Statistics extension for Numbas

Returns the variance of the array vector. By default, the population variance is calculated. Passing true to flag indicates to compute the sample variance instead.

`population_variance(array)`

Returns the population variance of the array vector.

`sample_variance(array)`

Returns the sample variance of the array vector.

`stdev(array)`

Returns the standard deviation of the array vector. By default, the population standard deviation is returned. Passing true to flag returns the sample standard deviation.

`population_stdev(array)`

Returns the population standard deviation of the array vector.

`sample_stdev(array)`

Returns the sample standard deviation of the array vector.

`meandev(array)`

Returns the mean absolute deviation of the array vector.

`meddev(array)`

Returns the median absolute deviation of the array vector.

`coeffvar(array)`

Returns the coefficient of variation of the array vector.

`quartiles(array)`

Returns the quartiles of the array vector.

Correlation of two samples

`covariance(array1, array2)`

Returns the covariance of the array1 and array2 vectors.

`corrcoeff(array1, array2)`

Returns the population correlation coefficient of the array1 and array2 vectors (Pearson's Rho).

`stdev(array, is_sample)`

Returns the standard deviation of the array vector. By default, the population standard deviation is returned. Passing true to flag returns the sample standard deviation.

`variance(array, is_sample)`

Returns the variance of the array vector. By default, the population variance is calculated. Passing true to flag indicates to compute the sample variance instead.

`mode(array)`

Returns the mode of the array vector. If there are multiple modes then mode() will return all of them.

Distributions

`betapdf(x, alpha, beta)`

Returns the value of x in the Beta distribution with parameters alpha and beta.

`betacdf(x, alpha, beta)`

Returns the value of x in the cdf for the Beta distribution with parameters alpha and beta.

`betainv(p, alpha, beta)`

Returns the value of p in the inverse of the cdf for the Beta distribution with parameters alpha and beta.

`betamean(alpha, beta)`

Returns the mean of the Beta distribution with parameters alpha and beta.

`betamedian(alpha, beta)`

Returns the median of the Beta distribution with parameters alpha and beta.

`betamode(alpha, beta)`

Returns the mode of the Beta distribution with parameters alpha and beta.

`betasample(alpha, beta)`

Returns a random number whose distribution is the Beta distribution with parameters alpha and beta.

`betavariance(alpha, beta)`

Returns the variance of the Beta distribution with parameters alpha and beta.

`centralFpdf(x, df1, df2)`

Given x in the range [0, infinity), returns the probability density of the (central) F distribution at x.

`centralFcdf(x, df1, df2)`

Given x in the range [0, infinity), returns the cumulative probability density of the central F distribution. That is, jStat.centralF.cdf(2.5, 10, 20) will return the probability that a number randomly selected from the central F distribution with df1 = 10 and df2 = 20 will be less than 2.5.

`centralFinv(p, df1, df2)`

Given p in [0, 1), returns the value of x for which the cumulative probability density of the central F distribution is p. That is, jStat.centralF.inv(p, df1, df2) = x if and only if jStat.centralF.inv(x, df1, df2) = p.

`centralFmean(df1, df2)`

Returns the mean of the (Central) F distribution.

`centralFmode(df1, df2)`

Returns the mode of the (Central) F distribution.

`centralFsample(df1, df2)`

Returns a random number whose distribution is the (Central) F distribution.

`centralFvariance(df1, df2)`

Returns the variance of the (Central) F distribution.

`cauchypdf(x, local, scale)`

Returns the value of x in the pdf of the Cauchy distribution with a location (median) of local and scale factor of scale.

`cauchycdf(x, local, scale)`

Returns the value of x in the cdf of the Cauchy distribution with a location (median) of local and scale factor of scale.

`cauchyinv(p, local, scale)`

Returns the value of p in the inverse of the cdf for the Cauchy distribution with a location (median) of local and scale factor of scale.

`cauchymedian(local, scale)`

Returns the value of the median for the Cauchy distribution with a location (median) of local and scale factor of scale.

`cauchymode(local, scale)`

Returns the value of the mode for the Cauchy distribution with a location (median) of local and scale factor of scale.

`cauchysample(local, scale)`

Returns a random number whose distribution is the Cauchy distribution with a location (median) of local and scale factor of scale.

`chisquarepdf(x, dof)`

Returns the value of x in the pdf of the Chi Square distribution with dof degrees of freedom.

`chisquarecdf(x, dof)`

Returns the value of x in the cdf of the Chi Square distribution with dof degrees of freedom.

`chisquareinv(p, dof)`

Returns the value of x in the inverse of the cdf for the Chi Square distribution with dof degrees of freedom.

