Verify distribution uniformity/Chi-squared test

Verify distribution uniformity/Chi-squared test
You are encouraged to solve this task according to the task description, using any language you may know.

Write a function to determine whether a given set of frequency counts could plausibly have come from a uniform distribution by using the ${\displaystyle \chi ^{2}}$ test with a significance level of 5%.

The function should return a boolean that is true if and only if the distribution is one that a uniform distribution (with appropriate number of degrees of freedom) may be expected to produce.

Note: normally a two-tailed test would be used for this kind of problem.

Reference

11l

Translation of: Python
V a = 12
V k1_factrl = 1.0
[Float] c
c.append(sqrt(2.0 * math:pi))
L(k) 1 .< a
c.append(exp(a - k) * (a - k) ^ (k - 0.5) / k1_factrl)
k1_factrl *= -k

F gamma_spounge(z)
V accm = :c[0]
L(k) 1 .< :a
accm += :c[k] / (z + k)
accm *= exp(-(z + :a)) * (z + :a) ^ (z + 0.5)
R accm / z

F GammaInc_Q(a, x)
V a1 = a - 1
V a2 = a - 2
F f0(t)
R t ^ @a1 * exp(-t)

F df0(t)
R (@a1 - t) * t ^ @a2 * exp(-t)

V y = a1
L f0(y) * (x - y) > 2.0e-8 & y < x
y += 0.3
I y > x
y = x

V h = 3.0e-4
V n = Int(y / h)
h = y / n
V hh = 0.5 * h
V gamax = h * sum(((n - 1 .< -1).step(-1).map(j -> @h * j)).map(t -> @f0(t) + @hh * @df0(t)))

R gamax / gamma_spounge(a)

F chi2UniformDistance(dataSet)
V expected = sum(dataSet) * 1.0 / dataSet.len
V cntrd = (dataSet.map(d -> d - @expected))
R sum(cntrd.map(x -> x * x)) / expected

F chi2Probability(dof, distance)
R 1.0 - GammaInc_Q(0.5 * dof, 0.5 * distance)

F chi2IsUniform(dataSet, significance)
V dof = dataSet.len - 1
V dist = chi2UniformDistance(dataSet)
R chi2Probability(dof, dist) > significance

V dset1 = [199809, 200665, 199607, 200270, 199649]
V dset2 = [522573, 244456, 139979,  71531,  21461]

L(ds) (dset1, dset2)
print(‘Data set: ’ds)
V dof = ds.len - 1
V distance = chi2UniformDistance(ds)
print(‘dof: #. distance: #.4’.format(dof, distance), end' ‘ ’)
V prob = chi2Probability(dof, distance)
print(‘probability: #.4’.format(prob), end' ‘ ’)
print(‘uniform?  ’(I chi2IsUniform(ds, 0.05) {‘Yes’} E ‘No’))
Output:
Data set: [199809, 200665, 199607, 200270, 199649]
dof: 4 distance: 4.1463 probability: 0.3866 uniform?  Yes
Data set: [522573, 244456, 139979, 71531, 21461]
dof: 4 distance: 790063.2759 probability: -1.5002e-8 uniform?  No


First, we specify a simple package to compute the Chi-Square Distance from the uniform distribution:

package Chi_Square is

type Flt is digits 18;
type Bins_Type is array(Positive range <>) of Natural;

function Distance(Bins: Bins_Type) return Flt;

end Chi_Square;


Next, we implement that package:

package body Chi_Square is

function Distance(Bins: Bins_Type) return Flt is
Sum: Natural := 0;
Expected: Flt;
Result: Flt;
begin
for I in Bins'Range loop
if Bins(I) < 5 then
end if;
Sum := Sum + Bins(I);
end loop;
raise Program_Error with "too many (almost) empty bins";
end if;

Expected := Flt(Sum) / Flt(Bins'Length);
Result := 0.0;
for I in Bins'Range loop
Result := Result + ((Flt(Bins(I)) - Expected)**2) / Expected;
end loop;
return Result;
end Distance;

end Chi_Square;


Finally, we actually implement the Chi-square test. We do not actually compute the Chi-square probability; rather we hardcode a table of values for 5% significance level, which has been picked from Wikipedia [1]:

with Ada.Text_IO, Ada.Command_Line, Chi_Square; use Ada.Text_IO;

procedure Test_Chi_Square is

package Ch2 renames Chi_Square; use Ch2;
package FIO is new Float_IO(Flt);

Bound_For_5_Per_Cent: constant array(Positive range <>) of Flt :=
( 1 => 3.84,   2 =>  5.99,  3 =>  7.82,  4 => 9.49,   5 =>  11.07,
6 => 12.59,  7 => 14.07,  8 => 15.51,  9 => 16.92, 10 =>  18.31);
-- picked from http://en.wikipedia.org/wiki/Chi-squared_distribution

Dist: Flt;

begin
for I in B'Range loop
end loop;
Dist := Distance(B);
Put("Degrees of Freedom:" & Integer'Image(B'Length-1) & ", Distance: ");
FIO.Put(Dist, Fore => 6, Aft => 2, Exp => 0);
if Dist <= Bound_For_5_Per_Cent(B'Length-1) then
Put_Line("; (apparently uniform)");
else
Put_Line("; (deviates significantly from uniform)");
end if;
end;

Output:
$./Test_Chi_Square 199809 200665 199607 200270 199649 Degrees of Freedom: 4, Distance: 4.15; (apparently uniform)$ ./Test_Chi_Square 522573 244456 139979 71531 21461
Degrees of Freedom: 4, Distance: 790063.28; (deviates significantly from uniform)

C

This first sections contains the functions required to compute the Chi-Squared probability. These are not needed if a library containing the necessary function is availabile (e.g. see Numerical Integration, Gamma function).

#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#ifndef M_PI
#define M_PI 3.14159265358979323846
#endif

typedef double (* Ifctn)( double t);
/* Numerical integration method */
double Simpson3_8( Ifctn f, double a, double b, int N)
{
int j;
double l1;
double h = (b-a)/N;
double h1 = h/3.0;
double sum = f(a) + f(b);

for (j=3*N-1; j>0; j--) {
l1 = (j%3)? 3.0 : 2.0;
sum += l1*f(a+h1*j) ;
}
return h*sum/8.0;
}

#define A 12
double Gamma_Spouge( double z )
{
int k;
static double cspace[A];
static double *coefs = NULL;
double accum;
double a = A;

if (!coefs) {
double k1_factrl = 1.0;
coefs = cspace;
coefs[0] = sqrt(2.0*M_PI);
for(k=1; k<A; k++) {
coefs[k] = exp(a-k) * pow(a-k,k-0.5) / k1_factrl;
k1_factrl *= -k;
}
}

accum = coefs[0];
for (k=1; k<A; k++) {
accum += coefs[k]/(z+k);
}
accum *= exp(-(z+a)) * pow(z+a, z+0.5);
return accum/z;
}

double aa1;
double f0( double t)
{
return  pow(t, aa1)*exp(-t);
}

double GammaIncomplete_Q( double a, double x)
{
double y, h = 1.5e-2;  /* approximate integration step size */

/* this cuts off the tail of the integration to speed things up */
y = aa1 = a-1;
while((f0(y) * (x-y) > 2.0e-8) && (y < x))   y += .4;
if (y>x) y=x;

return 1.0 - Simpson3_8( &f0, 0, y, (int)(y/h))/Gamma_Spouge(a);
}


This section contains the functions specific to the task.

double chi2UniformDistance( double *ds, int dslen)
{
double expected = 0.0;
double sum = 0.0;
int k;

for (k=0; k<dslen; k++)
expected += ds[k];
expected /= k;

for (k=0; k<dslen; k++) {
double x = ds[k] - expected;
sum += x*x;
}
return sum/expected;
}

double chi2Probability( int dof, double distance)
{
return GammaIncomplete_Q( 0.5*dof, 0.5*distance);
}

int chiIsUniform( double *dset, int dslen, double significance)
{
int dof = dslen -1;
double dist = chi2UniformDistance( dset, dslen);
return chi2Probability( dof, dist ) > significance;
}


Testing

int main(int argc, char **argv)
{
double dset1[] = { 199809., 200665., 199607., 200270., 199649. };
double dset2[] = { 522573., 244456., 139979.,  71531.,  21461. };
double *dsets[] = { dset1, dset2 };
int     dslens[] = { 5, 5 };
int k, l;
double  dist, prob;
int dof;

for (k=0; k<2; k++) {
printf("Dataset: [ ");
for(l=0;l<dslens[k]; l++)
printf("%.0f, ", dsets[k][l]);

printf("]\n");
dist = chi2UniformDistance(dsets[k], dslens[k]);
dof = dslens[k]-1;
printf("dof: %d  distance: %.4f", dof, dist);
prob = chi2Probability( dof, dist );
printf(" probability: %.6f", prob);
printf(" uniform? %s\n", chiIsUniform(dsets[k], dslens[k], 0.05)? "Yes":"No");
}
return 0;
}


