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 to true.

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 to false.

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, and Double.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 product-moment 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 median-unbiased 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

• John Clema
• Thomas Fankhauser
• Alan J. Salmoni

License

Sigma is released under the MIT License.