`chisquaremean(dof)`

Returns the value of the mean for the Chi Square distribution with dof degrees of freedom.

`chisquaremedian(dof)`

Returns the value of the median for the Chi Square distribution with dof degrees of freedom.

`chisquaremode(dof)`

Returns the value of the mode for the Chi Square distribution with dof degrees of freedom.

`chisquaresample(dof)`

Returns a random number whose distribution is the Chi Square distribution with dof degrees of freedom.

`chisquarevariance(dof)`

Returns the value of the variance for the Chi Square distribution with dof degrees of freedom.

`exponentialpdf(x, rate)`

Returns the value of x in the pdf of the Exponential distribution with the parameter rate (lambda).

`exponentialcdf(x, rate)`

Returns the value of x in the cdf of the Exponential distribution with the parameter rate (lambda).

`exponentialinv(p, rate)`

Returns the value of p in the inverse of the cdf for the Exponential distribution with the parameter rate (lambda).

`exponentialmean(rate)`

Returns the value of the mean for the Exponential distribution with the parameter rate (lambda).

`exponentialmedian(rate)`

Returns the value of the median for the Exponential distribution with the parameter rate (lambda)

`exponentialmode(rate)`

Returns the value of the mode for the Exponential distribution with the parameter rate (lambda).

`exponentialsample(rate)`

Returns a random number whose distribution is the Exponential distribution with the parameter rate (lambda).

`exponentialvariance(rate)`

Returns the value of the variance for the Exponential distribution with the parameter rate (lambda).

`gammapdf(x, shape, scale)`

Returns the value of x in the pdf of the Gamma distribution with the parameters shape (k) and scale (theta). Notice that if using the alpha beta convention, scale = 1/beta.

`gammacdf(x, shape, scale)`

Returns the value of x in the cdf of the Gamma distribution with the parameters shape (k) and scale (theta). Notice that if using the alpha beta convention, scale = 1/beta.

`gammainv(p, shape, scale)`

Returns the value of p in the inverse of the cdf for the Gamma distribution with the parameters shape (k) and scale (theta). Notice that if using the alpha beta convention, scale = 1/beta.

`gammamean(shape, scale)`

Returns the value of the mean for the Gamma distribution with the parameters shape (k) and scale (theta). Notice that if using the alpha beta convention, scale = 1/beta.

`gammamode(shape, scale)`

Returns the value of the mode for the Gamma distribution with the parameters shape (k) and scale (theta). Notice that if using the alpha beta convention, scale = 1/beta.

`gammasample(shape, scale)`

Returns a random number whose distribution is the Gamma distribution with the parameters shape (k) and scale (theta). Notice that if using the alpha beta convention, scale = 1/beta.

`gammavariance(shape, scale)`

Returns the value of the variance for the Gamma distribution with the parameters shape (k) and scale (theta). Notice that if using the alpha beta convention, scale = 1/beta.

`invgammapdf(x, shape, scale)`

Returns the value of x in the pdf of the Inverse-Gamma distribution with parametres shape (alpha) and scale (beta).

`invgammacdf(x, shape, scale)`

Returns the value of x in the cdf of the Inverse-Gamma distribution with parametres shape (alpha) and scale (beta).

`invgammainv(p, shape, scale)`

Returns the value of p in the inverse of the cdf for the Inverse-Gamma distribution with parametres shape (alpha) and scale (beta).

`invgammamean(shape, scale)`

Returns the value of the mean for the Inverse-Gamma distribution with parametres shape (alpha) and scale (beta).

`invgammamode(shape, scale)`

Returns the value of the mode for the Inverse-Gamma distribution with parametres shape (alpha) and scale (beta).

`invgammasample(shape, scale)`

Returns a random number whose distribution is the Inverse-Gamma distribution with parametres shape (alpha) and scale (beta).

`invgammavariance(shape, scale)`

Returns the value of the variance for the Inverse-Gamma distribution with parametres shape (alpha) and scale (beta).

`kumaraswamypdf(x, alpha, beta)`

Returns the value of x in the pdf of the Kumaraswamy distribution with parameters a and b.

`kumaraswamycdf(x, alpha, beta)`

Returns the value of x in the cdf of the Kumaraswamy distribution with parameters alpha and beta.

`kumaraswamyinv(p, alpha, beta)`

Returns the value of p in the inverse of the pdf for the Kumaraswamy distribution with parametres alpha and beta.

`kumaraswamymean(alpha, beta)`

Returns the value of the mean of the Kumaraswamy distribution with parameters alpha and beta.

