# simple_statistics.js

- # simple_statistics.js

- ¶

      /* global module */

- ¶

  # simple-statistics

  A simple, literate statistics system. The code below uses the Javascript module pattern, eventually assigning `simple-statistics` to `ss` in browsers or the `exports` object for node.js

      (function() {
          var ss = {};

          if (typeof module !== 'undefined') {

- ¶

  Assign the `ss` object to exports, so that you can require it in node.js

              module.exports = ss;
          } else {

- ¶

  Otherwise, in a browser, we assign `ss` to the window object, so you can simply refer to it as `ss`.

              this.ss = ss;
          }

- ¶

  # Linear Regression

  Simple linear regression is a simple way to find a fitted line between a set of coordinates.

          function linear_regression() {
              var linreg = {},
                  data = [];

- ¶

  Assign data to the model. Data is assumed to be an array.

              linreg.data = function(x) {
                  if (!arguments.length) return data;
                  data = x.slice();
                  return linreg;
              };

- ¶

  Calculate the slope and y-intercept of the regression line by calculating the least sum of squares

              linreg.mb = function() {
                  var m, b;

- ¶

  Store data length in a local variable to reduce repeated object property lookups

                  var data_length = data.length;

- ¶

  if there’s only one point, arbitrarily choose a slope of 0 and a y-intercept of whatever the y of the initial point is

                  if (data_length === 1) {
                      m = 0;
                      b = data[0][1];
                  } else {

- ¶

  Initialize our sums and scope the `m` and `b` variables that define the line.

                      var sum_x = 0, sum_y = 0,
                          sum_xx = 0, sum_xy = 0;

- ¶

  Use local variables to grab point values with minimal object property lookups

                      var point, x, y;

- ¶

  Gather the sum of all x values, the sum of all y values, and the sum of x^2 and (x*y) for each value.

  In math notation, these would be SS_x, SS_y, SS_xx, and SS_xy

                      for (var i = 0; i < data_length; i++) {
                          point = data[i];
                          x = point[0];
                          y = point[1];

                          sum_x += x;
                          sum_y += y;

                          sum_xx += x * x;
                          sum_xy += x * y;
                      }

- ¶

  `m` is the slope of the regression line

                      m = ((data_length * sum_xy) - (sum_x * sum_y)) /
                          ((data_length * sum_xx) - (sum_x * sum_x));

- ¶

  `b` is the y-intercept of the line.

                      b = (sum_y / data_length) - ((m * sum_x) / data_length);
                  }

- ¶

  Return both values as an object.

                  return { m: m, b: b };
              };

- ¶

  a shortcut for simply getting the slope of the regression line

              linreg.m = function() {
                  return linreg.mb().m;
              };

- ¶

  a shortcut for simply getting the y-intercept of the regression line.

              linreg.b = function() {
                  return linreg.mb().b;
              };

- ¶

  ## Fitting The Regression Line

  This is called after `.data()` and returns the equation `y = f(x)` which gives the position of the regression line at each point in `x`.

              linreg.line = function() {

- ¶

  Get the slope, `m`, and y-intercept, `b`, of the line.

                  var mb = linreg.mb(),
                      m = mb.m,
                      b = mb.b;

- ¶

  Return a function that computes a `y` value for each x value it is given, based on the values of `b` and `a` that we just computed.

                  return function(x) {
                      return b + (m * x);
                  };
              };

              return linreg;
          }

- ¶

  # R Squared

  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.

          function r_squared(data, f) {
              if (data.length < 2) return 1;

- ¶

  Compute the average y value for the actual data set in order to compute the total sum of squares

              var sum = 0, average;
              for (var i = 0; i < data.length; i++) {
                  sum += data[i][1];
              }
              average = sum / data.length;

- ¶

  Compute the total sum of squares - the squared difference between each point and the average of all points.

              var sum_of_squares = 0;
              for (var j = 0; j < data.length; j++) {
                  sum_of_squares += Math.pow(average - data[j][1], 2);
              }

- ¶

  Finally estimate the error: the squared difference between the estimate and the actual data value at each point.

              var err = 0;
              for (var k = 0; k < data.length; k++) {
                  err += Math.pow(data[k][1] - f(data[k][0]), 2);
              }

- ¶

  As the error grows larger, its ratio to the sum of squares increases and the r squared value grows lower.

              return 1 - (err / sum_of_squares);
          }

- ¶

  # Bayesian Classifier

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

          function bayesian() {

- ¶

  The `bayes_model` object is what will be exposed by this closure, with all of its extended methods, and will have access to all scope variables, like `total_count`.

              var bayes_model = {},

- ¶

  The number of items that are currently classified in the model

                  total_count = 0,

- ¶

  Every item classified in the model

                  data = {};

- ¶

  ## Train

  Train the classifier with a new item, which has a single dimension of Javascript literal keys and values.

              bayes_model.train = function(item, category) {

- ¶

  If the data object doesn’t have any values for this category, create a new object for it.

                  if (!data[category]) data[category] = {};

- ¶

  Iterate through each key in the item.

                  for (var k in item) {
                      var v = item[k];

- ¶

  Initialize the nested object `data[category][k][item[k]]` with an object of keys that equal 0.

                      if (data[category][k] === undefined) data[category][k] = {};
                      if (data[category][k][v] === undefined) data[category][k][v] = 0;

- ¶

  And increment the key for this key/value combination.

                      data[category][k][item[k]]++;
                  }

- ¶

  Increment the number of items classified

                  total_count++;
              };

- ¶

  ## Score

  Generate a score of how well this item matches all possible categories based on its attributes

              bayes_model.score = function(item) {

- ¶

  Initialize an empty array of odds per category.

                  var odds = {}, category;

- ¶

  Iterate through each key in the item, then iterate through each category that has been used in previous calls to `.train()`

                  for (var k in item) {
                      var v = item[k];
                      for (category in data) {

- ¶

  Create an empty object for storing key - value combinations for this category.

                          if (odds[category] === undefined) odds[category] = {};

- ¶

  If this item doesn’t even have a property, it counts for nothing, but if it does have the property that we’re looking for from the item to categorize, it counts based on how popular it is versus the whole population.

                          if (data[category][k]) {
                              odds[category][k + '_' + v] = (data[category][k][v] || 0) / total_count;
                          } else {
                              odds[category][k + '_' + v] = 0;
                          }
                      }
                  }

- ¶

  Set up a new object that will contain sums of these odds by category

                  var odds_sums = {};

                  for (category in odds) {

- ¶

  Tally all of the odds for each category-combination pair - the non-existence of a category does not add anything to the score.

                      for (var combination in odds[category]) {
                          if (odds_sums[category] === undefined) odds_sums[category] = 0;
                          odds_sums[category] += odds[category][combination];
                      }
                  }

                  return odds_sums;
              };

- ¶

  Return the completed model.

              return bayes_model;
          }

- ¶

  # sum

  is simply the result of adding all numbers together, starting from zero.