C++

#include <iostream>
#include <vector>
#include <cmath>
#include <iomanip>

void print_vector(const std::vector<int32_t>& list) {
std::cout << "[";
for ( uint64_t i = 0; i < list.size() - 1; ++i ) {
std::cout << list[i] << ", ";
}
std::cout << list.back() << "]" << std::endl;
}

bool is_significant(const double p_value, const double significance_level) {
return p_value > significance_level;
}

// The normalised lower incomplete gamma function.
double gamma_cdf(const double aX, const double aK) {
double result = 0.0;
for ( uint32_t m = 0; m <= 99; ++m ) {
result += pow(aX, m) / tgamma(aK + m + 1);
}
result *= pow(aX, aK) * exp(-aX);
return std::isnan(result) ? 1.0 : result;
}

// The cumulative probability function of the Chi-squared distribution.
double cdf(const double aX, const double aK) {
if ( aX > 1'000 && aK < 100 ) {
return 1.0;
}
return ( aX > 0.0 && aK > 0.0 ) ? gamma_cdf(aX / 2, aK / 2) : 0.0;
}

void chi_squared_test(const std::vector<int32_t>& observed) {
double sum = 0.0;
for ( uint64_t i = 0; i < observed.size(); ++i ) {
sum += observed[i];
}
const double expected = sum / observed.size();

const int32_t degree_freedom = observed.size() - 1;

double test_statistic = 0.0;
for ( uint64_t i = 0; i < observed.size(); ++i ) {
test_statistic += pow(observed[i] - expected, 2) / expected;
}

const double p_value = 1.0 - cdf(test_statistic, degree_freedom);

std::cout << "\nUniform distribution test" << std::setprecision(6) << std::endl;
std::cout << "    observed values   : "; print_vector(observed);
std::cout << "    expected value    : " << expected << std::endl;
std::cout << "    degrees of freedom: " << degree_freedom << std::endl;
std::cout << "    test statistic    : " << test_statistic << std::endl;
std::cout.setf(std::ios::fixed);
std::cout << "    p-value           : " << p_value << std::endl;
std::cout.unsetf(std::ios::fixed);
std::cout << "    is 5% significant?: " << std::boolalpha << is_significant(p_value, 0.05) << std::endl;
}

int main() {
const std::vector<std::vector<int32_t>> datasets = { { 199809, 200665, 199607, 200270, 199649 },
{ 522573, 244456, 139979, 71531, 21461 } };
for ( std::vector<int32_t> dataset : datasets ) {
chi_squared_test(dataset);
}
}

Output:

Uniform distribution test
observed values   : [199809, 200665, 199607, 200270, 199649]
expected value    : 200000
degrees of freedom: 4
test statistic    : 4.14628
p-value           : 0.386571
is 5% significant?: true

Uniform distribution test
observed values   : [522573, 244456, 139979, 71531, 21461]
expected value    : 200000
degrees of freedom: 4
test statistic    : 790063
p-value           : 0.000000
is 5% significant?: false


D

import std.stdio, std.algorithm, std.mathspecial;

real x2Dist(T)(in T[] data) pure nothrow @safe @nogc {
immutable avg = data.sum / data.length;
immutable sqs = reduce!((a, b) => a + (b - avg) ^^ 2)(0.0L, data);
return sqs / avg;
}

real x2Prob(in real dof, in real distance) pure nothrow @safe @nogc {
return gammaIncompleteCompl(dof / 2, distance / 2);
}

bool x2IsUniform(T)(in T[] data, in real significance=0.05L)
pure nothrow @safe @nogc {
return x2Prob(data.length - 1.0L, x2Dist(data)) > significance;
}

void main() {
immutable dataSets = [[199809, 200665, 199607, 200270, 199649],
[522573, 244456, 139979,  71531,  21461]];
writefln(" %4s %12s  %12s %8s   %s",
"dof", "distance", "probability", "Uniform?", "dataset");
foreach (immutable ds; dataSets) {
immutable dof = ds.length - 1;
immutable dist = ds.x2Dist;
immutable prob = x2Prob(dof, dist);
writefln("%4d %12.3f  %12.8f    %5s    %6s",
dof, dist, prob, ds.x2IsUniform ? "YES" : "NO", ds);
}
}

Output:
  dof     distance   probability Uniform?   dataset
4        4.146    0.38657083      YES    [199809, 200665, 199607, 200270, 199649]
4   790063.276    0.00000000       NO    [522573, 244456, 139979,  71531,  21461]

Elixir

Translation of: Ruby
defmodule Verify do
defp gammaInc_Q(a, x) do
a1 = a-1
f0  = fn t -> :math.pow(t, a1) * :math.exp(-t) end
df0 = fn t -> (a1-t) * :math.pow(t, a-2) * :math.exp(-t) end
y = while_loop(f0, x, a1)
n = trunc(y / 3.0e-4)
h = y / n
hh = 0.5 * h
sum = Enum.reduce(n-1 .. 0, 0, fn j,sum ->
t = h * j
sum + f0.(t) + hh * df0.(t)
end)
h * sum / gamma_spounge(a, make_coef)
end

defp while_loop(f, x, y) do
if f.(y)*(x-y) > 2.0e-8 and y < x, do: while_loop(f, x, y+0.3), else: min(x, y)
end

@a  12
defp make_coef do
coef0 = [:math.sqrt(2.0 * :math.pi)]
{_, coef} = Enum.reduce(1..@a-1, {1.0, coef0}, fn k,{k1_factrl,c} ->
h = :math.exp(@a-k) * :math.pow(@a-k, k-0.5) / k1_factrl
{-k1_factrl*k, [h | c]}
end)
Enum.reverse(coef) |> List.to_tuple
end

defp gamma_spounge(z, coef) do
accm = Enum.reduce(1..@a-1, elem(coef,0), fn k,res -> res + elem(coef,k) / (z+k) end)
accm * :math.exp(-(z+@a)) * :math.pow(z+@a, z+0.5) / z
end

def chi2UniformDistance(dataSet) do
expected = Enum.sum(dataSet) / length(dataSet)
Enum.reduce(dataSet, 0, fn d,sum -> sum + (d-expected)*(d-expected) end) / expected
end

def chi2Probability(dof, distance) do
1.0 - gammaInc_Q(0.5*dof, 0.5*distance)
end

def chi2IsUniform(dataSet, significance\\0.05) do
dof = length(dataSet) - 1
dist = chi2UniformDistance(dataSet)
chi2Probability(dof, dist) > significance
end
end

dsets = [ [ 199809, 200665, 199607, 200270, 199649 ],
[ 522573, 244456, 139979,  71531,  21461 ] ]

Enum.each(dsets, fn ds ->
IO.puts "Data set:#{inspect ds}"
dof = length(ds) - 1
IO.puts "  degrees of freedom: #{dof}"
distance = Verify.chi2UniformDistance(ds)
:io.fwrite "  distance:           ~.4f~n", [distance]
:io.fwrite "  probability:        ~.4f~n", [Verify.chi2Probability(dof, distance)]
:io.fwrite "  uniform?            ~s~n", [(if Verify.chi2IsUniform(ds), do: "Yes", else: "No")]
end)

Output:
Data set:[199809, 200665, 199607, 200270, 199649]
degrees of freedom: 4
distance:           4.1463
probability:        0.3866
uniform?            Yes
Data set:[522573, 244456, 139979, 71531, 21461]
degrees of freedom: 4
distance:           790063.2759
probability:        -0.0000
uniform?            No


Fortran

Works with: Fortran version 95

Instead of implementing the chi-squared distribution by ourselves, we bind to GNU Scientific Library; so we need a module to interface to the function we need (gsl_cdf_chisq_Q)

module gsl_mini_bind_m

use iso_c_binding
implicit none
private

public :: p_value

interface
function gsl_cdf_chisq_q(x, nu) bind(c, name='gsl_cdf_chisq_Q')
import
real(c_double), value :: x
real(c_double), value :: nu
real(c_double) :: gsl_cdf_chisq_q
end function gsl_cdf_chisq_q
end interface

contains

!> Get p-value from chi-square distribution
real function p_value(x, df)
real, intent(in) :: x
integer, intent(in) :: df

p_value = real(gsl_cdf_chisq_q(real(x, c_double), real(df, c_double)))

end function p_value

end module gsl_mini_bind_m


program chi2test

use gsl_mini_bind_m, only: p_value
implicit none

real :: dset1(5) = [199809., 200665., 199607., 200270., 199649.]
real :: dset2(5) = [522573., 244456., 139979., 71531., 21461.]

real :: dist, prob
integer :: dof

write (*, '(A)', advance='no') "Dataset 1:"
write (*, '(5(F12.4,:,1x))') dset1

dist = chisq(dset1)
dof = size(dset1) - 1
write (*, '(A,I4,A,F12.4)') 'dof: ', dof, '   chisq: ', dist
prob = p_value(dist, dof)
write (*, '(A,F12.4)') 'probability: ', prob
write (*, '(A,L)') 'uniform? ', prob > 0.05