`kumaraswamymedian(alpha, beta)`

Returns the value of the median of the Kumaraswamy distribution with parameters alpha and beta.

`kumaraswamymode(alpha, beta)`

Returns the value of the mode of the Kumaraswamy distribution with parameters alpha and beta.

`kumaraswamyvariance(alpha, beta)`

Returns the value of the variance of the Kumaraswamy distribution with parameters alpha and beta.

`lognormalpdf(x, mu, sigma)`

Returns the value of x in the pdf of the Log-normal distribution with paramters mu (mean) and sigma (standard deviation).

`lognormalcdf(x, mu, sigma)`

Returns the value of x in the cdf of the Log-normal distribution with paramters mu (mean) and sigma (standard deviation).

`lognormalinv(p, mu, sigma)`

Returns the value of x in the inverse of the cdf for the Log-normal distribution with paramters mu (mean of the Normal distribution) and sigma (standard deviation of the Normal distribution).

`lognormalmean(mu, sigma)`

Returns the value of the mean for the Log-normal distribution with paramters mu (mean of the Normal distribution) and sigma (standard deviation of the Normal distribution).

`lognormalmedian(mu, sigma)`

Returns the value of the median for the Log-normal distribution with paramters mu (mean of the Normal distribution) and sigma (standard deviation of the Normal distribution).

`lognormalmode(mu, sigma)`

Returns the value of the mode for the Log-normal distribution with paramters mu (mean of the Normal distribution) and sigma (standard deviation of the Normal distribution).

`lognormalsample(mu, sigma)`

Returns a random number whose distribution is the Log-normal distribution with paramters mu (mean of the Normal distribution) and sigma (standard deviation of the Normal distribution).

`lognormalvariance(mu, sigma)`

Returns the value of the variance for the Log-normal distribution with paramters mu (mean of the Normal distribution) and sigma (standard deviation of the Normal distribution).

`normalpdf(x, mean, std)`

Returns the value of x in the pdf of the Normal distribution with parameters mean and std (standard deviation).

`normalcdf(x, mean, std)`

Returns the value of x in the cdf of the Normal distribution with parameters mean and std (standard deviation).

`normalinv(p, mean, std)`

Returns the value of p in the inverse cdf for the Normal distribution with parameters mean and std (standard deviation).

`normalmean(mean, std)`

Returns the value of the mean for the Normal distribution with parameters mean and std (standard deviation).

`normalmedian(mean, std)`

Returns the value of the median for the Normal distribution with parameters mean and std (standard deviation).

`normalmode(mean, std)`

Returns the value of the mode for the Normal distribution with parameters mean and std (standard deviation).

`normalsample(mean, std)`

Returns a random number whose distribution is the Normal distribution with parameters mean and std (standard deviation).

`normalvariance(mean, std)`

Returns the value of the variance for the Normal distribution with parameters mean and std (standard deviation).

`paretopdf(x, scale, shape)`

Returns the value of x in the pdf of the Pareto distribution with parameters scale (xm) and shape (alpha).

`paretocdf(x, scale, shape)`

Returns the value of x in the cdf of the Pareto distribution with parameters scale (xm) and shape (alpha).

`paretoinv(p, scale, shape)`

Returns the inverse of the Pareto distribution with probability p, scale, shape.

`paretomean(scale, shape)`

Returns the value of the mean of the Pareto distribution with parameters scale (xm) and shape (alpha).

`paretomedian(scale, shape)`

Returns the value of the median of the Pareto distribution with parameters scale (xm) and shape (alpha).

`paretomode(scale, shape)`

Returns the value of the mode of the Pareto distribution with parameters scale (xm) and shape (alpha).

`paretovariance(scale, shape)`

Returns the value of the variance of the Pareto distribution with parameters scale (xm) and shape (alpha).

`studenttpdf(x, dof)`

Returns the value of x in the pdf of the Student's T distribution with dof degrees of freedom.

`studenttcdf(x, dof)`

Returns the value of x in the cdf of the Student's T distribution with dof degrees of freedom.

`studenttinv(p, dof)`

Returns the value of p in the inverse of the cdf for the Student's T distribution with dof degrees of freedom.

`studenttmean(dof)`

Returns the value of the mean of the Student's T distribution with dof degrees of freedom.

`studenttmedian(dof)`

Returns the value of the median of the Student's T distribution with dof degrees of freedom.

`studenttmode(dof)`

Returns the value of the mode of the Student's T distribution with dof degrees of freedom.

`studenttsample(dof)`

Returns a random number whose distribution is the Student's T distribution with dof degrees of freedom.

