## Basic Descriptive Statistics

### min

The min is the lowest number in the array. This runs on `O(n)`, linear time in respect to the array

min(x: Array<number>): number
Parameters
x `(Array<number>)` sample of one or more data points
Returns
`number`: minimum value
Throws
• Error: if the the length of x is less than one
Example
`min([1, 5, -10, 100, 2]); // => -10`

### max

This computes the maximum number in an array.

This runs on `O(n)`, linear time in respect to the array

max(x: Array<number>): number
Parameters
x `(Array<number>)` sample of one or more data points
Returns
`number`: maximum value
Throws
• Error: if the the length of x is less than one
Example
```max([1, 2, 3, 4]);
// => 4```

### sum

Our default sum is the Kahan-Babuska algorithm. This method is an improvement over the classical Kahan summation algorithm. It aims at computing the sum of a list of numbers while correcting for floating-point errors. Traditionally, sums are calculated as many successive additions, each one with its own floating-point roundoff. These losses in precision add up as the number of numbers increases. This alternative algorithm is more accurate than the simple way of calculating sums by simple addition.

This runs on `O(n)`, linear time in respect to the array.

sum(x: Array<number>): number
Parameters
x `(Array<number>)` input
Returns
`number`: sum of all input numbers
Example
`sum([1, 2, 3]); // => 6`

### sumSimple

The simple sum of an array is the result of adding all numbers together, starting from zero.

This runs on `O(n)`, linear time in respect to the array

sumSimple(x: Array<number>): number
Parameters
x `(Array<number>)` input
Returns
`number`: sum of all input numbers
Example
`sumSimple([1, 2, 3]); // => 6`

### quantile

The quantile: this is a population quantile, since we assume to know the entire dataset in this library. This is an implementation of the Quantiles of a Population algorithm from wikipedia.

Sample is a one-dimensional array of numbers, and p is either a decimal number from 0 to 1 or an array of decimal numbers from 0 to 1. In terms of a k/q quantile, p = k/q - it's just dealing with fractions or dealing with decimal values. When p is an array, the result of the function is also an array containing the appropriate quantiles in input order

quantile(x: Array<number>, p: (Array<number> | number)): number
Parameters
x `(Array<number>)` sample of one or more numbers
p `((Array<number> | number))` the desired quantile, as a number between 0 and 1
Returns
`number`: quantile
Example
`quantile([3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20], 0.5); // => 9`

### product

The product of an array is the result of multiplying all numbers together, starting using one as the multiplicative identity.

This runs on `O(n)`, linear time in respect to the array

product(x: Array<number>): number
Parameters
x `(Array<number>)` input
Returns
`number`: product of all input numbers
Example
`product([1, 2, 3, 4]); // => 24`

## Sorted Basic Descriptive Statistics

These are special versions of methods that assume your input is sorted. This assumptions lets them run a lot faster, usually in O(1).

### minSorted

The minimum is the lowest number in the array. With a sorted array, the first element in the array is always the smallest, so this calculation can be done in one step, or constant time.

minSorted(x: Array<number>): number
Parameters
x `(Array<number>)` input
Returns
`number`: minimum value
Example
`minSorted([-100, -10, 1, 2, 5]); // => -100`

### maxSorted

The maximum is the highest number in the array. With a sorted array, the last element in the array is always the largest, so this calculation can be done in one step, or constant time.

maxSorted(x: Array<number>): number
Parameters
x `(Array<number>)` input
Returns
`number`: maximum value
Example
`maxSorted([-100, -10, 1, 2, 5]); // => 5`

### quantileSorted

This is the internal implementation of quantiles: when you know that the order is sorted, you don't need to re-sort it, and the computations are faster.

quantileSorted(x: Array<number>, p: number): number
Parameters
x `(Array<number>)` sample of one or more data points
p `(number)` desired quantile: a number between 0 to 1, inclusive
Returns
`number`: quantile value
Throws
• Error: if p ix outside of the range from 0 to 1
• Error: if x is empty
Example
`quantileSorted([3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20], 0.5); // => 9`

## Measures of central tendency

These are different ways to identifying centers or locations of a distribution.

### mean

The mean, also known as average, is the sum of all values over the number of values. This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

This runs on `O(n)`, linear time in respect to the array

mean(x: Array<number>): number
Parameters
x `(Array<number>)` sample of one or more data points
Returns
`number`: mean
Throws
• Error: if the the length of x is less than one
Example
`mean([0, 10]); // => 5`