! Lazy copy/past :|
write (*, '(/A)', advance='no') "Dataset 2:"
write (*, '(5(F12.4,:,1x))') dset2

dist = chisq(dset2)
dof = size(dset2) - 1
write (*, '(A,I4,A,F12.4)') 'dof: ', dof, '   chisq: ', dist
prob = p_value(dist, dof)
write (*, '(A,F12.4)') 'probability: ', prob
write (*, '(A,L)') 'uniform? ', prob > 0.05

contains

!> Get chi-square value for a set of data ds
real function chisq(ds)
real, intent(in) :: ds(:)

real :: expected, summa

expected = sum(ds)/size(ds)
summa = sum((ds - expected)**2)
chisq = summa/expected

end function chisq

end program chi2test


Output:

Dataset 1: 199809.0000  200665.0000  199607.0000  200270.0000  199649.0000
dof:    4   chisq:       4.1463
probability:       0.3866
uniform? T

Dataset 2: 522573.0000  244456.0000  139979.0000   71531.0000   21461.0000
dof:    4   chisq:  790063.2500
probability:       0.0000
uniform? F

Go

Translation of: C

Go has a nice gamma function in the library. Otherwise, it's mostly a port from C. Note, this implementation of the incomplete gamma function works for these two test cases, but, I believe, has serious limitations. See talk page.

package main

import (
"fmt"
"math"
)

type ifctn func(float64) float64

func simpson38(f ifctn, a, b float64, n int) float64 {
h := (b - a) / float64(n)
h1 := h / 3
sum := f(a) + f(b)
for j := 3*n - 1; j > 0; j-- {
if j%3 == 0 {
sum += 2 * f(a+h1*float64(j))
} else {
sum += 3 * f(a+h1*float64(j))
}
}
return h * sum / 8
}

func gammaIncQ(a, x float64) float64 {
aa1 := a - 1
var f ifctn = func(t float64) float64 {
return math.Pow(t, aa1) * math.Exp(-t)
}
y := aa1
h := 1.5e-2
for f(y)*(x-y) > 2e-8 && y < x {
y += .4
}
if y > x {
y = x
}
return 1 - simpson38(f, 0, y, int(y/h/math.Gamma(a)))
}

func chi2ud(ds []int) float64 {
var sum, expected float64
for _, d := range ds {
expected += float64(d)
}
expected /= float64(len(ds))
for _, d := range ds {
x := float64(d) - expected
sum += x * x
}
return sum / expected
}

func chi2p(dof int, distance float64) float64 {
return gammaIncQ(.5*float64(dof), .5*distance)
}

const sigLevel = .05

func main() {
for _, dset := range [][]int{
{199809, 200665, 199607, 200270, 199649},
{522573, 244456, 139979, 71531, 21461},
} {
utest(dset)
}
}

func utest(dset []int) {
fmt.Println("Uniform distribution test")
var sum int
for _, c := range dset {
sum += c
}
fmt.Println(" dataset:", dset)
fmt.Println(" samples:                      ", sum)
fmt.Println(" categories:                   ", len(dset))

dof := len(dset) - 1
fmt.Println(" degrees of freedom:           ", dof)

dist := chi2ud(dset)
fmt.Println(" chi square test statistic:    ", dist)

p := chi2p(dof, dist)
fmt.Println(" p-value of test statistic:    ", p)

sig := p < sigLevel
fmt.Printf(" significant at %2.0f%% level?      %t\n", sigLevel*100, sig)
fmt.Println(" uniform?                      ", !sig, "\n")
}


Output:

Uniform distribution test
dataset: [199809 200665 199607 200270 199649]
samples:                       1000000
categories:                    5
degrees of freedom:            4
chi square test statistic:     4.14628
p-value of test statistic:     0.3865708330827673
significant at  5% level?      false
uniform?                       true

Uniform distribution test
dataset: [522573 244456 139979 71531 21461]
samples:                       1000000
categories:                    5
degrees of freedom:            4
chi square test statistic:     790063.27594
p-value of test statistic:     2.3528290427066167e-11
significant at  5% level?      true
uniform?                       false


Hy

(import
[scipy.stats [chisquare]]
[collections [Counter]])

(defn uniform? [f repeats &optional [alpha .05]]
"Call 'f' 'repeats' times and do a chi-squared test for uniformity
of the resulting discrete distribution. Return false iff the
null hypothesis of uniformity is rejected for the test with
size 'alpha'."
(<= alpha (second (chisquare
(.values (Counter (take repeats (repeatedly f))))))))


Examples of use:

(import [random [randint]])

(for [f [
(fn [] (randint 1 10))
(fn [] (if (randint 0 1) (randint 1 9) (randint 1 10)))]]
(print (uniform? f 5000)))


J

Solution (Tacit):

require 'stats/base'

countCats=: #@~.                    NB. counts the number of unique items
getExpected=: #@] % [               NB. divides no of items by category count
getObserved=: #/.~@]                NB. counts frequency for each category
calcX2=: [: +/ *:@(getObserved - getExpected) % getExpected   NB. calculates test statistic
calcDf=: <:@[                       NB. calculates degrees of freedom for uniform distribution

NB.*isUniform v Tests (5%) whether y is uniformly distributed
NB. result is: boolean describing if distribution y is uniform
NB. y is: distribution to test
NB. x is: optionally specify number of categories possible
isUniform=: (countCats $: ]) : (0.95 > calcDf chisqcdf :: 1: calcX2)  Solution (Explicit): require 'stats/base' NB.*isUniformX v Tests (5%) whether y is uniformly distributed NB. result is: boolean describing if distribution y is uniform NB. y is: distribution to test NB. x is: optionally specify number of categories possible isUniformX=: verb define (#~. y) isUniformX y : signif=. 0.95 NB. set significance level expected=. (#y) % x NB. number of items divided by the category count observed=. #/.~ y NB. frequency count for each category X2=. +/ (*: observed - expected) % expected NB. the test statistic degfreedom=. <: x NB. degrees of freedom signif > degfreedom chisqcdf :: 1: X2 )  Example Usage:  FairDistrib=: 1e6 ?@$ 5
UnfairDistrib=: (9.5e5 ?@$5) , (5e4 ?@$ 4)
isUniformX FairDistrib
1
isUniformX UnfairDistrib
0
isUniform 4 4 4 5 5 5 5 5 5 5     NB. uniform if only 2 categories possible
1
4 isUniform 4 4 4 5 5 5 5 5 5 5   NB. not uniform if 4 categories possible
0


Java

Translation of: D
Works with: Java version 8
import static java.lang.Math.pow;
import java.util.Arrays;
import static java.util.Arrays.stream;
import org.apache.commons.math3.special.Gamma;

public class Test {

static double x2Dist(double[] data) {
double avg = stream(data).sum() / data.length;
double sqs = stream(data).reduce(0, (a, b) -> a + pow((b - avg), 2));
return sqs / avg;
}

static double x2Prob(double dof, double distance) {
return Gamma.regularizedGammaQ(dof / 2, distance / 2);
}

static boolean x2IsUniform(double[] data, double significance) {
return x2Prob(data.length - 1.0, x2Dist(data)) > significance;
}

public static void main(String[] a) {
double[][] dataSets = {{199809, 200665, 199607, 200270, 199649},
{522573, 244456, 139979, 71531, 21461}};

System.out.printf(" %4s %12s  %12s %8s   %s%n",
"dof", "distance", "probability", "Uniform?", "dataset");

for (double[] ds : dataSets) {
int dof = ds.length - 1;
double dist = x2Dist(ds);
double prob = x2Prob(dof, dist);
System.out.printf("%4d %12.3f  %12.8f    %5s    %6s%n",
dof, dist, prob, x2IsUniform(ds, 0.05) ? "YES" : "NO",
Arrays.toString(ds));
}
}
}

  dof     distance   probability Uniform?   dataset
4        4,146    0,38657083      YES    [199809.0, 200665.0, 199607.0, 200270.0, 199649.0]
4   790063,276    0,00000000       NO    [522573.0, 244456.0, 139979.0, 71531.0, 21461.0]

jq

Works with: jq

Also works with gojq, the Go implementation of jq.

This entry uses a two-tailed test, as is appropriate for this type of problem as illustrated by the last example. The test is based on the assumption that the sample size is large enough for the χ2 distribution to be used.

