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README
σ (sigma)  statistics library written in Swift
This library is a collection of functions that perform statistical calculations in Swift. It can be used in Swift apps for Apple devices and in open source Swift programs on other platforms.
 average
 centralMoment
 covariancePopulation
 covarianceSample
 coefficientOfVariationSample
 frequencies
 kurtosisA
 kurtosisB
 max
 median
 medianHigh
 medianLow
 min
 normalDistribution
 normalDensity
 normalQuantile
 pearson
 percentile
 quantiles
 rank
 skewnessA
 skewnessB
 standardDeviationPopulation
 standardDeviationSample
 standardErrorOfTheMean
 sum
 uniqueValues
 variancePopulation
 varianceSample
Setup
There are four ways you can add Sigma to your project.
Add source (iOS 7+)
Simply add SigmaDistrib.swift file to your project.
Setup with Carthage (iOS 8+)
Alternatively, add github "evgenyneu/SigmaSwiftStatistics" ~> 9.0
to your Cartfile and run carthage update
.
Setup with CocoaPods (iOS 8+)
If you are using CocoaPods add this text to your Podfile and run pod install
.
use_frameworks!
target 'Your target name'
pod 'SigmaSwiftStatistics', '~> 9.0'
Setup with Swift Package Manager
 In Xcode 11+ select File > Packages > Add Package Dependency....
 Enter this project's URL: https://github.com/evgenyneu/SigmaSwiftStatistics.git
Legacy Swift versions
Setup a previous version of the library if you use an older version of Swift.
Usage
Add import SigmaSwiftStatistics
to your source code unless you used the file setup method.
Average / mean
Computes arithmetic mean of values in the array.
Note:
 Returns nil for an empty array.
 Same as AVERAGE in Microsoft Excel and Google Docs Sheets.
Formula
A = Σ(x) / n
Where:
 n is the number of values.
Sigma.average([1, 3, 8])
// Result: 4
Central moment
Computes central moment of the dataset.
Note:
 Returns nil for an empty array.
 Same as in Wolfram Alpha and "moments" R package.
Formula
Σ(x  m)^{k} / n
Where:
 m is the sample mean.
 k is the order of the moment (0, 1, 2, 3, ...).
 n is the sample size.
Sigma.centralMoment([3, 1, 1, 4.1, 4.1, 0.7], order: 3)
// Result: 1.5999259259
Covariance of a population
Computes covariance of the entire population between two variables: x and y.
Note:
 Returns nil if arrays x and y have different number of values.
 Returns nil for empty arrays.
 Same as COVAR and COVARIANCE.P functions in Microsoft Excel and COVAR in Google Docs Sheets.
Formula
cov(x,y) = Σ(x  mx)(y  my) / n
Where:
 mx is the population mean of the first variable.
 my is the population mean of the second variable.
 n is the total number of values.
let x = [1, 2, 3.5, 3.7, 8, 12]
let y = [0.5, 1, 2.1, 3.4, 3.4, 4]
Sigma.covariancePopulation(x: x, y: y)
// Result: 4.19166666666667
Covariance of a sample
Computes sample covariance between two variables: x and y.
Note:
 Returns nil if arrays x and y have different number of values.
 Returns nil for empty arrays or arrays containing a single element.
 Same as COVARIANCE.S function in Microsoft Excel.
Formula
cov(x,y) = Σ(x  mx)(y  my) / (n  1)
Where:
 mx is the sample mean of the first variable.
 my is the sample mean of the second variable.
 n is the total number of values.
let x = [1, 2, 3.5, 3.7, 8, 12]
let y = [0.5, 1, 2.1, 3.4, 3.4, 4]
Sigma.covarianceSample(x: x, y: y)
// Result: 5.03
Coefficient of variation of a sample
Computes coefficient of variation based on a sample.
Note:
 Returns nil when the array is empty or contains a single value.
 Returns
Double.infinity
if the mean is zero.  Same as in Wolfram Alfa and in "raster" R package (expressed as a percentage in "raster").
Formula
CV = s / m
Where:
 s is the sample standard deviation.
 m is the mean.