When adding a new value to a list, one does not have to necessary recompute the mean of the list in linear time. They can instead use this function to compute the new mean by providing the current mean, the number of elements in the list that produced it and the new value to add.

addToMean(mean: number, n: number, newValue: number): number
Since: 2.5.0
Parameters
mean `(number)` current mean
n `(number)` number of items in the list
newValue `(number)` the added value
Returns
`number`: the new mean
Example
`addToMean(14, 5, 53); // => 20.5`

### mode

The mode is the number that appears in a list the highest number of times. There can be multiple modes in a list: in the event of a tie, this algorithm will return the most recently seen mode.

This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

This runs on `O(nlog(n))` because it needs to sort the array internally before running an `O(n)` search to find the mode.

mode(x: Array<number>): number
Parameters
x `(Array<number>)` input
Returns
`number`: mode
Example
`mode([0, 0, 1]); // => 0`

### modeSorted

The mode is the number that appears in a list the highest number of times. There can be multiple modes in a list: in the event of a tie, this algorithm will return the most recently seen mode.

This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

This runs in `O(n)` because the input is sorted.

modeSorted(sorted: Array<number>): number
Parameters
sorted `(Array<number>)` a sample of one or more data points
Returns
`number`: mode
Throws
• Error: if sorted is empty
Example
`modeSorted([0, 0, 1]); // => 0`

### modeFast

The mode is the number that appears in a list the highest number of times. There can be multiple modes in a list: in the event of a tie, this algorithm will return the most recently seen mode.

modeFast uses a Map object to keep track of the mode, instead of the approach used with `mode`, a sorted array. As a result, it is faster than `mode` and supports any data type that can be compared with `==`. It also requires a JavaScript environment with support for Map, and will throw an error if Map is not available.

This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

modeFast(x: Array<any>): any?
Parameters
x `(Array<any>)` a sample of one or more data points
Returns
`any?`: mode
Throws
Example
`modeFast(['rabbits', 'rabbits', 'squirrels']); // => 'rabbits'`

### median

The median is the middle number of a list. This is often a good indicator of 'the middle' when there are outliers that skew the `mean()` value. This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

The median isn't necessarily one of the elements in the list: the value can be the average of two elements if the list has an even length and the two central values are different.

median(x: Array<number>): number
Parameters
x `(Array<number>)` input
Returns
`number`: median value
Example
`median([10, 2, 5, 100, 2, 1]); // => 3.5`

### medianSorted

The median is the middle number of a list. This is often a good indicator of 'the middle' when there are outliers that skew the `mean()` value. This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

The median isn't necessarily one of the elements in the list: the value can be the average of two elements if the list has an even length and the two central values are different.

medianSorted(sorted: Array<number>): number
Parameters
sorted `(Array<number>)` input
Returns
`number`: median value
Example
`medianSorted([10, 2, 5, 100, 2, 1]); // => 52.5`

### harmonicMean

The Harmonic Mean is a mean function typically used to find the average of rates. This mean is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of the input numbers.

This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

This runs on `O(n)`, linear time in respect to the array.

harmonicMean(x: Array<number>): number
Parameters
x `(Array<number>)` sample of one or more data points
Returns
`number`: harmonic mean
Throws
• Error: if x is empty
• Error: if x contains a negative number
Example
`harmonicMean([2, 3]).toFixed(2) // => '2.40'`

### geometricMean

The Geometric Mean is a mean function that is more useful for numbers in different ranges.

This is the nth root of the input numbers multiplied by each other.

The geometric mean is often useful for proportional growth: given growth rates for multiple years, like 80%, 16.66% and 42.85%, a simple mean will incorrectly estimate an average growth rate, whereas a geometric mean will correctly estimate a growth rate that, over those years, will yield the same end value.

This runs on `O(n)`, linear time in respect to the array

geometricMean(x: Array<number>): number
Parameters
x `(Array<number>)` sample of one or more data points
Returns
`number`: geometric mean
Throws
• Error: if x is empty
• Error: if x contains a negative number
Example
```var growthRates = [1.80, 1.166666, 1.428571];
var averageGrowth = ss.geometricMean(growthRates);
var averageGrowthRates = [averageGrowth, averageGrowth, averageGrowth];
var startingValue = 10;
var startingValueMean = 10;
growthRates.forEach(function(rate) {
startingValue *= rate;
});
averageGrowthRates.forEach(function(rate) {
startingValueMean *= rate;
});
startingValueMean === startingValue;```