The implementation of Chi2_cdf here uses the recursion relation for the gamma function and should be both fast, accurate and quite robust. For an industrial-strength algorithm, see e.g. https://people.sc.fsu.edu/~jburkardt/c_src/asa239/asa239.c

Generic Functions

def round($dec): if type == "string" then . else pow(10;$dec) as $m | . *$m | floor / $m end; # sum of squares def ss(s): reduce s as$x (0; . + ($x *$x));

# Cumulative density function of the chi-squared distribution with $k # degrees of freedom # The recursion formula for gamma is used for efficiency and robustness. def Chi2_cdf($x; $k): if$x == 0 then 0
elif $x > (1e3 *$k) then 1
else 1e-15 as $tol # for example | { s: 0, m: 0, term: (1 / ((($k/2)+1)|gamma)) }
| until (.term|length < $tol; # length here is abs .s += .term | .m += 1 | .term *= (($x/2) / (($k/2) + .m )) ) | .s * ( ((-$x/2) + ($k/2)*(($x/2)|log)) | exp)
end ;

# Input: array of frequencies
def chi2UniformDistance:
(add / length) as $expected | ss(.[] -$expected) / $expected; # Input: a number # Output: an indication of the probability of observing this value or higher # assuming the value is drawn from a chi-squared distribution with$dof degrees
#   of freedom
def chi2Probability($dof): (1 - Chi2_cdf(.;$dof))
| if . < 1e-10 then "< 1e-10"
else .
end;

# Input: array of frequencies
# Output: result of a two-tailed test based on the chi-squared statistic
# assuming the sample size is large enough
def chiIsUniform($significance): (length - 1) as$dof
| chi2UniformDistance
| Chi2_cdf(.; $dof) as$cdf
| if $cdf then ($significance/2) as $s |$cdf > $s and$cdf < (1-$s) else false end; def dsets: [ [199809, 200665, 199607, 200270, 199649], [522573, 244456, 139979, 71531, 21461], [19,14,6,18,7,5,1], # low entropy [9,11,9,10,15,11,5], # high entropy [20,20,20] # made-up ]; def task: dsets[] | "Dataset: \(.)", ( chi2UniformDistance as$dist
| (length - 1) as $dof | "DOF: \($dof)  D (Distance): \($dist)", " Estimated probability of observing a value >= D: \($dist|chi2Probability($dof)|round(2))", " Uniform? \( (select(chiIsUniform(0.05)) | "Yes") // "No" )\n" ) ; task Dataset: [199809,200665,199607,200270,199649] DOF: 4 D (Distance): 4.14628 Estimated probability of observing a value >= D: 0.38 Uniform? Yes Dataset: [522573,244456,139979,71531,21461] DOF: 4 D (Distance): 790063.27594 Estimated probability of observing a value >= D: < 1e-10 Uniform? No Dataset: [19,14,6,18,7,5,1] DOF: 6 D (Distance): 29.2 Estimated probability of observing a value >= D: 0 Uniform? No Dataset: [9,11,9,10,15,11,5] DOF: 6 D (Distance): 5.4 Estimated probability of observing a value >= D: 0.49 Uniform? Yes Dataset: [20,20,20] DOF: 2 D (Distance): 0 Estimated probability of observing a value >= D: 1 Uniform? No  Julia # v0.6 using Distributions function eqdist(data::Vector{T}, α::Float64=0.05)::Bool where T <: Real if ! (0 ≤ α ≤ 1); error("α must be in [0, 1]") end exp = mean(data) chisqval = sum((x - exp) ^ 2 for x in data) / exp pval = ccdf(Chisq(2), chisqval) return pval > α end data1 = [199809, 200665, 199607, 200270, 199649] data2 = [522573, 244456, 139979, 71531, 21461] for data in (data1, data2) println("Data:\n$data")
println("Hypothesis test: the original population is ", (eqdist(data) ? "" : "not "), "uniform.\n")
end

Output:
Data:
[199809, 200665, 199607, 200270, 199649]
Hypothesis test: the original population is uniform.

Data:
[522573, 244456, 139979, 71531, 21461]
Hypothesis test: the original population is not uniform.


Kotlin

This program reuses Kotlin code from the Gamma function and Numerical Integration tasks but otherwise is a translation of the C entry for this task.

// version 1.1.51

typealias Func = (Double) -> Double

fun gammaLanczos(x: Double): Double {
var xx = x
val p = doubleArrayOf(
0.99999999999980993,
676.5203681218851,
-1259.1392167224028,
771.32342877765313,
-176.61502916214059,
12.507343278686905,
-0.13857109526572012,
9.9843695780195716e-6,
1.5056327351493116e-7
)
val g = 7
if (xx < 0.5) return Math.PI / (Math.sin(Math.PI * xx) * gammaLanczos(1.0 - xx))
xx--
var a = p[0]
val t = xx + g + 0.5
for (i in 1 until p.size) a += p[i] / (xx + i)
return Math.sqrt(2.0 * Math.PI) * Math.pow(t, xx + 0.5) * Math.exp(-t) * a
}

fun integrate(a: Double, b: Double, n: Int, f: Func): Double {
val h = (b - a) / n
var sum = 0.0
for (i in 0 until n) {
val x = a + i * h
sum += (f(x) + 4.0 * f(x + h / 2.0) + f(x + h)) / 6.0
}
return sum * h
}

fun gammaIncompleteQ(a: Double, x: Double): Double {
val aa1 = a - 1.0
fun f0(t: Double) = Math.pow(t, aa1) * Math.exp(-t)
val h = 1.5e-2
var y = aa1
while ((f0(y) * (x - y) > 2.0e-8) && y < x) y += 0.4
if (y > x) y = x
return 1.0 - integrate(0.0, y, (y / h).toInt(), ::f0) / gammaLanczos(a)
}

fun chi2UniformDistance(ds: DoubleArray): Double {
val expected = ds.average()
val sum = ds.map { val x = it - expected; x * x }.sum()
return sum / expected
}

fun chi2Probability(dof: Int, distance: Double) =
gammaIncompleteQ(0.5 * dof, 0.5 * distance)

fun chiIsUniform(ds: DoubleArray, significance: Double):Boolean {
val dof = ds.size - 1
val dist = chi2UniformDistance(ds)
return chi2Probability(dof, dist) > significance
}

fun main(args: Array<String>) {
val dsets = listOf(
doubleArrayOf(199809.0, 200665.0, 199607.0, 200270.0, 199649.0),
doubleArrayOf(522573.0, 244456.0, 139979.0,  71531.0,  21461.0)
)
for (ds in dsets) {
println("Dataset: ${ds.asList()}") val dist = chi2UniformDistance(ds) val dof = ds.size - 1 print("DOF:$dof  Distance: ${"%.4f".format(dist)}") val prob = chi2Probability(dof, dist) print(" Probability:${"%.6f".format(prob)}")
val uniform = if (chiIsUniform(ds, 0.05)) "Yes" else "No"
println("  Uniform? $uniform\n") } }  Output: Dataset: [199809.0, 200665.0, 199607.0, 200270.0, 199649.0] DOF: 4 Distance: 4.1463 Probability: 0.386571 Uniform? Yes Dataset: [522573.0, 244456.0, 139979.0, 71531.0, 21461.0] DOF: 4 Distance: 790063.2759 Probability: 0.000000 Uniform? No  Mathematica/Wolfram Language This code explicity assumes a discrete uniform distribution since the chi square test is a poor test choice for continuous distributions and requires Mathematica version 2 or later discreteUniformDistributionQ[data_, {min_Integer, max_Integer}, confLevel_: .05] := If[$VersionNumber >= 8,
confLevel <= PearsonChiSquareTest[data, DiscreteUniformDistribution[{min, max}]],
Block[{v, k = max - min, n = Length@data},
v = (k + 1) (Plus @@ (((Length /@ Split[Sort@data]))^2))/n - n;
GammaRegularized[k/2, 0, v/2] <= 1 - confLevel]]

discreteUniformDistributionQ[data_] :=discreteUniformDistributionQ[data, data[[Ordering[data][[{1, -1}]]]]]


code used to create test data requires Mathematica version 6 or later

uniformData = RandomInteger[10, 100];
nonUniformData = Total@RandomInteger[10, {5, 100}];

{discreteUniformDistributionQ[uniformData],discreteUniformDistributionQ[nonUniformData]}

Output:
{True,False}

Nim

Translation of: Go

We use the gamma function from the “math” module. To simplify the code, we use also the “lenientops” module which provides mixed operations between floats ane integers.

import lenientops, math, stats, strformat, sugar

func simpson38(f: (float) -> float; a, b: float; n: int): float =
let h = (b - a) / n
let h1 = h / 3
var sum = f(a) + f(b)
for i in countdown(3 * n - 1, 1):
if i mod 3 == 0:
sum += 2 * f(a + h1 * i)
else:
sum += 3 * f(a + h1 * i)
result = h * sum / 8

func gammaIncQ(a, x: float): float =
let aa1 = a - 1
func f(t: float): float = pow(t, aa1) * exp(-t)
var y = aa1
let h = 1.5e-2
while f(y) * (x - y) > 2e-8 and y < x:
y += 0.4
if y > x: y = x
result = 1 - simpson38(f, 0, y, (y / h / gamma(a)).toInt)

func chi2ud(ds: openArray[int]): float =
let expected = mean(ds)
var s = 0.0
for d in ds:
let x = d.toFloat - expected
s += x * x
result = s / expected

func chi2p(dof: int; distance: float): float =
gammaIncQ(0.5 * dof, 0.5 * distance)

const SigLevel = 0.05

proc utest(dset: openArray[int]) =

echo "Uniform distribution test"
let s = sum(dset)
echo " dataset:", dset
echo " samples:                      ", s
echo " categories:                   ", dset.len

let dof = dset.len - 1
echo " degrees of freedom:           ", dof

let dist = chi2ud(dset)
echo " chi square test statistic:    ", dist

let p = chi2p(dof, dist)
echo " p-value of test statistic:    ", p

let sig = p < SigLevel
echo &" significant at {int(SigLevel * 100)}% level?      {sig}"
echo &" uniform?                      {not sig}\n"

for dset in [[199809, 200665, 199607, 200270, 199649],
[522573, 244456, 139979, 71531, 21461]]:
utest(dset)