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

          function sum(x) {
              var value = 0;
              for (var i = 0; i < x.length; i++) {
                  value += x[i];
              }
              return value;
          }

- ¶

  # mean

  is the sum over the number of values

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

          function mean(x) {

- ¶

  The mean of no numbers is null

              if (x.length === 0) return null;

              return sum(x) / x.length;
          }

- ¶

  # geometric mean

  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

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

          function geometric_mean(x) {

- ¶

  The mean of no numbers is null

              if (x.length === 0) return null;

- ¶

  the starting value.

              var value = 1;

              for (var i = 0; i < x.length; i++) {

- ¶

  the geometric mean is only valid for positive numbers

                  if (x[i] <= 0) return null;

- ¶

  repeatedly multiply the value by each number

                  value *= x[i];
              }

              return Math.pow(value, 1 / x.length);
          }

- ¶

  # harmonic mean

  a mean function typically used to find the average of rates

  this is the reciprocal of the arithmetic mean of the reciprocals of the input numbers

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

          function harmonic_mean(x) {

- ¶

  The mean of no numbers is null

              if (x.length === 0) return null;

              var reciprocal_sum = 0;

              for (var i = 0; i < x.length; i++) {

- ¶

  the harmonic mean is only valid for positive numbers

                  if (x[i] <= 0) return null;

                  reciprocal_sum += 1 / x[i];
              }

- ¶

  divide n by the the reciprocal sum

              return x.length / reciprocal_sum;
          }

- ¶

  # min

  This is simply the minimum number in the set.

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

          function min(x) {
              var value;
              for (var i = 0; i < x.length; i++) {

- ¶

  On the first iteration of this loop, min is undefined and is thus made the minimum element in the array

                  if (x[i] < value || value === undefined) value = x[i];
              }
              return value;
          }

- ¶

  # max

  This is simply the maximum number in the set.

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

          function max(x) {
              var value;
              for (var i = 0; i < x.length; i++) {

- ¶

  On the first iteration of this loop, max is undefined and is thus made the maximum element in the array

                  if (x[i] > value || value === undefined) value = x[i];
              }
              return value;
          }

- ¶

  # variance

  is the sum of squared deviations from the mean

  depends on `mean()`

          function variance(x) {

- ¶

  The variance of no numbers is null

              if (x.length === 0) return null;

              var mean_value = mean(x),
                  deviations = [];

- ¶

  Make a list of squared deviations from the mean.

              for (var i = 0; i < x.length; i++) {
                  deviations.push(Math.pow(x[i] - mean_value, 2));
              }

- ¶

  Find the mean value of that list

              return mean(deviations);
          }

- ¶

  # standard deviation

  is just the square root of the variance.

  depends on `variance()`

          function standard_deviation(x) {

- ¶

  The standard deviation of no numbers is null

              if (x.length === 0) return null;

              return Math.sqrt(variance(x));
          }

- ¶

  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.

  depends on `mean()`

          function sum_nth_power_deviations(x, n) {
              var mean_value = mean(x),
                  sum = 0;

              for (var i = 0; i < x.length; i++) {
                  sum += Math.pow(x[i] - mean_value, n);
              }

              return sum;
          }

- ¶

  # variance

  is the sum of squared deviations from the mean

  depends on `sum_nth_power_deviations`

          function sample_variance(x) {

- ¶

  The variance of no numbers is null

              if (x.length <= 1) return null;

              var sum_squared_deviations_value = sum_nth_power_deviations(x, 2);

- ¶

  Find the mean value of that list

              return sum_squared_deviations_value / (x.length - 1);
          }

- ¶

  # standard deviation

  is just the square root of the variance.

  depends on `sample_variance()`

          function sample_standard_deviation(x) {

- ¶

  The standard deviation of no numbers is null

              if (x.length <= 1) return null;

              return Math.sqrt(sample_variance(x));
          }

- ¶

  # covariance

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

  depends on `mean()`

          function sample_covariance(x, y) {

- ¶

  The two datasets must have the same length which must be more than 1

              if (x.length <= 1 || x.length != y.length){
                  return null;
              }

- ¶

  determine the mean of each dataset so that we can judge each value of the dataset fairly as the difference from the mean. this way, if one dataset is [1, 2, 3] and [2, 3, 4], their covariance does not suffer because of the difference in absolute values

              var xmean = mean(x),
                  ymean = mean(y),
                  sum = 0;

- ¶

  for each pair of values, the covariance increases when their difference from the mean is associated - if both are well above or if both are well below the mean, the covariance increases significantly.

              for (var i = 0; i < x.length; i++){
                  sum += (x[i] - xmean) * (y[i] - ymean);
              }

- ¶

  the covariance is weighted by the length of the datasets.

              return sum / (x.length - 1);
          }

- ¶

  # correlation

  Gets a measure of how correlated two datasets are, between -1 and 1

  depends on `sample_standard_deviation()` and `sample_covariance()`

          function sample_correlation(x, y) {
              var cov = sample_covariance(x, y),
                  xstd = sample_standard_deviation(x),
                  ystd = sample_standard_deviation(y);

              if (cov === null || xstd === null || ystd === null) {
                  return null;
              }

              return cov / xstd / ystd;
          }

- ¶

  # median

  The middle number of a list. This is often a good indicator of ‘the middle’ when there are outliers that skew the `mean()` value.

          function median(x) {

- ¶

  The median of an empty list is null

              if (x.length === 0) return null;

- ¶

  Sorting the array makes it easy to find the center, but use `.slice()` to ensure the original array `x` is not modified

              var sorted = x.slice().sort(function (a, b) { return a - b; });

- ¶

  If the length of the list is odd, it’s the central number

              if (sorted.length % 2 === 1) {
                  return sorted[(sorted.length - 1) / 2];

- ¶

  Otherwise, the median is the average of the two numbers at the center of the list

              } else {
                  var a = sorted[(sorted.length / 2) - 1];
                  var b = sorted[(sorted.length / 2)];
                  return (a + b) / 2;
              }
          }

- ¶

  # 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 implementation is inspired by science.js

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

          function mode(x) {

- ¶

  Handle edge cases: The median of an empty list is null

              if (x.length === 0) return null;
              else if (x.length === 1) return x[0];

- ¶

  Sorting the array lets us iterate through it below and be sure that every time we see a new number it’s new and we’ll never see the same number twice

              var sorted = x.slice().sort(function (a, b) { return a - b; });

- ¶

  This assumes it is dealing with an array of size > 1, since size 0 and 1 are handled immediately. Hence it starts at index 1 in the array.

              var last = sorted[0],

- ¶

  store the mode as we find new modes

                  value,

- ¶

  store how many times we’ve seen the mode

                  max_seen = 0,

- ¶

  how many times the current candidate for the mode has been seen

                  seen_this = 1;

- ¶

  end at sorted.length + 1 to fix the case in which the mode is the highest number that occurs in the sequence. the last iteration compares sorted[i], which is undefined, to the highest number in the series

              for (var i = 1; i < sorted.length + 1; i++) {

- ¶

  we’re seeing a new number pass by

                  if (sorted[i] !== last) {

- ¶

  the last number is the new mode since we saw it more often than the old one

                      if (seen_this > max_seen) {
                          max_seen = seen_this;
                          value = last;
                      }
                      seen_this = 1;
                      last = sorted[i];

- ¶

  if this isn’t a new number, it’s one more occurrence of the potential mode

                  } else { seen_this++; }
              }
              return value;
          }

- ¶

  # t-test

  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.