Sigma.coefficientOfVariationSample([1, 12, 19.5, 5, 3, 8])
// Result: 1.3518226672
Frequencies
Returns a dictionary with the keys containing the numbers from the input array and the values corresponding to the frequencies of those numbers.
Sigma.frequencies([1, 2, 3, 4, 5, 4, 4, 3, 5])
// Result: [2:1, 3:2, 4:3, 5:2, 1:1]
Kurtosis A
Returns the kurtosis of a series of numbers.
Note:
 Returns nil if the dataset contains less than 4 values.
 Returns nil if all the values in the dataset are the same.
 Same as KURT in Microsoft Excel and Google Docs Sheets.
Formula
Sigma.kurtosisA([2, 1, 3, 4.1, 19, 1.5])
// Result: 5.4570693277
Kurtosis B
Returns the kurtosis of a series of numbers.
Note:
 Returns nil if the dataset contains less than 2 values.
 Returns nil if all the values in the dataset are the same.
 Same as in Wolfram Alpha and "moments" R package.
Formula
Sigma.kurtosisB([2, 1, 3, 4.1, 19, 1.5])
// Result: 4.0138523409
Max
Returns the maximum value in the array.
Note: returns nil for an empty array.
Sigma.max([1, 8, 3])
// Result: 8
Median
Returns the median value from the array.
Note:
 Returns nil when the array is empty.
 Returns the mean of the two middle values if there is an even number of items in the array.
 Same as MEDIAN in Microsoft Excel and Google Docs Sheets.
Sigma.median([1, 12, 19.5, 3, 5])
// Result: 3
Median high
Returns the median value from the array.
Note:
 Returns nil when the array is empty.
 Returns the higher of the two middle values if there is an even number of items in the array.
Sigma.medianHigh([1, 12, 19.5, 10, 3, 5])
// Result: 10
Median low
Returns the median value from the array.
Note:
 Returns nil when the array is empty.
 Returns the lower of the two middle values if there is an even number of items in the array.
Sigma.medianLow([1, 12, 19.5, 10, 3, 5])
// Result: 3
Min
Returns the minimum value in the array.
Note: returns nil for an empty array.
Sigma.min([7, 2, 3])
// Result: 2
Normal distribution
Returns the normal distribution for the given values of x
, μ
and σ
. The returned value is the area under the normal curve to the left of the value x
.
Note:
 Returns nil if σ is zero or negative.
 Defaults:
μ = 0
,σ = 1
.  Same as NORM.S.DIST, NORM.DIST and NORMDIST Excel functions and NORMDIST function in Google Docs sheet with
cumulative
argument equal totrue
.
Sigma.normalDistribution(x: 1, μ: 0, σ: 1)
// Result: 0.1586552539314570
Normal density
Returns density of the normal function for the given values of x
, μ
and σ
.
Note:
 Returns nil if σ is zero or negative.
 Defaults:
μ = 0
,σ = 1
.  Same as NORM.S.DIST, NORM.DIST and NORMDIST Excel functions and NORMDIST function in Google Docs sheet with
cumulative
argument equal tofalse
.
Formula
Where:
 x is the input value of the normal density function.
 μ is the mean.
 σ is the standard deviation.
Sigma.normalDensity(x: 0, μ: 0, σ: 1)
// Result: 0.3989422804014327
Normal quantile
Returns the quantile function for the normal distribution (the inverse of normal distribution). The p
argument is the probability, or the area under the normal curve to the left of the returned value.
Note:
 Returns nil if σ is zero or negative.
 Returns nil if p is negative or greater than one.
 Returns
Double.infinity
if p is zero, andDouble.infinity
if p is one.  Defaults:
μ = 0
,σ = 1
.  Same as NORM.INV, NORM.S.INV and NORMINV Excel functions and NORMINV, NORMSINV Google Docs sheet functions.
Sigma.normalQuantile(p: 0.025, μ: 0, σ: 1)
// 1.9599639845400538
Pearson correlation coefficient
Calculates the Pearson productmoment correlation coefficient between two variables: x and y.