Output:
Uniform distribution test
dataset:[199809, 200665, 199607, 200270, 199649]
samples:                      1000000
categories:                   5
degrees of freedom:           4
chi square test statistic:    4.14628
p-value of test statistic:    0.3865708330827673
significant at 5% level?      false
uniform?                      true

Uniform distribution test
dataset:[522573, 244456, 139979, 71531, 21461]
samples:                      1000000
categories:                   5
degrees of freedom:           4
chi square test statistic:    790063.27594
p-value of test statistic:    2.34864350190378e-11
significant at 5% level?      true
uniform?                      false

OCaml

This code needs to be compiled with library gsl.cma.

let sqr x = x *. x

let chi2UniformDistance distrib =
let count, len = Array.fold_left (fun (s, c) e -> s + e, succ c)
(0, 0) distrib in
let expected = float count /. float len in
let distance = Array.fold_left (fun s e ->
s +. sqr (float e -. expected) /. expected
) 0. distrib in
let dof = float (pred len) in
dof, distance

let chi2Proba dof distance =
Gsl_sf.gamma_inc_Q (0.5 *. dof) (0.5 *. distance)

let chi2IsUniform distrib significance =
let dof, distance = chi2UniformDistance distrib in
let likelihoodOfRandom = chi2Proba dof distance in
likelihoodOfRandom > significance

let _ =
List.iter (fun distrib ->
let dof, distance = chi2UniformDistance distrib in
Printf.printf "distribution ";
Array.iter (Printf.printf "\t%d") distrib;
Printf.printf "\tdistance %g" distance;
Printf.printf "\t[%g > 0.05]" (chi2Proba dof distance);
if chi2IsUniform distrib 0.05 then Printf.printf " fair\n"
else Printf.printf " unfair\n"
)
[
[| 199809; 200665; 199607; 200270; 199649 |];
[| 522573; 244456; 139979; 71531; 21461 |]
]


Output

distribution    199809  200665  199607  200270  199649  distance 4.14628        [0.386571 > 0.05] fair
distribution    522573  244456  139979  71531   21461   distance 790063 [0 > 0.05] unfair


PARI/GP

The solution is very easy in GP since PARI includes a good incomplete gamma implementation; the sum function is also useful for clarity. Most of the code is just for displaying results.

The sample data for the test was taken from Go.

cumChi2(chi2,dof)={
my(g=gamma(dof/2));
incgam(dof/2,chi2/2,g)/g
};
test(v,alpha=.05)={
my(chi2,p,s=sum(i=1,#v,v[i]),ave=s/#v);
print("chi^2 statistic: ",chi2=sum(i=1,#v,(v[i]-ave)^2)/ave);
print("p-value: ",p=cumChi2(chi2,#v-1));
if(p<alpha,
print("Significant at the alpha = "alpha" level: not uniform");
,
print("Not significant at the alpha = "alpha" level: uniform");
)
};

test([199809, 200665, 199607, 200270, 199649])
test([522573, 244456, 139979, 71531, 21461])

Perl

Translation of: Raku
use List::Util qw(sum reduce);
use constant pi => 3.14159265;

sub incomplete_G_series {
my($s,$z) = @_;
my $n = 10; push @numers,$z**$_ for 1..$n;
my @denoms = $s+1; push @denoms,$denoms[-1]*($s+$_) for 2..$n; my$M = 1;
$M +=$numers[$_-1]/$denoms[$_-1] for 1..$n;
$z**$s / $s * exp(-$z) * $M; } sub G_of_half { my($n) = @_;
if ($n % 2) { f(2*$_) / (4**$_ * f($_)) * sqrt(pi) for int ($n-1) / 2 } else { f(($n/2)-1) }
}

sub f { reduce { $a *$b } 1, 1 .. $_[0] } # factorial sub chi_squared_cdf { my($k, $x) = @_; my$f = $k < 20 ? 20 : 10; if ($x == 0)                  { 0.0 }
elsif ($x <$k + $f*sqrt($k)) { incomplete_G_series($k/2,$x/2) / G_of_half($k) } else { 1.0 } } sub chi_squared_test { my(@bins) = @_;$significance = 0.05;
my $n = @bins; my$N = sum @bins;
my $expected =$N / $n; my$chi_squared = sum map { ($_ -$expected)**2 / $expected } @bins; my$p_value = 1 - chi_squared_cdf($n-1,$chi_squared);
return $chi_squared,$p_value, $p_value >$significance ? 'True' : 'False';
}

for $dataset ([199809, 200665, 199607, 200270, 199649], [522573, 244456, 139979, 71531, 21461]) { printf "C2 = %10.3f, p-value = %.3f, uniform = %s\n", chi_squared_test(@$dataset);
}

Output:
C2 =      4.146, p-value = 0.387, uniform = True
C2 = 790063.276, p-value = 0.000, uniform = False

Phix

Translation of: Go
with javascript_semantics
function f(atom aa1, t)
return power(t, aa1) * exp(-t)
end function

function simpson38(atom aa1, a, b, integer n)
atom h := (b-a)/n,
h1 := h/3,
tot := f(aa1,a) + f(aa1,b)
for j=3*n-1 to 1 by -1 do
tot += (3-(mod(j,3)=0)) * f(aa1,a+h1*j)
end for
return h*tot/8
end function

--<copy of gamma from Gamma_function#Phix>
sequence c = repeat(0,12)

function gamma(atom z)
atom accm = c[1]
if accm=0 then
accm = sqrt(2*PI)
c[1] = accm
atom k1_factrl = 1  -- (k - 1)!*(-1)^k with 0!==1
for k=2 to 12 do
c[k] = exp(13-k)*power(13-k,k-1.5)/k1_factrl
k1_factrl *= -(k-1)
end for
end if
for k=2 to 12 do
accm += c[k]/(z+k-1)
end for
accm *= exp(-(z+12))*power(z+12,z+0.5) -- Gamma(z+1)
return accm/z
end function
--</copy of gamma>

function gammaIncQ(atom a, x)
atom aa1 := a-1,
y := aa1,
h := 1.5e-2
while f(aa1,y)*(x-y) > 2e-8 and y < x do
y += 0.4
end while
if y > x then y = x end if
return 1 - simpson38(aa1,0,y,floor(y/h/gamma(a)))
end function

function chi2ud(sequence ds)
atom expected = sum(ds)/length(ds),
tot = sum(sq_power(sq_sub(ds,expected),2))
end function

function chi2p(integer dof, atom distance)
return gammaIncQ(0.5*dof,0.5*distance)
end function

constant sigLevel = 0.05

procedure utest(sequence dset)
printf(1,"Uniform distribution test\n")
integer tot = sum(dset),
dof := length(dset)-1
atom dist := chi2ud(dset),
p := chi2p(dof, dist)
bool sig := p < sigLevel
printf(1," dataset: %v\n",{dset})
printf(1," samples:                   %d\n", tot)
printf(1," categories:                %d\n", length(dset))
printf(1," degrees of freedom:        %d\n", dof)
printf(1," chi square test statistic: %g\n", dist)
printf(1," p-value of test statistic: %g\n", p)
printf(1," significant at %.0f%% level?   %t\n", {sigLevel*100, sig})
printf(1," uniform?                   %t\n",not sig)
end procedure

utest({199809, 200665, 199607, 200270, 199649})
utest({522573, 244456, 139979, 71531, 21461})

Output:
Uniform distribution test
dataset: {199809,200665,199607,200270,199649}
samples:                   1000000
categories:                5
degrees of freedom:        4
chi square test statistic: 4.14628
p-value of test statistic: 0.386571
significant at 5% level?   false
uniform?                   true
Uniform distribution test
dataset: {522573,244456,139979,71531,21461}
samples:                   1000000
categories:                5
degrees of freedom:        4
chi square test statistic: 790063
p-value of test statistic: 2.35282e-11
significant at 5% level?   true
uniform?                   false


Python

Python: Using only standard libraries

Implements the Chi Square Probability function with an integration. I'm sure there are better ways to do this. Compare to OCaml implementation.