  Depends on `standard_deviation()` and `mean()`

          function t_test(sample, x) {

- ¶

  The mean of the sample

              var sample_mean = mean(sample);

- ¶

  The standard deviation of the sample

              var sd = standard_deviation(sample);

- ¶

  Square root the length of the sample

              var rootN = Math.sqrt(sample.length);

- ¶

  Compute the known value against the sample, returning the t value

              return (sample_mean - x) / (sd / rootN);
          }

- ¶

  # 2-sample t-test

  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 `sample_x` and `sample_y` are equal to each other.

  Depends on `sample_variance()` and `mean()`

          function t_test_two_sample(sample_x, sample_y, difference) {
              var n = sample_x.length,
                  m = sample_y.length;

- ¶

  If either sample doesn’t actually have any values, we can’t compute this at all, so we return `null`.

              if (!n || !m) return null ;

- ¶

  default difference (mu) is zero

              if (!difference) difference = 0;

              var meanX = mean(sample_x),
                  meanY = mean(sample_y);

              var weightedVariance = ((n - 1) * sample_variance(sample_x) +
                  (m - 1) * sample_variance(sample_y)) / (n + m - 2);

              return (meanX - meanY - difference) /
                  Math.sqrt(weightedVariance * (1 / n + 1 / m));
          }

- ¶

  # 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.

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

          function chunk(sample, chunkSize) {

- ¶

  a list of result chunks, as arrays in an array

              var output = [];

- ¶

  `chunkSize` must be zero or higher - otherwise the loop below, in which we call `start += chunkSize`, will loop infinitely. So, we’ll detect and return null in that case to indicate invalid input.

              if (chunkSize <= 0) {
                  return null;
              }

- ¶

  `start` is the index at which `.slice` will start selecting new array elements

              for (var start = 0; start < sample.length; start += chunkSize) {

- ¶

  for each chunk, slice that part of the array and add it to the output. The `.slice` function does not change the original array.

                  output.push(sample.slice(start, start + chunkSize));
              }
              return output;
          }

- ¶

  # shuffle_in_place

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

          function shuffle_in_place(sample, randomSource) {

- ¶

  a custom random number source can be provided if you want to use a fixed seed or another random number generator, like random-js

              randomSource = randomSource || Math.random;

- ¶

  store the current length of the sample to determine when no elements remain to shuffle.

              var length = sample.length;

- ¶

  temporary is used to hold an item when it is being swapped between indices.

              var temporary;

- ¶

  The index to swap at each stage.

              var index;

- ¶

  While there are still items to shuffle

              while (length > 0) {

- ¶

  chose a random index within the subset of the array that is not yet shuffled

                  index = Math.floor(randomSource() * length--);

- ¶

  store the value that we’ll move temporarily

                  temporary = sample[length];

- ¶

  swap the value at `sample[length]` with `sample[index]`

                  sample[length] = sample[index];
                  sample[index] = temporary;
              }

              return sample;
          }

- ¶

  # shuffle

  A Fisher-Yates shuffle is a fast way to create a random permutation of a finite set.

          function shuffle(sample, randomSource) {

- ¶

  slice the original array so that it is not modified

              sample = sample.slice();

- ¶

  and then shuffle that shallow-copied array, in place

              return shuffle_in_place(sample.slice(), randomSource);
          }

- ¶

  # sample

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

          function sample(array, n, randomSource) {

- ¶

  shuffle the original array using a fisher-yates shuffle

              var shuffled = shuffle(array, randomSource);

- ¶

  and then return a subset of it - the first `n` elements.

              return shuffled.slice(0, n);
          }

- ¶

  # quantile

  This is a population quantile, since we assume to know the entire dataset in this library. Thus I’m trying to follow 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

          function quantile(sample, p) {

- ¶

  We can’t derive quantiles from an empty list

              if (sample.length === 0) return null;

- ¶

  Sort a copy of the array. We’ll need a sorted array to index the values in sorted order.

              var sorted = sample.slice().sort(function (a, b) { return a - b; });

              if (p.length) {

- ¶

  Initialize the result array

                  var results = [];

- ¶

  For each requested quantile

                  for (var i = 0; i < p.length; i++) {
                      results[i] = quantile_sorted(sorted, p[i]);
                  }
                  return results;
              } else {
                  return quantile_sorted(sorted, p);
              }
          }

- ¶

  # quantile

  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 much faster.

          function quantile_sorted(sample, p) {
              var idx = (sample.length) * p;
              if (p < 0 || p > 1) {
                  return null;
              } else if (p === 1) {

- ¶

  If p is 1, directly return the last element

                  return sample[sample.length - 1];
              } else if (p === 0) {

- ¶

  If p is 0, directly return the first element

                  return sample[0];
              } else if (idx % 1 !== 0) {

- ¶

  If p is not integer, return the next element in array

                  return sample[Math.ceil(idx) - 1];
              } else if (sample.length % 2 === 0) {

- ¶

  If the list has even-length, we’ll take the average of this number and the next value, if there is one

                  return (sample[idx - 1] + sample[idx]) / 2;
              } else {

- ¶

  Finally, in the simple case of an integer value with an odd-length list, return the sample value at the index.

                  return sample[idx];
              }
          }

- ¶

  # Interquartile range

  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.

          function iqr(sample) {

- ¶

  We can’t derive quantiles from an empty list

              if (sample.length === 0) return null;

- ¶

  Interquartile range is the span between the upper quartile, at `0.75`, and lower quartile, `0.25`

              return quantile(sample, 0.75) - quantile(sample, 0.25);
          }

- ¶

  # Median Absolute Deviation

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

          function mad(x) {

- ¶

  The mad of nothing is null

              if (!x || x.length === 0) return null;

              var median_value = median(x),
                  median_absolute_deviations = [];