`studenttvariance(dof)`

Returns the value of the variance for the Student's T distribution with dof degrees of freedom.

`weibullpdf(x, scale, shape)`

Returns the value x in the pdf for the Weibull distribution with parameters scale (lambda) and shape (k).

`weibullcdf(x, scale, shape)`

Returns the value x in the cdf for the Weibull distribution with parameters scale (lambda) and shape (k).

`weibullinv(p, scale, shape)`

Returns the value of x in the inverse of the cdf for the Weibull distribution with parameters scale (lambda) and shape (k).

`weibullmean(scale, shape)`

Returns the value of the mean of the Weibull distribution with parameters scale (lambda) and shape (k).

`weibullmedian(scale, shape)`

Returns the value of the median of the Weibull distribution with parameters scale (lambda) and shape (k).

`weibullmode(scale, shape)`

Returns the mode of the Weibull distribution with parameters scale (lambda) and shape (k).

`weibullsample(scale, shape)`

Returns a random number whose distribution is the Weibull distribution with parameters scale (lambda) and shape (k).

`weibullvariance(scale, shape)`

Returns the variance of the Weibull distribution with parameters scale (lambda) and shape (k).

`uniformpdf(x, a, b)`

Returns the value of x in the pdf of the Uniform distribution from a to b.

`uniformcdf(x, a, b)`

Returns the value of x in the cdf of the Uniform distribution from a to b.

`uniforminv(p, a, b)`

Returns the inverse of the uniform.cdf function; i.e. the value of x for which uniform.cdf(x, a, b) == p.

`uniformmean(a, b)`

Returns the value of the mean of the Uniform distribution from a to b.

`uniformmedian(a, b)`

Returns the value of the median of the Uniform distribution from a to b.

`uniformmode(a, b)`

Returns the value of the mode of the Uniform distribution from a to b.

`uniformsample(a, b)`

Returns a random number whose distribution is the Uniform distribution from a to b.

`uniformvariance(a, b)`

Returns the variance of the Uniform distribution from a to b.

`binomialpdf(x, n, p)`

Returns the value of k in the pdf of the Binomial distribution with parameters n and p.

`binomialcdf(x, n, p)`

Returns the value of k in the cdf of the Binomial distribution with parameters n and p.

`geometricpdf(x, p)`

`geometriccdf(x, p)`

`geometricmean(p)`

`geometricmedian(p)`

`geometricmode(p)`

`geometricsample(p)`

`geometricvariance(p)`

`negbinpdf(x, r, p)`

Returns the value of k in the pdf of the Negative Binomial distribution with parameters n and p.

`negbincdf(x, r, p)`

Returns the value of x in the cdf of the Negative Binomial distribution with parameters n and p.

`hypgeompdf(x, population_size, success_rate, num_draws)`

Returns the value of k in the pdf of the Hypergeometric distribution with parameters N (the population size), m (the success rate), and n (the number of draws).

`hypgeomcdf(x, population_size, success_rate, num_draws)`

Returns the value of x in the cdf of the Hypergeometric distribution with parameters N (the population size), m (the success rate), and n (the number of draws).

`poissonpdf(x, l)`

Returns the value of k in the pdf of the Poisson distribution with parameter l (lambda).

`poissoncdf(x, l)`

Returns the value of x in the cdf of the Poisson distribution with parameter l (lambda).

`poissonmean(l)`

`poissonsample(l)`

Returns a random number whose distribution is the Poisson distribution with rate parameter l (lamda)

`poissonvariance(l)`

`triangularpdf(x, a, b, c)`

Returns the value of x in the pdf of the Triangular distribution with the parameters a, b, and c.

`triangularcdf(x, a, b, c)`

Returns the value of x in the cdf of the Triangular distribution with the parameters a, b, and c.

`triangularinv(p, a, b, c)`

`triangularmean(a, b, c)`

Returns the value of the mean of the Triangular distribution with the parameters a, b, and c.

`triangularmedian(a, b, c)`

Returns the value of the median of the Triangular distribution with the parameters a, b, and c.

`triangularmode(a, b, c)`

Returns the value of the mode of the Triangular distribution with the parameters a, b, and c.

`triangularsample(a, b, c)`

Returns a random number whose distribution is the Triangular distribution with the parameters a, b, and c.

`triangularvariance(a, b, c)`

Returns the value of the variance of the Triangular distribution with the parameters a, b, and c.

Statistical Tests

`zScore(value, mean, sd)`

Returns the z-score of value given the data from array. flag===true denotes use of the sample standard deviation.