import math
import random

def GammaInc_Q( a, x):
a1 = a-1
a2 = a-2
def f0( t ):
return t**a1*math.exp(-t)

def df0(t):
return (a1-t)*t**a2*math.exp(-t)

y = a1
while f0(y)*(x-y) >2.0e-8 and y < x: y += .3
if y > x: y = x

h = 3.0e-4
n = int(y/h)
h = y/n
hh = 0.5*h
gamax = h * sum( f0(t)+hh*df0(t) for t in ( h*j for j in xrange(n-1, -1, -1)))

return gamax/gamma_spounge(a)

c = None
def gamma_spounge( z):
global c
a = 12

if c is None:
k1_factrl = 1.0
c = []
c.append(math.sqrt(2.0*math.pi))
for k in range(1,a):
c.append( math.exp(a-k) * (a-k)**(k-0.5) / k1_factrl )
k1_factrl *= -k

accm = c[0]
for k in range(1,a):
accm += c[k] / (z+k)
accm *= math.exp( -(z+a)) * (z+a)**(z+0.5)
return accm/z;

def chi2UniformDistance( dataSet ):
expected = sum(dataSet)*1.0/len(dataSet)
cntrd = (d-expected for d in dataSet)
return sum(x*x for x in cntrd)/expected

def chi2Probability(dof, distance):
return 1.0 - GammaInc_Q( 0.5*dof, 0.5*distance)

def chi2IsUniform(dataSet, significance):
dof = len(dataSet)-1
dist = chi2UniformDistance(dataSet)
return chi2Probability( dof, dist ) > significance

dset1 = [ 199809, 200665, 199607, 200270, 199649 ]
dset2 = [ 522573, 244456, 139979,  71531,  21461 ]

for ds in (dset1, dset2):
print "Data set:", ds
dof = len(ds)-1
distance =chi2UniformDistance(ds)
print "dof: %d distance: %.4f" % (dof, distance),
prob = chi2Probability( dof, distance)
print "probability: %.4f"%prob,
print "uniform? ", "Yes"if chi2IsUniform(ds,0.05) else "No"


Output:

Data set: [199809, 200665, 199607, 200270, 199649]
dof: 4 distance: 4.146280 probability: 0.3866 uniform?  Yes
Data set: [522573, 244456, 139979, 71531, 21461]
dof: 4 distance: 790063.275940 probability: 0.0000 uniform?  No

Python: Using scipy

This uses the library routine scipy.stats.chisquare.

from scipy.stats import chisquare

if __name__ == '__main__':
dataSets = [[199809, 200665, 199607, 200270, 199649],
[522573, 244456, 139979,  71531,  21461]]
print(f"{'Distance':^12} {'pvalue':^12} {'Uniform?':^8} {'Dataset'}")
for ds in dataSets:
dist, pvalue = chisquare(ds)
uni = 'YES' if pvalue > 0.05 else 'NO'
print(f"{dist:12.3f} {pvalue:12.8f} {uni:^8} {ds}")

Output:
  Distance      pvalue    Uniform? Dataset
4.146   0.38657083   YES    [199809, 200665, 199607, 200270, 199649]
790063.276   0.00000000    NO    [522573, 244456, 139979, 71531, 21461]

R

R being a statistical computating language, the chi-squared test is built in with the function "chisq.test"


dset1=c(199809,200665,199607,200270,199649)
dset2=c(522573,244456,139979,71531,21461)

chi2IsUniform<-function(dataset,significance=0.05){
chi2IsUniform=(chisq.test(dataset)$p.value>significance) } for (ds in list(dset1,dset2)){ print(c("Data set:",ds)) print(chisq.test(ds)) print(paste("uniform?",chi2IsUniform(ds))) }  Output: [1] "Data set:" "199809" "200665" "199607" "200270" "199649" Chi-squared test for given probabilities data: ds X-squared = 4.1463, df = 4, p-value = 0.3866 [1] "uniform? TRUE" [1] "Data set:" "522573" "244456" "139979" "71531" "21461" Chi-squared test for given probabilities data: ds X-squared = 790063.3, df = 4, p-value < 2.2e-16 [1] "uniform? FALSE"  Racket #lang racket (require racket/flonum (planet williams/science:4:5/science) (only-in (planet williams/science/unsafe-ops-utils) real->float)) ; (chi^2-goodness-of-fit-test observed expected df) ; Given: observed, a sequence of observed frequencies ; expected, a sequence of expected frequencies ; df, the degrees of freedom ; Result: P-value = 1-chi^2cdf(X^2,df) , the p-value (define (chi^2-goodness-of-fit-test observed expected df) (define X^2 (for/sum ([o observed] [e expected]) (/ (sqr (- o e)) e))) (- 1.0 (chi-squared-cdf X^2 df))) (define (is-uniform? rand n α) ; Use significance level α to test whether ; n small random numbers generated by rand ; have a uniform distribution. ; Observed values: (define o (make-vector 10 0)) ; generate n random integers from 0 to 9. (for ([_ (+ n 1)]) (define r (rand 10)) (vector-set! o r (+ (vector-ref o r) 1))) ; Expected values: (define ex (make-vector 10 (/ n 10))) ; Calculate the P-value: (define P (chi^2-goodness-of-fit-test o ex (- n 1))) ; If the P-value is larger than α we accept the ; hypothesis that the numbers are distributed uniformly. (> P α)) ; Test whether the builtin generator is uniform: (is-uniform? random 1000 0.05) ; Test whether the constant generator fails: (is-uniform? (λ(_) 5) 1000 0.05)  Output: #t #f  Raku (formerly Perl 6) For the incomplete gamma function we use a series expansion related to Kummer's confluent hypergeometric function (see http://en.wikipedia.org/wiki/Incomplete_gamma_function#Evaluation_formulae). The gamma function is calculated in closed form, as we only need its value at integers and half integers. sub incomplete-γ-series($s, $z) { my \numers =$z X** 1..*;
my \denoms = [\*] $s X+ 1..*; my$M = 1 + [+] (numers Z/ denoms) ... * < 1e-6;
$z**$s / $s * exp(-$z) * $M; } sub postfix:<!>(Int$n) { [*] 2..$n } sub Γ-of-half(Int$n where * > 0) {
($n %% 2) ?? (($_-1)!                            given  $n div 2) !! ((2*$_)! / (4**$_ *$_!) * sqrt(pi) given ($n-1) div 2); } # degrees of freedom constrained due to numerical limitations sub chi-squared-cdf(Int$k where 1..200, $x where * >= 0) { my$f = $k < 20 ?? 20 !! 10; given$x {
when 0                    { 0.0 }
when * < $k +$f*sqrt($k) { incomplete-γ-series($k/2, $x/2) / Γ-of-half($k) }
default                   { 1.0 }
}
}

sub chi-squared-test(@bins, :$significance = 0.05) { my$n = +@bins;
my $N = [+] @bins; my$expected = $N /$n;
my $chi-squared = [+] @bins.map: { ($^bin - $expected)**2 /$expected }
my $p-value = 1 - chi-squared-cdf($n-1, $chi-squared); return (:$chi-squared, :$p-value, :uniform($p-value > $significance)); } for [< 199809 200665 199607 200270 199649 >], [< 522573 244456 139979 71531 21461 >] ->$dataset
{
my %t = chi-squared-test($dataset); say 'data: ',$dataset;
say "χ² = {%t<chi-squared>}, p-value = {%t<p-value>.fmt('%.4f')}, uniform = {%t<uniform>}";
}

Output:
data: 199809 200665 199607 200270 199649
χ² = 4.14628, p-value = 0.3866, uniform = True
data: 522573 244456 139979 71531 21461
χ² = 790063.27594, p-value = 0.0000, uniform = False

REXX

Translation of: Go

Programming notes:

The use of the   pow   was elided as it can just be replaced with   t**(a-1).

The   gamma   was replaced with a simple version.   The argument for   gamma   is (in the cases used herein)   always
positive,   and is either an integer,   or a number which is a multiple of   1/2,   both of these cases can be calculated with
a straight─forward calculation.

