard deviationsabove or below the mean of the normal distribution:(x-mean)/stdev.

Added in version 3.9.

Instances ofNormalDist support addition, subtraction,multiplication and division by a constant. These operationsare used for translation and scaling. For example:

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

Dividing a constant by an instance ofNormalDist is not supportedbecause the result wouldn’t be normally distributed.

Since normal distributions arise from additive effects of independentvariables, it is possible toadd and subtract two independent normallydistributed random variablesrepresented as instances ofNormalDist. 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

Added in version 3.8.

Examples and Recipes¶

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

>>>list(map(round,sat.quantiles()))[928, 1060, 1192]>>>list(map(round,sat.quantiles(n=10)))[810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]

>>>defmodel(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]

>>>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>>>frommathimportsqrt>>>round(NormalDist(mu=n*p,sigma=sqrt(n*p*q)).cdf(k+0.5),4)0.8402>>># Exact solution using the cumulative binomial distribution>>>frommathimportcomb,fsum>>>round(fsum(comb(n,r)*p**r*q**(n-r)forrinrange(k+1)),4)0.8402>>># Approximation using a simulation>>>fromrandomimportseed,binomialvariate>>>seed(8675309)>>>mean(binomialvariate(n,p)<=kforiinrange(10_000))0.8406

>>>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])

>>>ht=6.0# height>>>wt=130# weight>>>fs=8# foot size

>>>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))

>>>'male'ifposterior_male>posterior_femaleelse'female''female'

Kernel density estimation¶

It is possible to estimate a continuous probability distributionfrom a fixed number of discrete samples.

The basic idea is to smooth the data usinga kernel function such as anormal distribution, triangular distribution, or uniform distribution.The degree of smoothing is controlled by a scaling parameter,h,which is called thebandwidth.

fromrandomimportchoice,randomdefkde_normal(data,h):"Create a continuous probability distribution from discrete samples."# Smooth the data with a normal distribution kernel scaled by h.K_h=NormalDist(0.0,h)defpdf(x):'Probability density function.  P(x <= X < x+dx) / dx'returnsum(K_h.pdf(x-x_i)forx_iindata)/len(data)defcdf(x):'Cumulative distribution function.  P(X <= x)'returnsum(K_h.cdf(x-x_i)forx_iindata)/len(data)defrand():'Random selection from the probability distribution.'returnchoice(data)+K_h.inv_cdf(random())returnpdf,cdf,rand

Wikipedia has an examplewhere we can use thekde_normal() recipe to generate and plota probability density function estimated from a small sample:

>>>sample=[-2.1,-1.3,-0.4,1.9,5.1,6.2]>>>pdf,cdf,rand=kde_normal(sample,h=1.5)>>>xarr=[i/100foriinrange(-750,1100)]>>>yarr=[pdf(x)forxinxarr]

The points inxarr andyarr can be used to make a PDF plot:

Resamplethe data to produce 100 new selections:

>>>new_selections=[rand()foriinrange(100)]

Determine the probability of a new selection being below2.0:

>>>round(cdf(2.0),4)0.5794

Add a new sample data point and find the new CDF at2.0:

>>>sample.append(4.9)>>>round(cdf(2.0),4)0.5005