- ¶

  Make a list of absolute deviations from the median

              for (var i = 0; i < x.length; i++) {
                  median_absolute_deviations.push(Math.abs(x[i] - median_value));
              }

- ¶

  Find the median value of that list

              return median(median_absolute_deviations);
          }

- ¶

  ## Compute Matrices for Jenks

  Compute the matrices required for Jenks breaks. These matrices can be used for any classing of data with `classes <= n_classes`

          function jenksMatrices(data, n_classes) {

- ¶

  in the original implementation, these matrices are referred to as `LC` and `OP`

  - lower_class_limits (LC): optimal lower class limits
  - variance_combinations (OP): optimal variance combinations for all classes

              var lower_class_limits = [],
                  variance_combinations = [],

- ¶

  loop counters

                  i, j,

- ¶

  the variance, as computed at each step in the calculation

                  variance = 0;

- ¶

  Initialize and fill each matrix with zeroes

              for (i = 0; i < data.length + 1; i++) {
                  var tmp1 = [], tmp2 = [];

- ¶

  despite these arrays having the same values, we need to keep them separate so that changing one does not change the other

                  for (j = 0; j < n_classes + 1; j++) {
                      tmp1.push(0);
                      tmp2.push(0);
                  }
                  lower_class_limits.push(tmp1);
                  variance_combinations.push(tmp2);
              }

              for (i = 1; i < n_classes + 1; i++) {
                  lower_class_limits[1][i] = 1;
                  variance_combinations[1][i] = 0;

- ¶

  in the original implementation, 9999999 is used but since Javascript has `Infinity`, we use that.

                  for (j = 2; j < data.length + 1; j++) {
                      variance_combinations[j][i] = Infinity;
                  }
              }

              for (var l = 2; l < data.length + 1; l++) {

- ¶

  `SZ` originally. this is the sum of the values seen thus far when calculating variance.

                  var sum = 0,

- ¶

  `ZSQ` originally. the sum of squares of values seen thus far

                      sum_squares = 0,

- ¶

  `WT` originally. This is the number of

                      w = 0,

- ¶

  `IV` originally

                      i4 = 0;

- ¶

  in several instances, you could say `Math.pow(x, 2)` instead of `x * x`, but this is slower in some browsers introduces an unnecessary concept.

                  for (var m = 1; m < l + 1; m++) {

- ¶

  `III` originally

                      var lower_class_limit = l - m + 1,
                          val = data[lower_class_limit - 1];

- ¶

  here we’re estimating variance for each potential classing of the data, for each potential number of classes. `w` is the number of data points considered so far.

                      w++;

- ¶

  increase the current sum and sum-of-squares

                      sum += val;
                      sum_squares += val * val;

- ¶

  the variance at this point in the sequence is the difference between the sum of squares and the total x 2, over the number of samples.

                      variance = sum_squares - (sum * sum) / w;

                      i4 = lower_class_limit - 1;

                      if (i4 !== 0) {
                          for (j = 2; j < n_classes + 1; j++) {

- ¶

  if adding this element to an existing class will increase its variance beyond the limit, break the class at this point, setting the `lower_class_limit` at this point.

                              if (variance_combinations[l][j] >=
                                  (variance + variance_combinations[i4][j - 1])) {
                                  lower_class_limits[l][j] = lower_class_limit;
                                  variance_combinations[l][j] = variance +
                                      variance_combinations[i4][j - 1];
                              }
                          }
                      }
                  }

                  lower_class_limits[l][1] = 1;
                  variance_combinations[l][1] = variance;
              }

- ¶

  return the two matrices. for just providing breaks, only `lower_class_limits` is needed, but variances can be useful to evaluate goodness of fit.

              return {
                  lower_class_limits: lower_class_limits,
                  variance_combinations: variance_combinations
              };
          }

- ¶

  ## Pull Breaks Values for Jenks

  the second part of the jenks recipe: take the calculated matrices and derive an array of n breaks.

          function jenksBreaks(data, lower_class_limits, n_classes) {

              var k = data.length - 1,
                  kclass = [],
                  countNum = n_classes;

- ¶

  the calculation of classes will never include the upper and lower bounds, so we need to explicitly set them

              kclass[n_classes] = data[data.length - 1];
              kclass[0] = data[0];

- ¶

  the lower_class_limits matrix is used as indices into itself here: the `k` variable is reused in each iteration.

              while (countNum > 1) {
                  kclass[countNum - 1] = data[lower_class_limits[k][countNum] - 2];
                  k = lower_class_limits[k][countNum] - 1;
                  countNum--;
              }

              return kclass;
          }

- ¶

  # Jenks natural breaks optimization

  Implementations: 1 (python), 2 (buggy), 3 (works)

  Depends on `jenksBreaks()` and `jenksMatrices()`

          function jenks(data, n_classes) {

              if (n_classes > data.length) return null;

- ¶

  sort data in numerical order, since this is expected by the matrices function

              data = data.slice().sort(function (a, b) { return a - b; });

- ¶

  get our basic matrices

              var matrices = jenksMatrices(data, n_classes),

- ¶

  we only need lower class limits here

                  lower_class_limits = matrices.lower_class_limits;

- ¶

  extract n_classes out of the computed matrices

              return jenksBreaks(data, lower_class_limits, n_classes);

          }

- ¶

  # Skewness

  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.

  Depends on `sum_nth_power_deviations()` and `sample_standard_deviation`

          function sample_skewness(x) {

- ¶

  The skewness of less than three arguments is null

              if (x.length < 3) return null;

              var n = x.length,
                  cubed_s = Math.pow(sample_standard_deviation(x), 3),
                  sum_cubed_deviations = sum_nth_power_deviations(x, 3);

              return n * sum_cubed_deviations / ((n - 1) * (n - 2) * cubed_s);
          }

- ¶

  # Standard Normal Table

  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 taken from http://en.wikipedia.org/wiki/Standard_normal_table The table used is the cumulative, and not cumulative from 0 to mean (even though the latter has 5 digits precision, instead of 4).

