Statistics
For the following functions, when the input is a collection, it can take the following forms:
- Multiple arguments, e.g.
["Mean", 1, 2, 3] - A list of numbers, e.g.
["Mean", ["List", 1, 2, 3]] - A matrix, e.g.
["Mean", ["List", ["List", 1, 2], ["List", 3, 4]]] - A range, e.g.
["Mean", ["Range", 1, 10]] - A linear space:
["Mean", ["Linspace", 1, 5, 10]]
Functions
Mean(collection)
Evaluate to the arithmetic mean of a collection of numbers.
The arithmetic mean is the average of the list of numbers. The mean is calculated by dividing the sum of the numbers by the number of numbers in the list.
The formula for the mean of a list of numbers is
\bar{x} = \frac{1}{n}
\sum\_{i=1}^n x_i
n is the number of numbers in the list, and
x_i is the i-th number in the list.["Mean", ["List", 7, 8, 3.1, 12, 77]]
// 21.02
Mean(distribution)
When the argument is a probability distribution, evaluate to the mean of the distribution, as a symbolic expression when the parameters are symbolic.
["Mean", ["NormalDistribution", "mu", "sigma"]]
// mu
Median(collection)
Evaluate to the median of a collection of numbers.
The median is the value separating the higher half from the lower half of a data sample. For a list of numbers sorted in ascending order, the median is the middle value of the list. If the list has an odd number of elements, the median is the middle element. If the list has an even number of elements, the median is the average of the two middle elements.
["Median", ["List", 1, 2, 3, 4, 5]]
// 3
Mode(collection)
Evaluate to the mode of a collection of numbers.
The mode is the value that appears most often in a list of numbers. A list of
numbers can have more than one mode. If there are two modes, the list is called
bimodal. For example \lbrack 2, 5, 5, 3, 2\rbrack. If there are three
modes, the list is called trimodal. If there are more than three modes, the
list is called multimodal.
["Mode", ["List", 1, 2, 2, 3, 4, 4, 5, 5]]
// 2
Variance(collection)
Evaluate to the variance of a collection of numbers.
The variance is a measure of the amount of variation or dispersion of a set of values. A low variance indicates that the values tend to be close to the mean of the set, while a high variance indicates that the values are spread out over a wider range.
The formula for the variance of a list of numbers is
\frac{1}{n} \sum_{i=1}^n(x_i - \mu)^2
where \mu is the mean of the list.
Variance(distribution)
When the argument is a probability distribution, evaluate to the variance of the distribution.
["Variance", ["PoissonDistribution", "lambda"]]
// lambda
PopulationVariance(collection)
Evaluate to the population variance of a collection of numbers.
The population variance is the variance calculated by dividing the sum of squared differences from the mean by the number of elements in the population.
The formula for the population variance is
\frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2
where N is the size of the population, and \mu is the population mean.
StandardDeviation(collection)
Evaluate to the standard deviation of a collection of numbers.
The standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.
The formula for the standard deviation of a collection of numbers is
\sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2}
where \mu is the mean of the list.
StandardDeviation(distribution)
When the argument is a probability distribution, evaluate to the standard deviation of the distribution.
["StandardDeviation", ["NormalDistribution", "mu", "sigma"]]
// sigma
PopulationStandardDeviation(collection)
Evaluate to the population standard deviation of a collection of numbers.
The population standard deviation is the square root of the population variance.
The formula for the population standard deviation is
\sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2}
where N is the size of the population, and \mu is the population mean.
Skewness(collection)
Evaluate to the skewness of a list of numbers.
The skewness is a measure of the asymmetry of the distribution of a real-valued random variable about its mean. The skewness value can be positive or negative, or undefined.
The formula for the skewness of a collection of numbers is:
\frac{1}{n}
\sum_{i=1}^n \left(\frac{x_i - \mu}{\sigma}\right)^3
where \mu is the mean of the collection, and \sigma is the
standard deviation of the collection.
Kurtosis(collection)
Evaluate to the kurtosis of a collection of numbers.
