# Statistics The 35 definitions of the statistics library, each with its Epsil spelling, its MathJSON name, its signature and its full description. Each definition is listed under its Epsil spelling (the MathJSON name when it has none), with its signature in the engine's type syntax. The [Standard Library](/library/) page is the one-page index of every category. ## Definitions ### betaRegularized MathJSON `BetaRegularized` · `(complex | infinity, complex | infinity, complex | infinity) -> number` Regularized incomplete beta function I_x(a, b) ### binCounts MathJSON `BinCounts` · `(collection, list | number) -> list` Count the number of elements falling into each bin. ```epsil binCounts([1, 2, 2, 3], 3) // ➔ [1,2,1] ``` ### binomialDistribution MathJSON `BinomialDistribution` · `(integer<0..>, real<0..1>) -> expression` Binomial distribution: number of successes in n independent trials, each with success probability p. ### cdf MathJSON `CDF` · `(distribution, real | signed_infinity) -> nan | real<0..1>` Cumulative distribution function P(X ≤ x) of a distribution. ### correlation MathJSON `Correlation` · `(collection, collection?) -> nan | real<-1..1>` Pearson's correlation coefficient of paired data, given as two equal-length collections or one collection of (x, y) pairs. ### covariance MathJSON `Covariance` · `(collection, collection?) -> nan | real` Sample covariance (n − 1 denominator) of paired data, given as two equal-length collections or one collection of (x, y) pairs. ### erf MathJSON `Erf` · `(complex | signed_infinity) -> complex` Gauss error function ### erfInv MathJSON `ErfInv` · `(complex | infinity) -> number` Inverse of the error function ### erfc MathJSON `Erfc` · `(complex | signed_infinity) -> complex` Complementary error function: 1 - Erf(x) ### erfi MathJSON `Erfi` · `(complex | signed_infinity) -> complex | signed_infinity` Imaginary error function: -i·Erf(i·x) ### exponentialDistribution MathJSON `ExponentialDistribution` · `(real<0<..>) -> expression` Exponential distribution with rate parameter λ. ### findFit MathJSON `FindFit` · `(any, any, any, any) -> dictionary` Nonlinear least-squares fit of a model to data. FindFit(data, model, params, vars): fit `model` (an expression in `vars` and the parameters) to `data`, a list of (x…, y) tuples or a plain list of y values. Each parameter spec is a bare symbol, (a, a0), or (a, a0, lo, hi) with box constraints. Returns a record {parameters, converged, residualNorm, iterations}. The joint form takes a list of models and matching datasets sharing parameters. ### gammaRegularized MathJSON `GammaRegularized` · `(complex | infinity, complex | infinity) -> number` Regularized upper incomplete gamma function Q(a, z) = Γ(a, z)/Γ(a) ### histogram MathJSON `Histogram` · `(collection, list | number) -> list>` Compute a histogram of the values in a collection. Returns a list of (bin start, count) tuples. ```epsil histogram([1, 2, 2, 3], 3) // ➔ [(1, 1),(1.6666666666666665, 2),(2.333333333333333, 1)] ``` ### interquartileRange MathJSON `InterquartileRange` · `((collection | number)+) -> +oo | nan | real<0..>` Interquartile range (Q3 - Q1) of a collection. ### kurtosis MathJSON `Kurtosis` · `((collection | number)+) -> nan | real` Kurtosis of a collection of numbers. ### linearRegression MathJSON `LinearRegression` · `(any+) -> tuple` Least-squares linear fit b0 + b1·x. Returns Tuple(b0, b1), or the fitted expression if a trailing variable symbol is given. ### mean MathJSON `Mean` · `((collection | distribution | number)+) -> number` Arithmetic mean (average) of a collection of numbers. ### median MathJSON `Median` · `((collection | number)+) -> nan | real | signed_infinity` Median of a collection of numbers. ```epsil median([3, 1, 4, 2]) // ➔ 5/2 ``` ### mode MathJSON `Mode` · `((collection | number)+) -> nan | real | signed_infinity` Most frequently occurring value in a collection. ```epsil mode([1, 2, 2, 3]) // ➔ 2 ``` ### normalDistribution MathJSON `NormalDistribution` · `(real, real<0<..>) -> expression` Normal (Gaussian) distribution with mean μ and standard deviation σ. ### pdf MathJSON `PDF` · `(distribution, real | signed_infinity) -> nan | real<0..>` Probability density (continuous) or mass (discrete) function of a distribution, evaluated at x. ### poissonDistribution MathJSON `PoissonDistribution` · `(real<0<..>) -> expression` Poisson distribution with rate parameter λ. ### polynomialFit MathJSON `PolynomialFit` · `(any+) -> list` Least-squares polynomial fit of the given degree. Returns the ascending coefficient List(c0, …, c_deg), or the fitted expression if a trailing variable symbol is given. ### populationCovariance MathJSON `PopulationCovariance` · `(collection, collection?) -> nan | real` Population covariance (n denominator) of paired data, given as two equal-length collections or one collection of (x, y) pairs. ### populationStandardDeviation MathJSON `PopulationStandardDeviation` · `((collection | number)+) -> nan | real<0..>` Population Standard Deviation of a collection of numbers. ### populationVariance MathJSON `PopulationVariance` · `((collection | number)+) -> nan | real<0..>` Population variance of a collection of numbers. ### quantile MathJSON `Quantile` · `(collection | distribution, real<0..1>) -> nan | real | signed_infinity` Quantile (inverse CDF): the least x with CDF(x) ≥ p, for p in [0, 1]. The first argument may also be a data collection, in which case the empirical quantile is returned. ### quartiles MathJSON `Quartiles` · `((collection | number)+) -> tuple` Lower quartile, median, and upper quartile of a collection. Uses the Moore–McCabe (exclusive-hinges) convention: the sample is split at its median, and Q1/Q3 are the medians of the lower/upper halves with the overall median excluded from both halves when the sample size is odd. ```epsil quartiles([1, 2, 3, 4, 5]) // ➔ (3/2, 3, 9/2) ``` ### randomSample MathJSON `RandomSample` · `((T, number) random -> T where T: string) & ((indexed_collection, number) random -> list)` RandomSample(xs, k): a list of k elements drawn from the indexed collection `xs`, without replacement. "Without replacement" is over POSITIONS, not values: on a multiset, repeats are expected — RandomSample([1, 1, 2], 2) can return [1, 1]. Sampling a string yields a string. Wrap the call in `WithRandomSeed(seed, ...)` to make it deterministic. ### skewness MathJSON `Skewness` · `((collection | number)+) -> nan | real` Skewness of a collection of numbers. ### slidingWindow MathJSON `SlidingWindow` · `((S, integer, integer?) -> list where S: string) & ((collection, integer, integer?) -> list)` Return overlapping sliding windows of fixed size over the collection. ```epsil slidingWindow([1, 2, 3, 4], 2) // ➔ [[1,2],[2,3],[3,4]] ``` ```epsil slidingWindow("abcd", 2) // ➔ ["ab","bc","cd"] ``` ### standardDeviation MathJSON `StandardDeviation` · `((collection | distribution | number)+) -> nan | real<0..>` Sample Standard Deviation of a collection of numbers. ### uniformDistribution MathJSON `UniformDistribution` · `(real, real) -> expression` Continuous uniform distribution on the interval [a, b]. ### variance MathJSON `Variance` · `((collection | distribution | number)+) -> nan | real<0..>` Sample variance of a collection of numbers.