          var standard_normal_table = [
              /*  z      0.00    0.01    0.02    0.03    0.04    0.05    0.06    0.07    0.08    0.09 */
              /* 0.0 */
              0.5000, 0.5040, 0.5080, 0.5120, 0.5160, 0.5199, 0.5239, 0.5279, 0.5319, 0.5359,
              /* 0.1 */
              0.5398, 0.5438, 0.5478, 0.5517, 0.5557, 0.5596, 0.5636, 0.5675, 0.5714, 0.5753,
              /* 0.2 */
              0.5793, 0.5832, 0.5871, 0.5910, 0.5948, 0.5987, 0.6026, 0.6064, 0.6103, 0.6141,
              /* 0.3 */
              0.6179, 0.6217, 0.6255, 0.6293, 0.6331, 0.6368, 0.6406, 0.6443, 0.6480, 0.6517,
              /* 0.4 */
              0.6554, 0.6591, 0.6628, 0.6664, 0.6700, 0.6736, 0.6772, 0.6808, 0.6844, 0.6879,
              /* 0.5 */
              0.6915, 0.6950, 0.6985, 0.7019, 0.7054, 0.7088, 0.7123, 0.7157, 0.7190, 0.7224,
              /* 0.6 */
              0.7257, 0.7291, 0.7324, 0.7357, 0.7389, 0.7422, 0.7454, 0.7486, 0.7517, 0.7549,
              /* 0.7 */
              0.7580, 0.7611, 0.7642, 0.7673, 0.7704, 0.7734, 0.7764, 0.7794, 0.7823, 0.7852,
              /* 0.8 */
              0.7881, 0.7910, 0.7939, 0.7967, 0.7995, 0.8023, 0.8051, 0.8078, 0.8106, 0.8133,
              /* 0.9 */
              0.8159, 0.8186, 0.8212, 0.8238, 0.8264, 0.8289, 0.8315, 0.8340, 0.8365, 0.8389,
              /* 1.0 */
              0.8413, 0.8438, 0.8461, 0.8485, 0.8508, 0.8531, 0.8554, 0.8577, 0.8599, 0.8621,
              /* 1.1 */
              0.8643, 0.8665, 0.8686, 0.8708, 0.8729, 0.8749, 0.8770, 0.8790, 0.8810, 0.8830,
              /* 1.2 */
              0.8849, 0.8869, 0.8888, 0.8907, 0.8925, 0.8944, 0.8962, 0.8980, 0.8997, 0.9015,
              /* 1.3 */
              0.9032, 0.9049, 0.9066, 0.9082, 0.9099, 0.9115, 0.9131, 0.9147, 0.9162, 0.9177,
              /* 1.4 */
              0.9192, 0.9207, 0.9222, 0.9236, 0.9251, 0.9265, 0.9279, 0.9292, 0.9306, 0.9319,
              /* 1.5 */
              0.9332, 0.9345, 0.9357, 0.9370, 0.9382, 0.9394, 0.9406, 0.9418, 0.9429, 0.9441,
              /* 1.6 */
              0.9452, 0.9463, 0.9474, 0.9484, 0.9495, 0.9505, 0.9515, 0.9525, 0.9535, 0.9545,
              /* 1.7 */
              0.9554, 0.9564, 0.9573, 0.9582, 0.9591, 0.9599, 0.9608, 0.9616, 0.9625, 0.9633,
              /* 1.8 */
              0.9641, 0.9649, 0.9656, 0.9664, 0.9671, 0.9678, 0.9686, 0.9693, 0.9699, 0.9706,
              /* 1.9 */
              0.9713, 0.9719, 0.9726, 0.9732, 0.9738, 0.9744, 0.9750, 0.9756, 0.9761, 0.9767,
              /* 2.0 */
              0.9772, 0.9778, 0.9783, 0.9788, 0.9793, 0.9798, 0.9803, 0.9808, 0.9812, 0.9817,
              /* 2.1 */
              0.9821, 0.9826, 0.9830, 0.9834, 0.9838, 0.9842, 0.9846, 0.9850, 0.9854, 0.9857,
              /* 2.2 */
              0.9861, 0.9864, 0.9868, 0.9871, 0.9875, 0.9878, 0.9881, 0.9884, 0.9887, 0.9890,
              /* 2.3 */
              0.9893, 0.9896, 0.9898, 0.9901, 0.9904, 0.9906, 0.9909, 0.9911, 0.9913, 0.9916,
              /* 2.4 */
              0.9918, 0.9920, 0.9922, 0.9925, 0.9927, 0.9929, 0.9931, 0.9932, 0.9934, 0.9936,
              /* 2.5 */
              0.9938, 0.9940, 0.9941, 0.9943, 0.9945, 0.9946, 0.9948, 0.9949, 0.9951, 0.9952,
              /* 2.6 */
              0.9953, 0.9955, 0.9956, 0.9957, 0.9959, 0.9960, 0.9961, 0.9962, 0.9963, 0.9964,
              /* 2.7 */
              0.9965, 0.9966, 0.9967, 0.9968, 0.9969, 0.9970, 0.9971, 0.9972, 0.9973, 0.9974,
              /* 2.8 */
              0.9974, 0.9975, 0.9976, 0.9977, 0.9977, 0.9978, 0.9979, 0.9979, 0.9980, 0.9981,
              /* 2.9 */
              0.9981, 0.9982, 0.9982, 0.9983, 0.9984, 0.9984, 0.9985, 0.9985, 0.9986, 0.9986,
              /* 3.0 */
              0.9987, 0.9987, 0.9987, 0.9988, 0.9988, 0.9989, 0.9989, 0.9989, 0.9990, 0.9990
          ];

- ¶

  # 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

          function cumulative_std_normal_probability(z) {

- ¶

  Calculate the position of this value.

              var absZ = Math.abs(z),

- ¶

  Each row begins with a different significant digit: 0.5, 0.6, 0.7, and so on. So the row is simply this value’s significant digit: 0.567 will be in row 0, so row=0, 0.643 will be in row 1, so row=10.

                  row = Math.floor(absZ * 10),
                  column = 10 * (Math.floor(absZ * 100) / 10 - Math.floor(absZ * 100 / 10)),
                  index = Math.min((row * 10) + column, standard_normal_table.length - 1);

- ¶

  The index we calculate must be in the table as a positive value, but we still pay attention to whether the input is positive or negative, and flip the output value as a last step.

              if (z >= 0) {
                  return standard_normal_table[index];
              } else {

- ¶

  due to floating-point arithmetic, values in the table with 4 significant figures can nevertheless end up as repeating fractions when they’re computed here.

                  return +(1 - standard_normal_table[index]).toFixed(4);
              }
          }

- ¶

  # Z-Score, or Standard Score

  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.

          function z_score(x, mean, standard_deviation) {
              return (x - mean) / standard_deviation;
          }

- ¶

  We use `ε`, epsilon, as a stopping criterion when we want to iterate until we’re “close enough”.

          var epsilon = 0.0001;

- ¶

  # 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.

          function factorial(n) {

- ¶

  factorial is mathematically undefined for negative numbers

              if (n < 0 ) { return null; }

- ¶

  typically you’ll expand the factorial function going down, like 5! = 5 4 3 2 1. This is going in the opposite direction, counting from 2 up to the number in question, and since anything multiplied by 1 is itself, the loop only needs to start at 2.

              var accumulator = 1;
              for (var i = 2; i <= n; i++) {

- ¶

  for each number up to and including the number `n`, multiply the accumulator my that number.

                  accumulator *= i;
              }
              return accumulator;
          }

- ¶

  # Bernoulli Distribution

  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.