The kurtosis is a measure of the "tailedness" of the distribution of a real-valued random variable. The kurtosis value can be positive or negative, or undefined.
The formula for the kurtosis of a collection of numbers is
\frac{1}{n} \sum_{i=1}^n \left(\frac{x_i - \mu}{\sigma}\right)^4
where \mu is the mean of the list, and \sigma is the standard
deviation of the list.
Covariance(xs:collection, ys:collection)
Covariance(points:collection)
Evaluate to the sample covariance of two data sets: either two
equal-length collections, or a single collection of (x, y) pairs.
\operatorname{cov}(x, y) = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})
Exact data gives an exact result.
["Covariance", ["List", 1, 2, 3, 4, 5], ["List", 2, 4, 5, 4, 5]]
// 3/2
PopulationCovariance(xs:collection, ys:collection)
PopulationCovariance(points:collection)
Evaluate to the population covariance of two data sets: the covariance
with an n denominator instead of n - 1.
\frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})
Correlation(xs:collection, ys:collection)
Correlation(points:collection)
Evaluate to the Pearson correlation coefficient of two data sets: a value
between -1 and 1 measuring the strength of their linear relationship.
r = \frac{\operatorname{cov}(x, y)}{\sigma_x \sigma_y}
Exact data gives an exact result.
["Correlation", ["List", 1, 2, 3, 4, 5], ["List", 2, 4, 5, 4, 5]]
// sqrt(15) / 5
LinearRegression(xs:collection, ys:collection)
LinearRegression(points:collection)
LinearRegression(..., variable:symbol)
Evaluate to the least-squares linear fit of the data, as the tuple
(b_0, b_1) of the intercept and slope of the fitted line
y = b_0 + b_1 x.
Exact data yields exact coefficients.
["LinearRegression", ["List", 1, 2, 3, 4, 5], ["List", 2, 4, 5, 4, 5]]
// (11/5, 3/5)
With a trailing variable argument, evaluate to the fitted expression in that variable instead, ready to plot alongside the data.
["LinearRegression", ["List", 1, 2, 3, 4, 5], ["List", 2, 4, 5, 4, 5], "x"]
// 3/5 x + 11/5
PolynomialFit(xs:collection, ys:collection, degree:number)
PolynomialFit(points:collection, degree:number)
PolynomialFit(..., variable:symbol)
Evaluate to the least-squares polynomial fit of the data at the given degree, as the list of coefficients, constant term first.
Exact data yields exact coefficients: points lying exactly on a polynomial recover it exactly.
["PolynomialFit",
["List", ["List", 0, 1], ["List", 1, 2], ["List", 2, 5], ["List", 3, 10]],
2]
// [1, 0, 1] (the polynomial 1 + x²)
With a trailing variable argument, evaluate to the fitted expression in that variable instead.
The degree must satisfy \deg \le \min(n - 1, 12) where n is the
number of data points.
Quartiles(collection)
Evaluate to the quartiles of a collection of numbers.
The quartiles are the three points that divide a collection of numbers into four equal groups, each group comprising a quarter of the collection.
["Quartiles", ["List", 1, 2, 3, 4, 5, 6, 7, 8]]
// [2.5, 4.5, 6.5]
InterquartileRange(collection)
Evaluate to the interquartile range (IQR) of a collection of numbers.
The interquartile range is the difference between the third quartile and the first quartile.
Histogram(collection, bins:number)
Evaluate to the histogram of a collection of numbers.
The histogram groups the data into a specified number of bins and counts the number of elements in each bin. The bins argument is an integer bin count or a list of explicit bin edges; a non-integer bin count is left unevaluated (a Desmos-style bin width should be translated to explicit bin edges).
The LaTeX \operatorname{histogram} parses to Histogram (likewise
\operatorname{pdf} and \operatorname{cdf} parse to PDF and CDF).
["Histogram", ["List", 1, 2, 2, 3, 4, 5, 5, 5], 3]
// [[1, 3], [2.3333…, 1], [3.6667…, 4]] (bin start, count)
BinCounts(collection, bins:number)
Evaluate to the bin counts of a collection of numbers.