Note:
 Returns nil if arrays x and y have different number of values.
 Returns nil for empty arrays.
 Same as CORREL and PEARSON functions in Microsoft Excel and Google Docs Sheets.
Formula
p(x,y) = cov(x,y) / (σx * σy)
Where:
 cov is the population covariance.
 σ is the population standard deviation.
let x = [1, 2, 3.5, 3.7, 8, 12]
let y = [0.5, 1, 2.1, 3.4, 3.4, 4]
Sigma.pearson(x: x, y: y)
// Result: 0.843760859352745
Percentile
Calculates the Percentile value for the given dataset.
Note:
 Returns nil when the
values
array is empty.  Returns nil when supplied
percentile
parameter is negative or greater than 1.  Same as PERCENTILE or PERCENTILE.INC in Microsoft Excel and PERCENTILE in Google Docs Sheets.
 Same as the 7th sample quantile method from the Hyndman and Fan paper (1996).
See the Percentile method documentation for more information.
// Calculate 40th percentile
Sigma.percentile([35, 20, 50, 40, 15], percentile: 0.4)
// Result: 29
// Same as
Sigma.quantiles.method7([35, 20, 50, 40, 15], probability: 0.4)
Quantiles
Collection of nine functions that calculate sample quantiles corresponding to the given probability. This is an implementation of the nine algorithms described in the Hyndman and Fan paper (1996). The documentation of the functions is based on R and Wikipedia.
Note:
 Returns nil if the dataset is empty.
 Returns nil if the probability is outside the [0, 1] range.
 Same as
quantile
function in R.
Quantile method 1
This method calculates quantiles using the inverse of the empirical distribution function.
Sigma.quantiles.method1([1, 12, 19.5, 5, 3, 8], probability: 0.5)
// Result: 3
Quantile method 2
This method uses inverted empirical distribution function with averaging.
Sigma.quantiles.method2([1, 12, 19.5, 5, 3, 8], probability: 0.5)
// Result: 5.5
Quantile method 3
Sigma.quantiles.method3([1, 12, 19.5, 5, 3, 8], probability: 0.5)
// Result: 3
Quantile method 4
The method uses linear interpolation of the empirical distribution function.
Sigma.quantiles.method4([1, 12, 19.5, 5, 3, 8], probability: 0.17)
// Result: 4.88
Quantile method 5
This method uses a piecewise linear function where the knots are the values midway through the steps of the empirical distribution function.
Sigma.quantiles.method5([1, 12, 19.5, 5, 3, 8], probability: 0.11)
// Result: 4.04
Quantile method 6
This method is implemented in Microsoft Excel (PERCENTILE.EXC), Minitab and SPSS. It uses linear interpolation of the expectations for the order statistics for the uniform distribution on [0,1].
Sigma.quantiles.method6([1, 12, 19.5, 5, 3, 8], probability: 0.1999)
// Result: 2.6042
Quantile method 7
This method is implemented in S, Microsoft Excel (PERCENTILE or PERCENTILE.INC) and Google Docs Sheets (PERCENTILE). It uses linear interpolation of the modes for the order statistics for the uniform distribution on [0, 1].
Sigma.quantiles.method7([1, 12, 19.5, 5, 3, 8], probability: 0.00001)
// Result: 4.9997
Quantile method 8
The quantiles returned by the method are approximately medianunbiased regardless of the distribution of x.
Sigma.quantiles.method8([1, 12, 19.5, 5, 3, 8], probability: 0.11)
// Result: 4.82
Quantile method 9
The quantiles returned by this method are approximately unbiased for the expected order statistics if x is normally distributed.
Sigma.quantiles.method9([1, 12, 19.5, 5, 3, 8], probability: 0.10001)
// Result: 4.999625
Rank
Returns the ranks of the values in the dataset.