`zScore(value, array)`

Returns the z-score of value given the data from array. flag===true denotes use of the sample standard deviation.

`zTest(value, mean, sd, sides)`

Returns the p-value of value given the data from array. sides is an integer value 1 or 2 denoting a one or two sided z-test. If sides is not specified the test defaults to a two sided z-test. flag===true denotes the use of the sample standard deviation.

`zTest(zscore, sides)`

Returns the p-value of value given the data from array. sides is an integer value 1 or 2 denoting a one or two sided z-test. If sides is not specified the test defaults to a two sided z-test. flag===true denotes the use of the sample standard deviation.

`tScore(value, mean, sd, n)`

Returns the t-score of value given the data from array.

`tScore(value, array)`

Returns the t-score of value given the data from array.

`tTest(value, mean, sd, n, sides)`

Returns the p-value of value given the data in array. sides is an integer value 1 or 2 denoting a one or two sided t-test. If sides is not specified the test defaults to a two sided t-test.

`tTest(tscore, n, sides)`

Returns the p-value of value given the data in array. sides is an integer value 1 or 2 denoting a one or two sided t-test. If sides is not specified the test defaults to a two sided t-test.

`tTest(value, array, sides)`

Returns the p-value of value given the data in array. sides is an integer value 1 or 2 denoting a one or two sided t-test. If sides is not specified the test defaults to a two sided t-test.

`anovaFScore(array1, ..., arrayN)`

Returns the f-score of an ANOVA on the arrays.

`anovaFTest(array1, ..., arrayN)`

Returns the p-value of the f-statistic from the ANOVA test on the arrays.

`ftest(fscore, df1, df2)`

Returns the p-value for the fscore f-score with a df1 numerator degrees of freedom and a df2 denominator degrees of freedom.

Confidence intervals

`normalci(value, alpha, sd, n)`

Returns a 1-alpha confidence interval for value given a normal distribution in the data from array.

`normalci(value, alpha, array)`

Returns a 1-alpha confidence interval for value given a normal distribution in the data from array.

`tci(value, alpha, sd, n)`

Returns a 1-alpha confidence interval for value given the data from array.

`tci(value, alpha, array)`

Returns a 1-alpha confidence interval for value given the data from array.

Special functions

`betafn(x, y)`

Evaluates the Beta function at (x,y).

`betaln(x, y)`

Evaluates the log Beta function at (x,y).

`betacf(x, a, b)`

Returns the continued fraction for the incomplete Beta function with parameters a and b modified by Lentz's method evaluated at x.

`ibetainv(p, a, b)`

Returns the inverse of the incomplete Beta function evaluated at (p,a,b).

`ibeta(x, a, b)`

Returns the incomplete Beta function evaluated at (x,a,b).

`gammaln(x)`

Returns the Log-Gamma function evaluated at x.

`gammafn(x)`

Returns the Gamma function evaluated at x. This is sometimes called the 'complete' gamma function.

`gammap(a, x)`

Returns the lower incomplete gamma function evaluated at (a,x). This function is usually written with a lower case greek gamma character, and is one of the two incomplete gamma functions.

`factorialln(n)`

Returns the natural log factorial of n.

`factorial(n)`

Returns the factorial of n.

`combination(n, m)`

Returns the number of combinations of n, m.

`permutation(permutation)`

Returns the number of permutations of n, m.

`gammapinv(p, a)`

Returns the inverse of the lower regularized incomplete Gamma function evaluated at (p,a). This function is the inverse of lowerRegularizedGamma(x, a).

`erf(x)`

Returns the error function evaluated at x.

`erfc(x)`

Returns the complementary error function evaluated at x.

`erfcinv(p)`

Returns the inverse of the complementary error function evaluated at p.

`randn(n, m)`

Returns a normal deviate (mean 0 and standard deviation 1).

`randg(shape, n, m)`

Returns a Gamma deviate by the method of Marsaglia and Tsang.

Data binning

`bin(data, num_bins, [range])`

Put the list of numbers data into bins of equal size. The number of bins is given by num_bins. If range is given, then the bins span that range; otherwise they span the same range as the given data, with the first bin starting at the minimum value and the last bin ending at the maximum.

Examples:

bin([1,2,3,4,5,6,7], 3) → [ [1,2], [3,4], [5,6,7] ]
bin([1,2,3,4,5,6,7], 3, 1..10) → [ [1,2,3], [4,5,6], [7] ]
bin([1,7,9], 3) → [ [1], [], [7,9] ]

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Statistics extension for Numbas

Functions

Descriptive statistics of a list of numbers

Correlation of two samples

Distributions