Bin counts are the counts of the number of elements in each bin for a given number of bins.
["BinCounts", ["List", 1, 2, 2, 3, 4, 5, 5, 5], 3]
// [3, 1, 4]
SlidingWindow(collection, windowSize:number)
Evaluate to the sliding windows of a collection of numbers.
A sliding window is a moving subset of the data of a specified window size.
["SlidingWindow", ["List", 1, 2, 3, 4, 5], 3]
// [[1, 2, 3], [2, 3, 4], [3, 4, 5]]
Sample(collection, size:number)
Evaluate to a random sample of a specified size from a collection of numbers.
Sampling is done without replacement unless otherwise specified.
Rank(collection)
Evaluate to the rank of each element in a collection of numbers.
The rank is the position of each element in the sorted order of the collection.
Argsort(collection)
Evaluate to the indices that would sort a collection of numbers.
This returns a list of indices that sorts the collection.
Probability Distributions
A probability distribution is a first-class value constructed with one of
the distribution functions below. It can be assigned to a symbol, passed as an
argument, and queried with PDF, CDF, Quantile, Mean, Variance and
StandardDeviation.
These operators evaluate to exact closed forms: the result is an ordinary
expression that can be simplified, differentiated, compiled or plotted, and
numeric approximation (.N()) works at machine or arbitrary precision.
["CDF", ["NormalDistribution", 0, 1], "x"]
// 1/2 + Erf(x / sqrt(2)) / 2
NormalDistribution(mean, standardDeviation)
A normal (Gaussian) distribution with the given mean \mu and standard
deviation \sigma > 0.
Note that the second parameter is the standard deviation, not the variance, following the convention of most computer algebra and statistics systems.
BinomialDistribution(n, p)
A binomial distribution: the number of successes in n \ge 0 independent
trials, each succeeding with probability 0 \le p \le 1.
PoissonDistribution(lambda)
A Poisson distribution: the number of events occurring in a fixed interval
when events occur independently at a mean rate of \lambda > 0.
UniformDistribution(a, b)
A continuous uniform distribution on the interval [a, b], with
a < b.
ExponentialDistribution(lambda)
An exponential distribution: the waiting time between events occurring at
a mean rate of \lambda > 0 (the rate parameterization; the mean is
1/\lambda).
PDF(distribution, x)
Evaluate to the probability density function of distribution at x.
For a discrete distribution, this is the probability mass function, and it
is 0 at numeric non-integer points.
The result is a closed-form expression: with a symbolic x it is the density formula itself, ready to plot or integrate.
["PDF", ["BinomialDistribution", 4, ["Rational", 1, 2]], 2]
// 3/8
CDF(distribution, x)
Evaluate to the cumulative distribution function of distribution at x: the probability that a random value is less than or equal to x.
The result is exact: a normal CDF is expressed with Erf, and discrete CDFs
with GammaRegularized/BetaRegularized. Use N() for a numeric value.
["N", ["CDF", ["NormalDistribution", 0, 1], 1]]
// 0.8413447460685429
Quantile(distribution, p:number)
Evaluate to the quantile (inverse CDF) of distribution: the smallest
value x such that \operatorname{CDF}(x) \ge p, for 0 \le p \le 1.
Quantile(dist, 0.5) is the median of the distribution.
Quantile(collection, p:number)
When the first argument is a collection of numbers, evaluate to the
empirical quantile of the data, interpolating the sorted values. It is
consistent with Quartiles and Median: Quantile(xs, 1/4),
Quantile(xs, 1/2) and Quantile(xs, 3/4) equal the three quartiles, and
p = 0 and p = 1 give the minimum and maximum.
["Quantile", ["PoissonDistribution", 9], 0.95]
// 14
For continuous distributions the result is a closed form
(Quantile(NormalDistribution(mu, sigma), p) is
\mu + \sigma\sqrt2\operatorname{erf}^{-1}(2p-1)); for discrete
distributions it is computed by exact search when p is numeric.