          function bernoulli_distribution(p) {

- ¶

  Check that `p` is a valid probability (0 ≤ p ≤ 1)

              if (p < 0 || p > 1 ) { return null; }

              return binomial_distribution(1, p);
          }

- ¶

  # Binomial Distribution

  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.

          function binomial_distribution(trials, probability) {

- ¶

  Check that `p` is a valid probability (0 ≤ p ≤ 1), that `n` is an integer, strictly positive.

              if (probability < 0 || probability > 1 ||
                  trials <= 0 || trials % 1 !== 0) {
                  return null;
              }

- ¶

  a probability mass function

              function probability_mass(x, trials, probability) {
                  return factorial(trials) /
                      (factorial(x) * factorial(trials - x)) *
                      (Math.pow(probability, x) * Math.pow(1 - probability, trials - x));
              }

- ¶

  We initialize `x`, the random variable, and `accumulator`, an accumulator for the cumulative distribution function to 0. `distribution_functions` is the object we’ll return with the `probability_of_x` and the `cumulative_probability_of_x`, as well as the calculated mean & variance. We iterate until the `cumulative_probability_of_x` is within `epsilon` of 1.0.

              var x = 0,
                  cumulative_probability = 0,
                  cells = {};

- ¶

  This algorithm iterates through each potential outcome, until the `cumulative_probability` is very close to 1, at which point we’ve defined the vast majority of outcomes

              do {
                  cells[x] = probability_mass(x, trials, probability);
                  cumulative_probability += cells[x];
                  x++;

- ¶

  when the cumulative_probability is nearly 1, we’ve calculated the useful range of this distribution

              } while (cumulative_probability < 1 - epsilon);

              return cells;
          }

- ¶

  # Poisson Distribution

  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`.

          function poisson_distribution(lambda) {

- ¶

  Check that lambda is strictly positive

              if (lambda <= 0) { return null; }

- ¶

  our current place in the distribution

              var x = 0,

- ¶

  and we keep track of the current cumulative probability, in order to know when to stop calculating chances.

                  cumulative_probability = 0,

- ¶

  the calculated cells to be returned

                  cells = {};

- ¶

  a probability mass function

              function probability_mass(x, lambda) {
                  return (Math.pow(Math.E, -lambda) * Math.pow(lambda, x)) /
                      factorial(x);
              }

- ¶

  This algorithm iterates through each potential outcome, until the `cumulative_probability` is very close to 1, at which point we’ve defined the vast majority of outcomes

              do {
                  cells[x] = probability_mass(x, lambda);
                  cumulative_probability += cells[x];
                  x++;

- ¶

  when the cumulative_probability is nearly 1, we’ve calculated the useful range of this distribution

              } while (cumulative_probability < 1 - epsilon);