### rootMeanSquare

The Root Mean Square (RMS) is a mean function used as a measure of the magnitude of a set of numbers, regardless of their sign. This is the square root of the mean of the squares of the input numbers. This runs on `O(n)`, linear time in respect to the array

rootMeanSquare(x: Array<number>): number
Parameters
x `(Array<number>)` a sample of one or more data points
Returns
`number`: root mean square
Throws
Example
`rootMeanSquare([-1, 1, -1, 1]); // => 1`

### sampleSkewness

Skewness is a measure of the extent to which a probability distribution of a real-valued random variable "leans" to one side of the mean. The skewness value can be positive or negative, or even undefined.

Implementation is based on the adjusted Fisher-Pearson standardized moment coefficient, which is the version found in Excel and several statistical packages including Minitab, SAS and SPSS.

sampleSkewness(x: Array<number>): number
Since: 4.1.0
Parameters
x `(Array<number>)` a sample of 3 or more data points
Returns
`number`: sample skewness
Throws
• Error: if x has length less than 3
Example
`sampleSkewness([2, 4, 6, 3, 1]); // => 0.590128656384365`

## Measures of dispersion

These are different ways of determining how spread out a distribution is.

### variance

The variance is the sum of squared deviations from the mean.

This is an implementation of variance, not sample variance: see the `sampleVariance` method if you want a sample measure.

variance(x: Array<number>): number
Parameters
x `(Array<number>)` a population of one or more data points
Returns
`number`: variance: a value greater than or equal to zero. zero indicates that all values are identical.
Throws
• Error: if x's length is 0
Example
`variance([1, 2, 3, 4, 5, 6]); // => 2.9166666666666665`

### sampleVariance

The sample variance is the sum of squared deviations from the mean. The sample variance is distinguished from the variance by the usage of Bessel's Correction: instead of dividing the sum of squared deviations by the length of the input, it is divided by the length minus one. This corrects the bias in estimating a value from a set that you don't know if full.

References:

sampleVariance(x: Array<number>): number
Parameters
x `(Array<number>)` a sample of two or more data points
Returns
`number`: sample variance
Throws
• Error: if the length of x is less than 2
Example
`sampleVariance([1, 2, 3, 4, 5]); // => 2.5`

### standardDeviation

The standard deviation is the square root of the variance. This is also known as the population standard deviation. It's useful for measuring the amount of variation or dispersion in a set of values.

Standard deviation is only appropriate for full-population knowledge: for samples of a population, sampleStandardDeviation is more appropriate.

standardDeviation(x: Array<number>): number
Parameters
x `(Array<number>)` input
Returns
`number`: standard deviation
Example
```variance([2, 4, 4, 4, 5, 5, 7, 9]); // => 4
standardDeviation([2, 4, 4, 4, 5, 5, 7, 9]); // => 2```

### sampleStandardDeviation

The sample standard deviation is the square root of the sample variance.

sampleStandardDeviation(x: Array<number>): number
Parameters
x `(Array<number>)` input array
Returns
`number`: sample standard deviation
Example
```sampleStandardDeviation([2, 4, 4, 4, 5, 5, 7, 9]).toFixed(2);
// => '2.14'```

### medianAbsoluteDeviation

The Median Absolute Deviation is a robust measure of statistical dispersion. It is more resilient to outliers than the standard deviation.

medianAbsoluteDeviation(x: Array<number>): number
Parameters
x `(Array<number>)` input array
Returns
`number`: median absolute deviation
Example
`medianAbsoluteDeviation([1, 1, 2, 2, 4, 6, 9]); // => 1`

### interquartileRange

The Interquartile range is a measure of statistical dispersion, or how scattered, spread, or concentrated a distribution is. It's computed as the difference between the third quartile and first quartile.

interquartileRange(x: Array<number>): number
Parameters
x `(Array<number>)` sample of one or more numbers
Returns
`number`: interquartile range: the span between lower and upper quartile, 0.25 and 0.75
Example
`interquartileRange([0, 1, 2, 3]); // => 2`

### sumNthPowerDeviations

The sum of deviations to the Nth power. When n=2 it's the sum of squared deviations. When n=3 it's the sum of cubed deviations.

sumNthPowerDeviations(x: Array<number>, n: number): number
Parameters
x `(Array<number>)`
n `(number)` power
Returns
`number`: sum of nth power deviations
Example
```var input = [1, 2, 3];
// since the variance of a set is the mean squared
// deviations, we can calculate that with sumNthPowerDeviations:
sumNthPowerDeviations(input, 2) / input.length;```

### zScore

The standard score is the number of standard deviations an observation or datum is above or below the mean. Thus, a positive standard score represents a datum above the mean, while a negative standard score represents a datum below the mean. It is a dimensionless quantity obtained by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation.

The z-score is only defined if one knows the population parameters; if one only has a sample set, then the analogous computation with sample mean and sample standard deviation yields the Student's t-statistic.

zScore(x: number, mean: number, standardDeviation: number): number
Parameters
x `(number)`
mean `(number)`
standardDeviation `(number)`
Returns
`number`: z score
Example
`zScore(78, 80, 5); // => -0.4`

## Similarity

### sampleCorrelation

The correlation is a measure of how correlated two datasets are, between -1 and 1

sampleCorrelation(x: Array<number>, y: Array<number>): number
Parameters
x `(Array<number>)` first input
y `(Array<number>)` second input
Returns
`number`: sample correlation
Example
```sampleCorrelation([1, 2, 3, 4, 5, 6], [2, 2, 3, 4, 5, 60]).toFixed(2);
// => '0.69'```

### sampleCovariance

Sample covariance of two datasets: how much do the two datasets move together? x and y are two datasets, represented as arrays of numbers.

sampleCovariance(x: Array<number>, y: Array<number>): number
Parameters
x `(Array<number>)` a sample of two or more data points
y `(Array<number>)` a sample of two or more data points
Returns
`number`: sample covariance
Throws
• Error: if x and y do not have equal lengths
• Error: if x or y have length of one or less
Example
`sampleCovariance([1, 2, 3, 4, 5, 6], [6, 5, 4, 3, 2, 1]); // => -3.5`

### rSquared

The R Squared value of data compared with a function `f` is the sum of the squared differences between the prediction and the actual value.