/*REXX program performs a chi─squared test to verify a given distribution is uniform.   */
numeric digits length( pi() )  - length(.)       /*enough decimal digs for calculations.*/
@.=;                                        @.1= 199809 200665 199607 200270 199649
@.2= 522573 244456 139979  71531  21461
do s=1  while @.s\=='';  call uTest @.s  /*invoke  uTest with a data set of #'s.*/
end   /*s*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
!:      procedure; parse arg x;  p=1;    do j=2  to x;   p= p*j;   end  /*j*/;    return p
chi2p:  procedure;  parse arg dof, distance;       return gammaI( dof/2,  distance/2 )
f:      parse arg t;   if t=0  then return 0;      return t ** (a-1)    *    exp(-t)
e:      e =2.718281828459045235360287471352662497757247093699959574966967627724; return e
pi:     pi=3.141592653589793238462643383279502884197169399375105820974944592308; return pi
/*──────────────────────────────────────────────────────────────────────────────────────*/
!!:     procedure; parse arg x;                   if x<2  then return 1;    p= x
do k=2+x//2  to x-1  by 2;   p= p*k;   end  /*k*/;          return p
/*──────────────────────────────────────────────────────────────────────────────────────*/
chi2ud: procedure: parse arg ds; sum=0;                       expect= 0
do j=1  for words(ds); expect= expect + word(ds, j)
end   /*j*/
expect = expect / words(ds)
do k=1  for words(ds)
sum= sum   +   (word(ds, k) - expect) **2
end   /*k*/
return sum / expect
/*──────────────────────────────────────────────────────────────────────────────────────*/
exp:    procedure; parse arg x; ix= x%1;  if abs(x-ix)>.5  then ix= ix + sign(x);  x= x-ix
z=1; _=1; w=z;    do j=1;  _= _*x/j;  z= (z + _)/1;  if z==w  then leave;      w=z
end  /*j*/;         if z\==0  then z= z * e()**ix;      return z
/*──────────────────────────────────────────────────────────────────────────────────────*/
gamma:  procedure; parse arg x; if datatype(x, 'W')  then return !(x-1) /*Int?  Use fact*/
n= trunc(x)                     /*at this point, X is pos and a multiple of 1/2.*/
d= !!(n+n - 1)                  /*compute the double factorial of:    2*n - 1.  */
if n//2  then p= -1             /*if  N  is  odd,   then use a negative unity.  */
else p=  1             /*if  N  is even,   then use a positive unity.  */
if x>0   then return p * d * sqrt(pi()) / (2**n)
return p * (2**n) * sqrt(pi()) / d
/*──────────────────────────────────────────────────────────────────────────────────────*/
gammaI: procedure; parse arg a,x;  y= a-1;   do  while f(y)*(x-y) > 2e-8 & y<x;  y= y + .4
end  /*while*/
y= min(x, y)
return 1   -   simp38(0, y,   y / 0.015 / gamma(a-1) % 1)
/*──────────────────────────────────────────────────────────────────────────────────────*/
simp38: procedure; parse arg a, b, n;                h= (b-a) / n;        h1= h / 3
sum= f(a) + f(b)
do j=3*n-1   by -1   while j>0
if j//3 == 0  then sum= sum   +   2 * f(a + h1*j)
else sum= sum   +   3 * f(a + h1*j)
end   /*j*/
return h * sum / 8
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt:   procedure; parse arg x;  if x=0  then return 0; d=digits(); numeric digits; h= d+6
numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g "E" _ .;g=g *.5'e'_%2
do j=0  while h>9;      m.j=h;               h=h%2+1;       end  /*j*/
do k=j+5  to 0  by -1;  numeric digits m.k;  g=(g+x/g)*.5;  end  /*k*/; return g
/*──────────────────────────────────────────────────────────────────────────────────────*/
uTest:  procedure; parse arg dset;  sum= 0;   pad= left('', 11);      sigLev= 1/20  /*5%*/
say;   say '     '   center(" Uniform distribution test ", 75, '═')
#= words(dset);                                              sigPC= sigLev*100/1
do j=1  for #;      sum= sum  +  word(dset, j)
end   /*j*/
say pad "                  dataset: "  dset
say pad "                  samples: "  sum
say pad "               categories: "  #
say pad "       degrees of freedom: "  # - 1
dist= chi2ud(dset)
P= chi2p(# - 1,  dist)
sig = (abs(P) < dist * sigLev)
say pad "significant at " sigPC'%  level? '  word('no yes',    sig  + 1)
say pad "   is the dataset uniform? "        word('no yes', (\(sig))+ 1)
return

output   when using the default inputs:
      ════════════════════════ Uniform distribution test ════════════════════════
dataset:  199809 200665 199607 200270 199649
samples:  1000000
categories:  5
degrees of freedom:  4
significant at  5%  level?  no
is the dataset uniform?  yes

════════════════════════ Uniform distribution test ════════════════════════
dataset:  522573 244456 139979 71531 21461
samples:  1000000
categories:  5
degrees of freedom:  4
significant at  5%  level?  yes
is the dataset uniform?  no


Ruby

Translation of: Python
def gammaInc_Q(a, x)
a1, a2 = a-1, a-2
f0  = lambda {|t| t**a1 * Math.exp(-t)}
df0 = lambda {|t| (a1-t) * t**a2 * Math.exp(-t)}

y = a1
y += 0.3  while f0[y]*(x-y) > 2.0e-8 and y < x
y = x  if y > x

h = 3.0e-4
n = (y/h).to_i
h = y/n
hh = 0.5 * h
sum = 0
(n-1).step(0, -1) do |j|
t = h * j
sum += f0[t] + hh * df0[t]
end
h * sum / gamma_spounge(a)
end

A = 12
k1_factrl = 1.0
coef = [Math.sqrt(2.0*Math::PI)]
COEF = (1...A).each_with_object(coef) do |k,c|
c << Math.exp(A-k) * (A-k)**(k-0.5) / k1_factrl
k1_factrl *= -k
end

def gamma_spounge(z)
accm = (1...A).inject(COEF[0]){|res,k| res += COEF[k] / (z+k)}
accm * Math.exp(-(z+A)) * (z+A)**(z+0.5) / z
end

def chi2UniformDistance(dataSet)
expected = dataSet.inject(:+).to_f / dataSet.size
dataSet.map{|d|(d-expected)**2}.inject(:+) / expected
end

def chi2Probability(dof, distance)
1.0 - gammaInc_Q(0.5*dof, 0.5*distance)
end

def chi2IsUniform(dataSet, significance=0.05)
dof = dataSet.size - 1
dist = chi2UniformDistance(dataSet)
chi2Probability(dof, dist) > significance
end

dsets = [ [ 199809, 200665, 199607, 200270, 199649 ],
[ 522573, 244456, 139979,  71531,  21461 ] ]

for ds in dsets
puts "Data set:#{ds}"
dof = ds.size - 1
puts "  degrees of freedom: %d" % dof
distance = chi2UniformDistance(ds)
puts "  distance:           %.4f" % distance
puts "  probability:        %.4f" % chi2Probability(dof, distance)
puts "  uniform?            %s" % (chi2IsUniform(ds) ? "Yes" : "No")
end

Output:
Data set:[199809, 200665, 199607, 200270, 199649]
degrees of freedom: 4
distance:           4.1463
probability:        0.3866
uniform?            Yes
Data set:[522573, 244456, 139979, 71531, 21461]
degrees of freedom: 4
distance:           790063.2759
probability:        -0.0000
uniform?            No


Rust

use statrs::function::gamma::gamma_li;

fn chi_distance(dataset: &[u32]) -> f64 {
let expected = f64::from(dataset.iter().sum::<u32>()) / dataset.len() as f64;
dataset
.iter()
.fold(0., |acc, &elt| acc + (elt as f64 - expected).powf(2.))
/ expected
}

fn chi2_probability(dof: f64, distance: f64) -> f64 {
1. - gamma_li(dof * 0.5, distance * 0.5)
}

fn chi2_uniform(dataset: &[u32], significance: f64) -> bool {
let d = chi_distance(&dataset);
chi2_probability(dataset.len() as f64 - 1., d) > significance
}

fn main() {
let dsets = vec![
vec![199809, 200665, 199607, 200270, 199649],
vec![522573, 244456, 139979, 71531, 21461],
];

for ds in dsets {
println!("Data set: {:?}", ds);
let d = chi_distance(&ds);
print!("Distance: {:.6} ", d);
print!(
"Chi2 probability: {:.6} ",
chi2_probability(ds.len() as f64 - 1., d)
);
print!("Uniform? {}\n", chi2_uniform(&ds, 0.05));
}
}

Output:
Data set: [199809, 200665, 199607, 200270, 199649]
Distance: 4.146280 Chi2 probability: 0.386571 Uniform? true
Data set: [522573, 244456, 139979, 71531, 21461]
Distance: 790063.275940 Chi2 probability: 0.000000 Uniform? false


Scala

Output:
See it yourself by running in your browser Scastie (remote JVM).
Works with: Scala version 2.13
import org.apache.commons.math3.special.Gamma.regularizedGammaQ

object ChiSquare extends App {
private val dataSets: Seq[Seq[Double]] =
Seq(
Seq(199809, 200665, 199607, 200270, 199649),
Seq(522573, 244456, 139979, 71531, 21461)
)

private def χ2IsUniform(data: Seq[Double], significance: Double) =
χ2Prob(data.size - 1.0, χ2Dist(data)) > significance

private def χ2Dist(data: Seq[Double]) = {
val avg = data.sum / data.size

data.reduce((a, b) => a + math.pow(b - avg, 2)) / avg
}

private def χ2Prob(dof: Double, distance: Double) =
regularizedGammaQ(dof / 2, distance / 2)

printf(" %4s %10s  %12s %8s   %s%n",
"d.f.", "χ²distance", "χ²probability", "Uniform?", "dataset")
dataSets.foreach { ds =>
val (dist, dof) = (χ2Dist(ds), ds.size - 1)

printf("%4d %11.3f  %13.8f    %5s    %6s%n",
dof, dist, χ2Prob(dof.toDouble, dist), if (χ2IsUniform(ds, 0.05)) "YES" else "NO", ds.mkString(", "))
}
}