Note:
Receives an optional
ties
parameter that determines how the ranks for the equal values ('ties') are calculated. Default value is.average
. Possible values:.average
: uses the average rank. Same as RANK.AVG in Microsoft Excel and Google Docs Sheets..min
,.max
: uses the minimum/maximum rank. The value.min
is the same as RANK and RANK.EQ in Microsoft Excel and Google Docs Sheets..first
,.last
: the ranks are incremented/decremented.
Same as rank function in R.
Sigma.rank([2, 3, 6, 5, 3], ties: .average)
// Result: [1.0, 2.5, 5.0, 4.0, 2.5]
Skewness A
Returns the skewness of the dataset.
Note:
 Returns nil if the dataset contains less than 3 values.
 Returns nil if all the values in the dataset are the same.
 Same as SKEW in Microsoft Excel and Google Docs Sheets.
Formula
Sigma.skewnessA([4, 2.1, 8, 21, 1])
// Result: 1.6994131524
Skewness B
Returns the skewness of the dataset.
Note:
 Returns nil if the dataset contains less than 3 values.
 Returns nil if all the values in the dataset are the same.
 Same as in Wolfram Alpha, SKEW.P in Microsoft Excel and
skewness
function in "moments" R package.
Formula
Sigma.skewnessB([4, 2.1, 8, 21, 1])
// Result: 1.1400009992
Standard deviation of a population
Computes standard deviation of entire population.
Note:
 Returns nil for an empty array.
 Same as STDEVP and STDEV.P in Microsoft Excel and STDEVP in Google Docs Sheets.
Formula
σ = sqrt( Σ( (x  m)^{2} ) / n )
Where:
 m is the population mean.
 n is the population size.
Sigma.standardDeviationPopulation([1, 12, 19.5, 5, 3, 8])
// Result: 7.918420858282849
Standard deviation of a sample
Computes standard deviation based on a sample.
Note:
 Returns nil when the array is empty or contains a single value.
 Same as STDEV and STDEV.S in Microsoft Excel and STDEV in Google Docs Sheets.
Formula
s = sqrt( Σ( (x  m)^{2} ) / (n  1) )
Where:
 m is the sample mean.
 n is the sample size.
Sigma.standardDeviationSample([1, 12, 19.5, 5, 3, 8])
// Result: 8.674195447801869
Standard error of the mean
Computes standard error of the mean.
Note:
 Returns nil when the array is empty or contains a single value.
Formula
SE = s / sqrt(n)
Where:
 s is the sample standard deviation.
 n is the sample size.
Sigma.standardErrorOfTheMean([1, 12, 19.5, 5, 3, 8])
// Result: 3.5412254627
Sum
Computes sum of values from the array.
Sigma.sum([1, 3, 8])
// Result: 12
Unique values
Returns an unsorted array containing all values that occur within the input array without the duplicates.
Sigma.uniqueValues([2, 1, 3, 4, 5, 4, 3, 5])
// Result: [2, 3, 4, 5, 1]
Variance of a population
Computes variance of entire population.
Note:
 Returns nil when the array is empty.
 Same as VAR.P or VARPA in Microsoft Excel and VARP or VARPA in Google Docs Sheets.
Formula
σ^{2} = Σ( (x  m)^{2} ) / n
Where:
 m is the population mean.
 n is the population size.
Sigma.variancePopulation([1, 12, 19.5, 5, 3, 8])
// Result: 62.70138889
Variance of a sample
Computes variance based on a sample.
Note:
 Returns nil when the array is empty or contains a single value.
 Same as VAR, VAR.S or VARA in Microsoft Excel and VAR or VARA in Google Docs Sheets.
Formula
s^{2} = Σ( (x  m)^{2} ) / (n  1)
Where:
 m is the sample mean.
 n is the sample size.
Sigma.varianceSample([1, 12, 19.5, 5, 3, 8])
// Result: 75.24166667
Feedback is welcome
If you need help or want to extend the library feel free to create an issue or submit a pull request.
Help will always be given at Hogwarts to those who ask for it.
 J.K. Rowling, Harry Potter
Contributors
License
Sigma is released under the [MIT License](LICENSE).
*Note that all licence references and agreements mentioned in the SigmaSwiftStatistics README section above
are relevant to that project's source code only.