rSquared(x: Array<Array<number>>, func: Function): number
Parameters
x `(Array<Array<number>>)` input data: this should be doubly-nested
func `(Function)` function called on `[i]` values within the dataset
Returns
`number`: r-squared value
Example
```var samples = [[0, 0], [1, 1]];
var regressionLine = linearRegressionLine(linearRegression(samples));
rSquared(samples, regressionLine); // = 1 this line is a perfect fit```

## Linear Regression

### linearRegression

Simple linear regression is a simple way to find a fitted line between a set of coordinates. This algorithm finds the slope and y-intercept of a regression line using the least sum of squares.

linearRegression(data: Array<Array<number>>): Object
Parameters
data `(Array<Array<number>>)` an array of two-element of arrays, like `[[0, 1], [2, 3]]`
Returns
`Object`: object containing slope and intersect of regression line
Example
`linearRegression([[0, 0], [1, 1]]); // => { m: 1, b: 0 }`

### linearRegressionLine

Given the output of `linearRegression`: an object with `m` and `b` values indicating slope and intercept, respectively, generate a line function that translates x values into y values.

linearRegressionLine(mb: Object): Function
Parameters
mb `(Object)` object with `m` and `b` members, representing slope and intersect of desired line
Returns
`Function`: method that computes y-value at any given x-value on the line.
Example
```var l = linearRegressionLine(linearRegression([[0, 0], [1, 1]]));
l(0) // = 0
l(2) // = 2
linearRegressionLine({ b: 0, m: 1 })(1); // => 1
linearRegressionLine({ b: 1, m: 1 })(1); // => 2```

## Randomness

### shuffle

A Fisher-Yates shuffle is a fast way to create a random permutation of a finite set. This is a function around `shuffle_in_place` that adds the guarantee that it will not modify its input.

shuffle(x: Array, randomSource: Function): Array
Parameters
x `(Array)` sample of 0 or more numbers
randomSource ```(Function = Math.random)``` an optional entropy source that returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)
Returns
`Array`: shuffled version of input
Example
```var shuffled = shuffle([1, 2, 3, 4]);
shuffled; // = [2, 3, 1, 4] or any other random permutation```

### shuffleInPlace

A Fisher-Yates shuffle in-place - which means that it will change the order of the original array by reference.

This is an algorithm that generates a random permutation of a set.

shuffleInPlace(x: Array, randomSource: Function): Array
Parameters
x `(Array)` sample of one or more numbers
randomSource ```(Function = Math.random)``` an optional entropy source that returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)
Returns
`Array`: x
Example
```var x = [1, 2, 3, 4];
shuffleInPlace(x);
// x is shuffled to a value like [2, 1, 4, 3]```

### sampleWithReplacement

Sampling with replacement is a type of sampling that allows the same item to be picked out of a population more than once.

sampleWithReplacement(x: Array<any>, n: number, randomSource: Function): Array
Parameters
x `(Array<any>)` an array of any kind of value
n `(number)` count of how many elements to take
randomSource ```(Function = Math.random)``` an optional entropy source that returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)
Returns
`Array`: n sampled items from the population
Example
```var values = [1, 2, 3, 4];
sampleWithReplacement(values, 2); // returns 2 random values, like [2, 4];```

### sample

Create a simple random sample from a given array of `n` elements.

The sampled values will be in any order, not necessarily the order they appear in the input.

sample(x: Array<any>, n: number, randomSource: Function): Array
Parameters
x `(Array<any>)` input array. can contain any type
n `(number)` count of how many elements to take
randomSource ```(Function = Math.random)``` an optional entropy source that returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)
Returns
`Array`: subset of n elements in original array
Example
```var values = [1, 2, 4, 5, 6, 7, 8, 9];
sample(values, 3); // returns 3 random values, like [2, 5, 8];```

## Classifiers

### BayesianClassifier

Bayesian Classifier

This is a naïve bayesian classifier that takes singly-nested objects.

new BayesianClassifier()
Example
```var bayes = new BayesianClassifier();
bayes.train({
species: 'Cat'
}, 'animal');
var result = bayes.score({
species: 'Cat'
})
// result
// {
//   animal: 1
// }```
Instance Members
train(item, category)
score(item)

### PerceptronModel

This is a single-layer Perceptron Classifier that takes arrays of numbers and predicts whether they should be classified as either 0 or 1 (negative or positive examples).

new PerceptronModel()
Example
```// Create the model
var p = new PerceptronModel();
// Train the model with input with a diagonal boundary.
for (var i = 0; i < 5; i++) {
p.train([1, 1], 1);
p.train([0, 1], 0);
p.train([1, 0], 0);
p.train([0, 0], 0);
}
p.predict([0, 0]); // 0
p.predict([0, 1]); // 0
p.predict([1, 0]); // 0
p.predict([1, 1]); // 1```
Instance Members
predict(features)
train(features, label)

## Distributions

### bernoulliDistribution

The Bernoulli distribution is the probability discrete distribution of a random variable which takes value 1 with success probability `p` and value 0 with failure probability `q` = 1 - `p`. It can be used, for example, to represent the toss of a coin, where "1" is defined to mean "heads" and "0" is defined to mean "tails" (or vice versa). It is a special case of a Binomial Distribution where `n` = 1.

bernoulliDistribution(p: number): Array<number>
Parameters
p `(number)` input value, between 0 and 1 inclusive
Returns
`Array<number>`: values of bernoulli distribution at this point
Throws
• Error: if p is outside 0 and 1
Example
`bernoulliDistribution(0.3); // => [0.7, 0.3]`

### binomialDistribution

The Binomial Distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability `probability`. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when trials = 1, the Binomial Distribution is a Bernoulli Distribution.

binomialDistribution(trials: number, probability: number): Array<number>
Parameters
trials `(number)` number of trials to simulate
probability `(number)`
Returns
`Array<number>`: output

### poissonDistribution

The Poisson Distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.