              return cells;
          }

- ¶

  # 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).

          var chi_squared_distribution_table = {
              1: { 0.995:  0.00, 0.99:  0.00, 0.975:  0.00, 0.95:  0.00, 0.9:  0.02, 0.5:  0.45, 0.1:  2.71, 0.05:  3.84, 0.025:  5.02, 0.01:  6.63, 0.005:  7.88 },
              2: { 0.995:  0.01, 0.99:  0.02, 0.975:  0.05, 0.95:  0.10, 0.9:  0.21, 0.5:  1.39, 0.1:  4.61, 0.05:  5.99, 0.025:  7.38, 0.01:  9.21, 0.005: 10.60 },
              3: { 0.995:  0.07, 0.99:  0.11, 0.975:  0.22, 0.95:  0.35, 0.9:  0.58, 0.5:  2.37, 0.1:  6.25, 0.05:  7.81, 0.025:  9.35, 0.01: 11.34, 0.005: 12.84 },
              4: { 0.995:  0.21, 0.99:  0.30, 0.975:  0.48, 0.95:  0.71, 0.9:  1.06, 0.5:  3.36, 0.1:  7.78, 0.05:  9.49, 0.025: 11.14, 0.01: 13.28, 0.005: 14.86 },
              5: { 0.995:  0.41, 0.99:  0.55, 0.975:  0.83, 0.95:  1.15, 0.9:  1.61, 0.5:  4.35, 0.1:  9.24, 0.05: 11.07, 0.025: 12.83, 0.01: 15.09, 0.005: 16.75 },
              6: { 0.995:  0.68, 0.99:  0.87, 0.975:  1.24, 0.95:  1.64, 0.9:  2.20, 0.5:  5.35, 0.1: 10.65, 0.05: 12.59, 0.025: 14.45, 0.01: 16.81, 0.005: 18.55 },
              7: { 0.995:  0.99, 0.99:  1.25, 0.975:  1.69, 0.95:  2.17, 0.9:  2.83, 0.5:  6.35, 0.1: 12.02, 0.05: 14.07, 0.025: 16.01, 0.01: 18.48, 0.005: 20.28 },
              8: { 0.995:  1.34, 0.99:  1.65, 0.975:  2.18, 0.95:  2.73, 0.9:  3.49, 0.5:  7.34, 0.1: 13.36, 0.05: 15.51, 0.025: 17.53, 0.01: 20.09, 0.005: 21.96 },
              9: { 0.995:  1.73, 0.99:  2.09, 0.975:  2.70, 0.95:  3.33, 0.9:  4.17, 0.5:  8.34, 0.1: 14.68, 0.05: 16.92, 0.025: 19.02, 0.01: 21.67, 0.005: 23.59 },
              10: { 0.995:  2.16, 0.99:  2.56, 0.975:  3.25, 0.95:  3.94, 0.9:  4.87, 0.5:  9.34, 0.1: 15.99, 0.05: 18.31, 0.025: 20.48, 0.01: 23.21, 0.005: 25.19 },
              11: { 0.995:  2.60, 0.99:  3.05, 0.975:  3.82, 0.95:  4.57, 0.9:  5.58, 0.5: 10.34, 0.1: 17.28, 0.05: 19.68, 0.025: 21.92, 0.01: 24.72, 0.005: 26.76 },
              12: { 0.995:  3.07, 0.99:  3.57, 0.975:  4.40, 0.95:  5.23, 0.9:  6.30, 0.5: 11.34, 0.1: 18.55, 0.05: 21.03, 0.025: 23.34, 0.01: 26.22, 0.005: 28.30 },
              13: { 0.995:  3.57, 0.99:  4.11, 0.975:  5.01, 0.95:  5.89, 0.9:  7.04, 0.5: 12.34, 0.1: 19.81, 0.05: 22.36, 0.025: 24.74, 0.01: 27.69, 0.005: 29.82 },
              14: { 0.995:  4.07, 0.99:  4.66, 0.975:  5.63, 0.95:  6.57, 0.9:  7.79, 0.5: 13.34, 0.1: 21.06, 0.05: 23.68, 0.025: 26.12, 0.01: 29.14, 0.005: 31.32 },
              15: { 0.995:  4.60, 0.99:  5.23, 0.975:  6.27, 0.95:  7.26, 0.9:  8.55, 0.5: 14.34, 0.1: 22.31, 0.05: 25.00, 0.025: 27.49, 0.01: 30.58, 0.005: 32.80 },
              16: { 0.995:  5.14, 0.99:  5.81, 0.975:  6.91, 0.95:  7.96, 0.9:  9.31, 0.5: 15.34, 0.1: 23.54, 0.05: 26.30, 0.025: 28.85, 0.01: 32.00, 0.005: 34.27 },
              17: { 0.995:  5.70, 0.99:  6.41, 0.975:  7.56, 0.95:  8.67, 0.9: 10.09, 0.5: 16.34, 0.1: 24.77, 0.05: 27.59, 0.025: 30.19, 0.01: 33.41, 0.005: 35.72 },
              18: { 0.995:  6.26, 0.99:  7.01, 0.975:  8.23, 0.95:  9.39, 0.9: 10.87, 0.5: 17.34, 0.1: 25.99, 0.05: 28.87, 0.025: 31.53, 0.01: 34.81, 0.005: 37.16 },
              19: { 0.995:  6.84, 0.99:  7.63, 0.975:  8.91, 0.95: 10.12, 0.9: 11.65, 0.5: 18.34, 0.1: 27.20, 0.05: 30.14, 0.025: 32.85, 0.01: 36.19, 0.005: 38.58 },
              20: { 0.995:  7.43, 0.99:  8.26, 0.975:  9.59, 0.95: 10.85, 0.9: 12.44, 0.5: 19.34, 0.1: 28.41, 0.05: 31.41, 0.025: 34.17, 0.01: 37.57, 0.005: 40.00 },
              21: { 0.995:  8.03, 0.99:  8.90, 0.975: 10.28, 0.95: 11.59, 0.9: 13.24, 0.5: 20.34, 0.1: 29.62, 0.05: 32.67, 0.025: 35.48, 0.01: 38.93, 0.005: 41.40 },
              22: { 0.995:  8.64, 0.99:  9.54, 0.975: 10.98, 0.95: 12.34, 0.9: 14.04, 0.5: 21.34, 0.1: 30.81, 0.05: 33.92, 0.025: 36.78, 0.01: 40.29, 0.005: 42.80 },
              23: { 0.995:  9.26, 0.99: 10.20, 0.975: 11.69, 0.95: 13.09, 0.9: 14.85, 0.5: 22.34, 0.1: 32.01, 0.05: 35.17, 0.025: 38.08, 0.01: 41.64, 0.005: 44.18 },
              24: { 0.995:  9.89, 0.99: 10.86, 0.975: 12.40, 0.95: 13.85, 0.9: 15.66, 0.5: 23.34, 0.1: 33.20, 0.05: 36.42, 0.025: 39.36, 0.01: 42.98, 0.005: 45.56 },
              25: { 0.995: 10.52, 0.99: 11.52, 0.975: 13.12, 0.95: 14.61, 0.9: 16.47, 0.5: 24.34, 0.1: 34.28, 0.05: 37.65, 0.025: 40.65, 0.01: 44.31, 0.005: 46.93 },
              26: { 0.995: 11.16, 0.99: 12.20, 0.975: 13.84, 0.95: 15.38, 0.9: 17.29, 0.5: 25.34, 0.1: 35.56, 0.05: 38.89, 0.025: 41.92, 0.01: 45.64, 0.005: 48.29 },
              27: { 0.995: 11.81, 0.99: 12.88, 0.975: 14.57, 0.95: 16.15, 0.9: 18.11, 0.5: 26.34, 0.1: 36.74, 0.05: 40.11, 0.025: 43.19, 0.01: 46.96, 0.005: 49.65 },
              28: { 0.995: 12.46, 0.99: 13.57, 0.975: 15.31, 0.95: 16.93, 0.9: 18.94, 0.5: 27.34, 0.1: 37.92, 0.05: 41.34, 0.025: 44.46, 0.01: 48.28, 0.005: 50.99 },
              29: { 0.995: 13.12, 0.99: 14.26, 0.975: 16.05, 0.95: 17.71, 0.9: 19.77, 0.5: 28.34, 0.1: 39.09, 0.05: 42.56, 0.025: 45.72, 0.01: 49.59, 0.005: 52.34 },
              30: { 0.995: 13.79, 0.99: 14.95, 0.975: 16.79, 0.95: 18.49, 0.9: 20.60, 0.5: 29.34, 0.1: 40.26, 0.05: 43.77, 0.025: 46.98, 0.01: 50.89, 0.005: 53.67 },
              40: { 0.995: 20.71, 0.99: 22.16, 0.975: 24.43, 0.95: 26.51, 0.9: 29.05, 0.5: 39.34, 0.1: 51.81, 0.05: 55.76, 0.025: 59.34, 0.01: 63.69, 0.005: 66.77 },
              50: { 0.995: 27.99, 0.99: 29.71, 0.975: 32.36, 0.95: 34.76, 0.9: 37.69, 0.5: 49.33, 0.1: 63.17, 0.05: 67.50, 0.025: 71.42, 0.01: 76.15, 0.005: 79.49 },
              60: { 0.995: 35.53, 0.99: 37.48, 0.975: 40.48, 0.95: 43.19, 0.9: 46.46, 0.5: 59.33, 0.1: 74.40, 0.05: 79.08, 0.025: 83.30, 0.01: 88.38, 0.005: 91.95 },
              70: { 0.995: 43.28, 0.99: 45.44, 0.975: 48.76, 0.95: 51.74, 0.9: 55.33, 0.5: 69.33, 0.1: 85.53, 0.05: 90.53, 0.025: 95.02, 0.01: 100.42, 0.005: 104.22 },
              80: { 0.995: 51.17, 0.99: 53.54, 0.975: 57.15, 0.95: 60.39, 0.9: 64.28, 0.5: 79.33, 0.1: 96.58, 0.05: 101.88, 0.025: 106.63, 0.01: 112.33, 0.005: 116.32 },
              90: { 0.995: 59.20, 0.99: 61.75, 0.975: 65.65, 0.95: 69.13, 0.9: 73.29, 0.5: 89.33, 0.1: 107.57, 0.05: 113.14, 0.025: 118.14, 0.01: 124.12, 0.005: 128.30 },
              100: { 0.995: 67.33, 0.99: 70.06, 0.975: 74.22, 0.95: 77.93, 0.9: 82.36, 0.5: 99.33, 0.1: 118.50, 0.05: 124.34, 0.025: 129.56, 0.01: 135.81, 0.005: 140.17 }
          };