The Poisson Distribution is characterized by the strictly positive mean arrival or occurrence rate, `λ`.

poissonDistribution(lambda: number): Array<number>
Parameters
lambda `(number)` location poisson distribution
Returns
`Array<number>`: values of poisson distribution at that point

### chiSquaredDistributionTable

Percentage Points of the χ2 (Chi-Squared) Distribution

The χ2 (Chi-Squared) Distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation.

Values from Appendix 1, Table III of William W. Hines & Douglas C. Montgomery, "Probability and Statistics in Engineering and Management Science", Wiley (1980).

chiSquaredDistributionTable

### standardNormalTable

A standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of Φ (phi), which are the values of the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic is observed below, above, or between values on the standard normal distribution, and by extension, any normal distribution.

The probabilities are calculated using the Cumulative distribution function. The table used is the cumulative, and not cumulative from 0 to mean (even though the latter has 5 digits precision, instead of 4).

standardNormalTable

### tTest

This is to compute a one-sample t-test, comparing the mean of a sample to a known value, x.

in this case, we're trying to determine whether the population mean is equal to the value that we know, which is `x` here. usually the results here are used to look up a p-value, which, for a certain level of significance, will let you determine that the null hypothesis can or cannot be rejected.

tTest(x: Array<number>, expectedValue: number): number
Parameters
x `(Array<number>)` sample of one or more numbers
expectedValue `(number)` expected value of the population mean
Returns
`number`: value
Example
`tTest([1, 2, 3, 4, 5, 6], 3.385).toFixed(2); // => '0.16'`

### tTestTwoSample

This is to compute two sample t-test. Tests whether "mean(X)-mean(Y) = difference", ( in the most common case, we often have `difference == 0` to test if two samples are likely to be taken from populations with the same mean value) with no prior knowledge on standard deviations of both samples other than the fact that they have the same standard deviation.

Usually the results here are used to look up a p-value, which, for a certain level of significance, will let you determine that the null hypothesis can or cannot be rejected.

`diff` can be omitted if it equals 0.

This is used to confirm or deny a null hypothesis that the two populations that have been sampled into `sampleX` and `sampleY` are equal to each other.

tTestTwoSample(sampleX: Array<number>, sampleY: Array<number>, difference: number): (number | null)
Parameters
sampleX `(Array<number>)` a sample as an array of numbers
sampleY `(Array<number>)` a sample as an array of numbers
difference ```(number = 0)```
Returns
`(number | null)`: test result
Example
`tTestTwoSample([1, 2, 3, 4], [3, 4, 5, 6], 0); // => -2.1908902300206643`

### cumulativeStdNormalProbability

Cumulative Standard Normal Probability

Since probability tables cannot be printed for every normal distribution, as there are an infinite variety of normal distributions, it is common practice to convert a normal to a standard normal and then use the standard normal table to find probabilities.

You can use `.5 + .5 * errorFunction(x / Math.sqrt(2))` to calculate the probability instead of looking it up in a table.

cumulativeStdNormalProbability(z: number): number
Parameters
z `(number)`
Returns
`number`: cumulative standard normal probability

### kernelDensityEstimation

Kernel density estimation is a useful tool for, among other things, estimating the shape of the underlying probability distribution from a sample.

kernelDensityEstimation
Parameters
X `(any)` sample values
kernel `(any)` The kernel function to use. If a function is provided, it should return non-negative values and integrate to 1. Defaults to 'gaussian'.
bandwidthMethod `(any)` The "bandwidth selection" method to use, or a fixed bandwidth value. Defaults to "nrd", the commonly-used "normal reference distribution" rule-of-thumb .
Returns
`Function`: An estimated probability density function for the given sample. The returned function runs in `O(X.length)` .

## Errors

### errorFunction

Gaussian error function

The `errorFunction(x/(sd * Math.sqrt(2)))` is the probability that a value in a normal distribution with standard deviation sd is within x of the mean.

This function returns a numerical approximation to the exact value.

errorFunction(x: number): number
Parameters
x `(number)` input
Returns
`number`: error estimation
Example
`errorFunction(1).toFixed(2); // => '0.84'`

### inverseErrorFunction

The Inverse Gaussian error function returns a numerical approximation to the value that would have caused `errorFunction()` to return x.

inverseErrorFunction(x: number): number
Parameters
x `(number)` value of error function
Returns
`number`: estimated inverted value

### probit

The Probit is the inverse of cumulativeStdNormalProbability(), and is also known as the normal quantile function.

It returns the number of standard deviations from the mean where the p'th quantile of values can be found in a normal distribution. So, for example, probit(0.5 + 0.6827/2) ≈ 1 because 68.27% of values are normally found within 1 standard deviation above or below the mean.

probit(p: number): number
Parameters
p `(number)`
Returns
`number`: probit

## Breaks

Breaks methods split datasets into chunks. Often these are used for segmentation or visualization of a dataset. A method of computing breaks that splits data evenly can make for a better choropleth map, for instance, because each color will be represented equally.

### ckmeans

Ckmeans clustering is an improvement on heuristic-based clustering approaches like Jenks. The algorithm was developed in Haizhou Wang and Mingzhou Song as a dynamic programming approach to the problem of clustering numeric data into groups with the least within-group sum-of-squared-deviations.

Minimizing the difference within groups - what Wang & Song refer to as `withinss`, or within sum-of-squares, means that groups are optimally homogenous within and the data is split into representative groups. This is very useful for visualization, where you may want to represent a continuous variable in discrete color or style groups. This function can provide groups that emphasize differences between data.

Being a dynamic approach, this algorithm is based on two matrices that store incrementally-computed values for squared deviations and backtracking indexes.

This implementation is based on Ckmeans 3.4.6, which introduced a new divide and conquer approach that improved runtime from O(kn^2) to O(kn log(n)).

Unlike the original implementation, this implementation does not include any code to automatically determine the optimal number of clusters: this information needs to be explicitly provided.

### References

Ckmeans.1d.dp: Optimal k-means Clustering in One Dimension by Dynamic Programming Haizhou Wang and Mingzhou Song ISSN 2073-4859

from The R Journal Vol. 3/2, December 2011

ckmeans(x: Array<number>, nClusters: number): Array<Array<number>>
Parameters
x `(Array<number>)` input data, as an array of number values
nClusters `(number)` number of desired classes. This cannot be greater than the number of values in the data array.
Returns
`Array<Array<number>>`: clustered input
Throws
• Error: if the number of requested clusters is higher than the size of the data
Example
```ckmeans([-1, 2, -1, 2, 4, 5, 6, -1, 2, -1], 3);
// The input, clustered into groups of similar numbers.
//= [[-1, -1, -1, -1], [2, 2, 2], [4, 5, 6]]);```

### equalIntervalBreaks

Given an array of x, this will find the extent of the x and return an array of breaks that can be used to categorize the x into a number of classes. The returned array will always be 1 longer than the number of classes because it includes the minimum value.

equalIntervalBreaks(x: Array<number>, nClasses: number): Array<number>
Parameters
x `(Array<number>)` an array of number values
nClasses `(number)` number of desired classes
Returns
`Array<number>`: array of class break positions
Example
`equalIntervalBreaks([1, 2, 3, 4, 5, 6], 4); // => [1, 2.25, 3.5, 4.75, 6]`

## Utilities

### chunk

Split an array into chunks of a specified size. This function has the same behavior as PHP's array_chunk function, and thus will insert smaller-sized chunks at the end if the input size is not divisible by the chunk size.

`x` is expected to be an array, and `chunkSize` a number. The `x` array can contain any kind of data.

chunk(x: Array, chunkSize: number): Array<Array>
Parameters
x `(Array)` a sample
chunkSize `(number)` size of each output array. must be a positive integer
Returns
`Array<Array>`: a chunked array
Throws
• Error: if chunk size is less than 1 or not an integer
Example
```chunk([1, 2, 3, 4, 5, 6], 2);
// => [[1, 2], [3, 4], [5, 6]]```

### chiSquaredGoodnessOfFit

The χ2 (Chi-Squared) Goodness-of-Fit Test uses a measure of goodness of fit which is the sum of differences between observed and expected outcome frequencies (that is, counts of observations), each squared and divided by the number of observations expected given the hypothesized distribution. The resulting χ2 statistic, `chiSquared`, can be compared to the chi-squared distribution to determine the goodness of fit. In order to determine the degrees of freedom of the chi-squared distribution, one takes the total number of observed frequencies and subtracts the number of estimated parameters. The test statistic follows, approximately, a chi-square distribution with (k − c) degrees of freedom where `k` is the number of non-empty cells and `c` is the number of estimated parameters for the distribution.

chiSquaredGoodnessOfFit(data: Array<number>, distributionType: Function, significance: number): number
Parameters
data `(Array<number>)`
distributionType `(Function)` a function that returns a point in a distribution: for instance, binomial, bernoulli, or poisson
significance `(number)`
Returns
`number`: chi squared goodness of fit
Example
```// Data from Poisson goodness-of-fit example 10-19 in William W. Hines & Douglas C. Montgomery,
// "Probability and Statistics in Engineering and Management Science", Wiley (1980).
var data1019 = [
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
2, 2, 2, 2, 2, 2, 2, 2, 2,
3, 3, 3, 3
];
ss.chiSquaredGoodnessOfFit(data1019, ss.poissonDistribution, 0.05); //= false```

### epsilon

We use `ε`, epsilon, as a stopping criterion when we want to iterate until we're "close enough". Epsilon is a very small number: for simple statistics, that number is 0.0001

This is used in calculations like the binomialDistribution, in which the process of finding a value is iterative: it progresses until it is close enough.

Below is an example of using epsilon in gradient descent, where we're trying to find a local minimum of a function's derivative, given by the `fDerivative` method.

epsilon
Example
```// From calculation, we expect that the local minimum occurs at x=9/4
var x_old = 0;
// The algorithm starts at x=6
var x_new = 6;
var stepSize = 0.01;

function fDerivative(x) {
return 4 * Math.pow(x, 3) - 9 * Math.pow(x, 2);
}

// The loop runs until the difference between the previous
// value and the current value is smaller than epsilon - a rough
// meaure of 'close enough'
while (Math.abs(x_new - x_old) > ss.epsilon) {
x_old = x_new;
x_new = x_old - stepSize * fDerivative(x_old);
}

console.log('Local minimum occurs at', x_new);```

### factorial

A Factorial, usually written n!, is the product of all positive integers less than or equal to n. Often factorial is implemented recursively, but this iterative approach is significantly faster and simpler.