- ¶

  # χ2 (Chi-Squared) Goodness-of-Fit Test

  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, `chi_squared`, 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.

          function chi_squared_goodness_of_fit(data, distribution_type, significance) {

- ¶

  Estimate from the sample data, a weighted mean.

              var input_mean = mean(data),

- ¶

  Calculated value of the χ2 statistic.

                  chi_squared = 0,

- ¶

  Degrees of freedom, calculated as (number of class intervals - number of hypothesized distribution parameters estimated - 1)

                  degrees_of_freedom,

- ¶

  Number of hypothesized distribution parameters estimated, expected to be supplied in the distribution test. Lose one degree of freedom for estimating `lambda` from the sample data.

                  c = 1,

- ¶

  The hypothesized distribution. Generate the hypothesized distribution.

                  hypothesized_distribution = distribution_type(input_mean),
                  observed_frequencies = [],
                  expected_frequencies = [],
                  k;

- ¶

  Create an array holding a histogram from the sample data, of the form `{ value: numberOfOcurrences }`

              for (var i = 0; i < data.length; i++) {
                  if (observed_frequencies[data[i]] === undefined) {
                      observed_frequencies[data[i]] = 0;
                  }
                  observed_frequencies[data[i]]++;
              }

- ¶

  The histogram we created might be sparse - there might be gaps between values. So we iterate through the histogram, making sure that instead of undefined, gaps have 0 values.

              for (i = 0; i < observed_frequencies.length; i++) {
                  if (observed_frequencies[i] === undefined) {
                      observed_frequencies[i] = 0;
                  }
              }

- ¶

  Create an array holding a histogram of expected data given the sample size and hypothesized distribution.

              for (k in hypothesized_distribution) {
                  if (k in observed_frequencies) {
                      expected_frequencies[k] = hypothesized_distribution[k] * data.length;
                  }
              }

- ¶

  Working backward through the expected frequencies, collapse classes if less than three observations are expected for a class. This transformation is applied to the observed frequencies as well.

              for (k = expected_frequencies.length - 1; k >= 0; k--) {
                  if (expected_frequencies[k] < 3) {
                      expected_frequencies[k - 1] += expected_frequencies[k];
                      expected_frequencies.pop();

                      observed_frequencies[k - 1] += observed_frequencies[k];
                      observed_frequencies.pop();
                  }
              }

- ¶

  Iterate through the squared differences between observed & expected frequencies, accumulating the `chi_squared` statistic.

              for (k = 0; k < observed_frequencies.length; k++) {
                  chi_squared += Math.pow(
                      observed_frequencies[k] - expected_frequencies[k], 2) /
                      expected_frequencies[k];
              }

- ¶

  Calculate degrees of freedom for this test and look it up in the `chi_squared_distribution_table` in order to accept or reject the goodness-of-fit of the hypothesized distribution.

              degrees_of_freedom = observed_frequencies.length - c - 1;
              return chi_squared_distribution_table[degrees_of_freedom][significance] < chi_squared;
          }

- ¶

  # Mixin

  Mixin simple_statistics to a single Array instance if provided or the Array native object if not. This is an optional feature that lets you treat simple_statistics as a native feature of Javascript.

          function mixin(array) {
              var support = !!(Object.defineProperty && Object.defineProperties);
              if (!support) throw new Error('without defineProperty, simple-statistics cannot be mixed in');

- ¶

  only methods which work on basic arrays in a single step are supported

              var arrayMethods = ['median', 'standard_deviation', 'sum',
                  'sample_skewness',
                  'mean', 'min', 'max', 'quantile', 'geometric_mean',
                  'harmonic_mean'];

- ¶

  create a closure with a method name so that a reference like `arrayMethods[i]` doesn’t follow the loop increment

              function wrap(method) {
                  return function() {

- ¶

  cast any arguments into an array, since they’re natively objects

                      var args = Array.prototype.slice.apply(arguments);

- ¶

  make the first argument the array itself

                      args.unshift(this);

- ¶

  return the result of the ss method

                      return ss[method].apply(ss, args);
                  };
              }

- ¶

  select object to extend

              var extending;
              if (array) {

- ¶

  create a shallow copy of the array so that our internal operations do not change it by reference

                  extending = array.slice();
              } else {
                  extending = Array.prototype;
              }

- ¶

  for each array function, define a function that gets the array as the first argument. We use defineProperty because it allows these properties to be non-enumerable: `for (var in x)` loops will not run into problems with this implementation.

              for (var i = 0; i < arrayMethods.length; i++) {
                  Object.defineProperty(extending, arrayMethods[i], {
                      value: wrap(arrayMethods[i]),
                      configurable: true,
                      enumerable: false,
                      writable: true
                  });
              }

              return extending;
          }

          ss.linear_regression = linear_regression;
          ss.standard_deviation = standard_deviation;
          ss.r_squared = r_squared;
          ss.median = median;
          ss.mean = mean;
          ss.mode = mode;
          ss.min = min;
          ss.max = max;
          ss.sum = sum;
          ss.quantile = quantile;
          ss.quantile_sorted = quantile_sorted;
          ss.iqr = iqr;
          ss.mad = mad;

          ss.chunk = chunk;
          ss.shuffle = shuffle;
          ss.shuffle_in_place = shuffle_in_place;

          ss.sample = sample;

          ss.sample_covariance = sample_covariance;
          ss.sample_correlation = sample_correlation;
          ss.sample_variance = sample_variance;
          ss.sample_standard_deviation = sample_standard_deviation;
          ss.sample_skewness = sample_skewness;

          ss.geometric_mean = geometric_mean;
          ss.harmonic_mean = harmonic_mean;
          ss.variance = variance;
          ss.t_test = t_test;
          ss.t_test_two_sample = t_test_two_sample;

- ¶

  jenks

          ss.jenksMatrices = jenksMatrices;
          ss.jenksBreaks = jenksBreaks;
          ss.jenks = jenks;

          ss.bayesian = bayesian;

- ¶

  Distribution-related methods

          ss.epsilon = epsilon; // We make ε available to the test suite.
          ss.factorial = factorial;
          ss.bernoulli_distribution = bernoulli_distribution;
          ss.binomial_distribution = binomial_distribution;
          ss.poisson_distribution = poisson_distribution;
          ss.chi_squared_goodness_of_fit = chi_squared_goodness_of_fit;

- ¶

  Normal distribution

          ss.z_score = z_score;
          ss.cumulative_std_normal_probability = cumulative_std_normal_probability;
          ss.standard_normal_table = standard_normal_table;

- ¶

  Alias this into its common name

          ss.average = mean;
          ss.interquartile_range = iqr;
          ss.mixin = mixin;
          ss.median_absolute_deviation = mad;

      })(this);