factorial(n: number): number
Parameters
n `(number)` input, must be an integer number 1 or greater
Returns
`number`: factorial: n!
Throws
• Error: if n is less than 0 or not an integer
Example
`factorial(5); // => 120`

### gamma

Compute the gamma function of a value using Nemes' approximation. The gamma of n is equivalent to (n-1)!, but unlike the factorial function, gamma is defined for all real n except zero and negative integers (where NaN is returned). Note, the gamma function is also well-defined for complex numbers, though this implementation currently does not handle complex numbers as input values. Nemes' approximation is defined here as Theorem 2.2. Negative values use Euler's reflection formula for computation.

gamma(n: number): number
Parameters
n `(number)` Any real number except for zero and negative integers.
Returns
`number`: The gamma of the input value.
Example
```gamma(11.5); // 11899423.084037038
gamma(-11.5); // 2.29575810481609e-8
gamma(5); // 24```

### uniqueCountSorted

For a sorted input, counting the number of unique values is possible in constant time and constant memory. This is a simple implementation of the algorithm.

Values are compared with `===`, so objects and non-primitive objects are not handled in any special way.

uniqueCountSorted(x: Array<any>): number
Parameters
x `(Array<any>)` an array of any kind of value
Returns
`number`: count of unique values
Example
```uniqueCountSorted([1, 2, 3]); // => 3
uniqueCountSorted([1, 1, 1]); // => 1```

### bisect

Bisection method is a root-finding method that repeatedly bisects an interval to find the root.

This function returns a numerical approximation to the exact value.

bisect(func: Function, start: number, end: number, maxIterations: number, errorTolerance: number): number
Parameters
func `(Function)` input function
start `(number)` start of interval
end `(number)` end of interval
maxIterations `(number)` the maximum number of iterations
errorTolerance `(number)` the error tolerance
Returns
`number`: estimated root value
Throws
• TypeError: Argument func must be a function
Example
`bisect(Math.cos,0,4,100,0.003); // => 1.572265625`

### combinationsReplacement

Implementation of Combinations with replacement Combinations are unique subsets of a collection - in this case, k x from a collection at a time. 'With replacement' means that a given element can be chosen multiple times. Unlike permutation, order doesn't matter for combinations.

combinationsReplacement(x: Array, k: int): Array<Array>
Parameters
x `(Array)` any type of data
k `(int)` the number of objects in each group (without replacement)
Returns
`Array<Array>`: array of permutations
Example
`combinationsReplacement([1, 2], 2); // => [[1, 1], [1, 2], [2, 2]]`

### combinations

Implementation of Combinations Combinations are unique subsets of a collection - in this case, k x from a collection at a time. https://en.wikipedia.org/wiki/Combination

combinations(x: Array, k: int): Array<Array>
Parameters
x `(Array)` any type of data
k `(int)` the number of objects in each group (without replacement)
Returns
`Array<Array>`: array of permutations
Example
`combinations([1, 2, 3], 2); // => [[1,2], [1,3], [2,3]]`

### combineMeans

When combining two lists of values for which one already knows the means, one does not have to necessary recompute the mean of the combined lists in linear time. They can instead use this function to compute the combined mean by providing the mean & number of values of the first list and the mean & number of values of the second list.

combineMeans(mean1: number, n1: number, mean2: number, n2: number): number
Since: 3.0.0
Parameters
mean1 `(number)` mean of the first list
n1 `(number)` number of items in the first list
mean2 `(number)` mean of the second list
n2 `(number)` number of items in the second list
Returns
`number`: the combined mean
Example
`combineMeans(5, 3, 4, 3); // => 4.5`

### combineVariances

When combining two lists of values for which one already knows the variances, one does not have to necessary recompute the variance of the combined lists in linear time. They can instead use this function to compute the combined variance by providing the variance, mean & number of values of the first list and the variance, mean & number of values of the second list.

combineVariances(variance1: number, mean1: number, n1: number, variance2: number, mean2: number, n2: number): number
Since: 3.0.0
Parameters
variance1 `(number)` variance of the first list
mean1 `(number)` mean of the first list
n1 `(number)` number of items in the first list
variance2 `(number)` variance of the second list
mean2 `(number)` mean of the second list
n2 `(number)` number of items in the second list
Returns
`number`: the combined mean
Example
`combineVariances(14 / 3, 5, 3, 8 / 3, 4, 3); // => 47 / 12`

### extentSorted

The extent is the lowest & highest number in the array. With a sorted array, the first element in the array is always the lowest while the last element is always the largest, so this calculation can be done in one step, or constant time.

extentSorted(x: Array<number>): Array<number>
Parameters
x `(Array<number>)` input
Returns
`Array<number>`: minimum & maximum value
Example
`extentSorted([-100, -10, 1, 2, 5]); // => [-100, 5]`

### extent

This computes the minimum & maximum number in an array.

This runs on `O(n)`, linear time in respect to the array

extent(x: Array<number>): Array<number>
Parameters
x `(Array<number>)` sample of one or more data points
Returns
`Array<number>`: minimum & maximum value
Throws
• Error: if the the length of x is less than one
Example
```extent([1, 2, 3, 4]);
// => [1, 4]```

### permutationTest

Conducts a permutation test to determine if two data sets are significantly different from each other, using the difference of means between the groups as the test statistic. The function allows for the following hypotheses:

• two_tail = Null hypothesis: the two distributions are equal.
• greater = Null hypothesis: observations from sampleX tend to be smaller than those from sampleY.
• less = Null hypothesis: observations from sampleX tend to be greater than those from sampleY. Learn more about one-tail vs two-tail tests.
permutationTest(sampleX: Array<number>, sampleY: Array<number>, alternative: string, k: number): number
Parameters
sampleX `(Array<number>)` first dataset (e.g. treatment data)
sampleY `(Array<number>)` second dataset (e.g. control data)
alternative `(string)` alternative hypothesis, either 'two_sided' (default), 'greater', or 'less'
k `(number)` number of values in permutation distribution.
Returns
`number`: p-value The probability of observing the difference between groups (as or more extreme than what we did), assuming the null hypothesis.
Example
```var control = [2, 5, 3, 6, 7, 2, 5];
var treatment = [20, 5, 13, 12, 7, 2, 2];
permutationTest(control, treatment); // ~0.1324```

### permutationsHeap

Implementation of Heap's Algorithm for generating permutations.

permutationsHeap(elements: Array): Array<Array>
Parameters
elements `(Array)` any type of data
Returns
`Array<Array>`: array of permutations

### quantileRankSorted

This function returns the quantile in which one would find the given value in the given array. With a sorted array, leveraging binary search, we can find this information in logarithmic time.

quantileRankSorted(x: Array<number>, value: any): number
Parameters
x `(Array<number>)` input
value `(any)`
Returns
`number`: value value
Example
```quantileRankSorted([1, 2, 3, 4], 3); // => 0.75
quantileRankSorted([1, 2, 3, 3, 4], 3); // => 0.7
quantileRankSorted([1, 2, 3, 4], 6); // => 1
quantileRankSorted([1, 2, 3, 3, 5], 4); // => 0.8```

### quantileRank

This function returns the quantile in which one would find the given value in the given array. It will require to copy and sort your array beforehand, so if you know your array is already sorted, you would rather use `quantileRankSorted`.

quantileRank(x: Array<number>, value: any): number
Parameters
x `(Array<number>)` input
value `(any)`
Returns
`number`: value value
Example
```quantileRank([4, 3, 1, 2], 3); // => 0.75
quantileRank([4, 3, 2, 3, 1], 3); // => 0.7
quantileRank([2, 4, 1, 3], 6); // => 1
quantileRank([5, 3, 1, 2, 3], 4); // => 0.8```

### quickselect

Rearrange items in `arr` so that all items in `[left, k]` range are the smallest. The `k`-th element will have the `(k - left + 1)`-th smallest value in `[left, right]`.

Implements Floyd-Rivest selection algorithm https://en.wikipedia.org/wiki/Floyd-Rivest_algorithm

quickselect(arr: Array<number>, k: number, left: number?, right: number?): void
Parameters
arr `(Array<number>)` input array
k `(number)` pivot index
left `(number?)` left index
right `(number?)` right index
Returns
`void`: mutates input array
Example
```var arr = [65, 28, 59, 33, 21, 56, 22, 95, 50, 12, 90, 53, 28, 77, 39];
quickselect(arr, 8);
// = [39, 28, 28, 33, 21, 12, 22, 50, 53, 56, 59, 65, 90, 77, 95]```

### sampleKurtosis

Kurtosis is a measure of the heaviness of a distribution's tails relative to its variance. The kurtosis value can be positive or negative, or even undefined.

Implementation is based on Fisher's excess kurtosis definition and uses unbiased moment estimators. This is the version found in Excel and available in several statistical packages, including SAS and SciPy.

sampleKurtosis(x: Array<number>): number
Parameters
x `(Array<number>)` a sample of 4 or more data points
Returns
`number`: sample kurtosis
Throws
• Error: if x has length less than 4
Example
`sampleKurtosis([1, 2, 2, 3, 5]); // => 1.4555765595463122`

### subtractFromMean

When removing a value from a list, one does not have to necessary recompute the mean of the list in linear time. They can instead use this function to compute the new mean by providing the current mean, the number of elements in the list that produced it and the value to remove.

subtractFromMean(mean: number, n: number, value: number): number
Since: 3.0.0
Parameters
mean `(number)` current mean
n `(number)` number of items in the list
value `(number)` the value to remove
Returns
`number`: the new mean
Example
`subtractFromMean(20.5, 6, 53); // => 14`