Sidef

# Confluent hypergeometric function of the first kind F_1(a;b;z)
func F1(a, b, z, limit=100) {
sum(0..limit, {|k|
rising_factorial(a, k) / rising_factorial(b, k) * z**k / k!
})
}

func γ(a,x) {    # lower incomplete gamma function γ(a,x)
#a**(-1) * x**a * F1(a, a+1, -x)            # simpler formula
a**(-1) * x**a * exp(-x) * F1(1, a+1, x)    # slightly better convergence
}

func P(a,z) {    # regularized gamma function P(a,z)
γ(a,z) / Γ(a)
}

func chi_squared_cdf (k, x) {
var f = (k<20 ? 20 : 10)
given(x) {
when (0) { 0 }
case (. < (k + f*sqrt(k))) { P(k/2, x/2) }
else { 1 }
}
}

func chi_squared_test(arr, significance = 0.05) {
var n = arr.len
var N = arr.sum
var expected = N/n
var χ_squared = arr.sum_by {|v| (v-expected)**2 / expected }
var p_value = (1 - chi_squared_cdf(n-1, χ_squared))
[χ_squared, p_value, p_value > significance]
}

[
%n< 199809 200665 199607 200270 199649 >,
%n< 522573 244456 139979  71531  21461 >,
].each {|dataset|
var r = chi_squared_test(dataset)
say "data: #{dataset}"
say "χ² = #{r[0]}, p-value = #{r[1].round(-4)}, uniform = #{r[2]}\n"
}

Output:
data: [199809, 200665, 199607, 200270, 199649]
χ² = 4.14628, p-value = 0.3866, uniform = true

data: [522573, 244456, 139979, 71531, 21461]
χ² = 790063.27594, p-value = 0, uniform = false


Tcl

Works with: Tcl version 8.5
Library: Tcllib (Package: math::statistics)
package require Tcl 8.5
package require math::statistics

proc isUniform {distribution {significance 0.05}} {
set count [tcl::mathop::+ {*}[dict values $distribution]] set expected [expr {double($count) / [dict size $distribution]}] set X2 0.0 foreach value [dict values$distribution] {
set X2 [expr {$X2 + ($value - $expected)**2 /$expected}]
}
set degreesOfFreedom [expr {[dict size $distribution] - 1}] set likelihoodOfRandom [::math::statistics::incompleteGamma \ [expr {$degreesOfFreedom / 2.0}] [expr {$X2 / 2.0}]] expr {$likelihoodOfRandom > $significance} }  Testing: proc makeDistribution {operation {count 1000000}} { for {set i 0} {$i<$count} {incr i} {incr distribution([uplevel 1$operation])}
return [array get distribution]
}

set distFair [makeDistribution {expr int(rand()*5)}]
puts "distribution \"$distFair\" assessed as [expr [isUniform$distFair]?{fair}:{unfair}]"
set distUnfair [makeDistribution {expr int(rand()*rand()*5)}]
puts "distribution \"$distUnfair\" assessed as [expr [isUniform$distUnfair]?{fair}:{unfair}]"


Output:

distribution "0 199809 4 199649 1 200665 2 199607 3 200270" assessed as fair
distribution "4 21461 0 522573 1 244456 2 139979 3 71531" assessed as unfair

VBA

The built in worksheetfunction ChiSq_Dist of Excel VBA is used. Output formatted like R.

Private Function Test4DiscreteUniformDistribution(ObservationFrequencies() As Variant, Significance As Single) As Boolean
'Returns true if the observed frequencies pass the Pearson Chi-squared test at the required significance level.
Dim Total As Long, Ei As Long, i As Integer
Dim ChiSquared As Double, DegreesOfFreedom As Integer, p_value As Double
Debug.Print "[1] ""Data set:"" ";
For i = LBound(ObservationFrequencies) To UBound(ObservationFrequencies)
Total = Total + ObservationFrequencies(i)
Debug.Print ObservationFrequencies(i); " ";
Next i
DegreesOfFreedom = UBound(ObservationFrequencies) - LBound(ObservationFrequencies)
'This is exactly the number of different categories minus 1
Ei = Total / (DegreesOfFreedom + 1)
For i = LBound(ObservationFrequencies) To UBound(ObservationFrequencies)
ChiSquared = ChiSquared + (ObservationFrequencies(i) - Ei) ^ 2 / Ei
Next i
p_value = 1 - WorksheetFunction.ChiSq_Dist(ChiSquared, DegreesOfFreedom, True)
Debug.Print
Debug.Print "   Chi-squared test for given frequencies"
Debug.Print "X-squared ="; ChiSquared; ", ";
Debug.Print "df ="; DegreesOfFreedom; ", ";
Debug.Print "p-value = "; Format(p_value, "0.0000")
Test4DiscreteUniformDistribution = p_value > Significance
End Function
Public Sub test()
Dim O() As Variant
O = [{199809,200665,199607,200270,199649}]
Debug.Print "[1] ""Uniform? "; Test4DiscreteUniformDistribution(O, 0.05); """"
O = [{522573,244456,139979,71531,21461}]
Debug.Print "[1] ""Uniform? "; Test4DiscreteUniformDistribution(O, 0.05); """"
End Sub
{{out}
[1] "Data set:"  199809   200665   199607   200270   199649
Chi-squared test for given frequencies
X-squared = 4.14628 , df = 4 , p-value = 0.3866
[1] "Uniform? True"
[1] "Data set:"  522573   244456   139979   71531   21461
Chi-squared test for given frequencies
X-squared = 790063.27594 , df = 4 , p-value = 0.0000
[1] "Uniform? False"

V (Vlang)

Translation of: Go
import math

type Ifctn = fn(f64) f64

fn simpson38(f Ifctn, a f64, b f64, n int) f64 {
h := (b - a) / f64(n)
h1 := h / 3
mut sum := f(a) + f(b)
for j := 3*n - 1; j > 0; j-- {
if j%3 == 0 {
sum += 2 * f(a+h1*f64(j))
} else {
sum += 3 * f(a+h1*f64(j))
}
}
return h * sum / 8
}

fn gamma_inc_q(a f64, x f64) f64 {
aa1 := a - 1
f := Ifctn(fn[aa1](t f64) f64 {
return math.pow(t, aa1) * math.exp(-t)
})
mut y := aa1
h := 1.5e-2
for f(y)*(x-y) > 2e-8 && y < x {
y += .4
}
if y > x {
y = x
}
return 1 - simpson38(f, 0, y, int(y/h/math.gamma(a)))
}

fn chi2ud(ds []int) f64 {
mut sum, mut expected := 0.0,0.0
for d in ds {
expected += f64(d)
}
expected /= f64(ds.len)
for d in ds {
x := f64(d) - expected
sum += x * x
}
return sum / expected
}

fn chi2p(dof int, distance f64) f64 {
return gamma_inc_q(.5*f64(dof), .5*distance)
}

const sig_level = .05

fn main() {
for dset in [
[199809, 200665, 199607, 200270, 199649],
[522573, 244456, 139979, 71531, 21461],
] {
utest(dset)
}
}

fn utest(dset []int) {
println("Uniform distribution test")
mut sum := 0
for c in dset {
sum += c
}
println(" dataset: $dset") println(" samples:$sum")
println(" categories:                   $dset.len") dof := dset.len - 1 println(" degrees of freedom:$dof")

dist := chi2ud(dset)
println(" chi square test statistic:    $dist") p := chi2p(dof, dist) println(" p-value of test statistic:$p")

sig := p < sig_level
println(" significant at ${sig_level*100:2.0f}% level?$sig")
println(" uniform?                      ${!sig}\n") } Output: Uniform distribution test dataset: [199809 200665 199607 200270 199649] samples: 1000000 categories: 5 degrees of freedom: 4 chi square test statistic: 4.14628 p-value of test statistic: 0.3865708330827673 significant at 5% level? false uniform? true Uniform distribution test dataset: [522573 244456 139979 71531 21461] samples: 1000000 categories: 5 degrees of freedom: 4 chi square test statistic: 790063.27594 p-value of test statistic: 2.3528290427066167e-11 significant at 5% level? true uniform? false  Wren Translation of: Kotlin Library: Wren-math Library: Wren-fmt import "/math" for Math, Nums import "/fmt" for Fmt var integrate = Fn.new { |a, b, n, f| var h = (b - a) / n var sum = 0 for (i in 0...n) { var x = a + i*h sum = sum + (f.call(x) + 4 * f.call(x + h/2) + f.call(x + h)) / 6 } return sum * h } var gammaIncomplete = Fn.new { |a, x| var am1 = a - 1 var f0 = Fn.new { |t| t.pow(am1) * (-t).exp } var h = 1.5e-2 var y = am1 while ((f0.call(y) * (x - y) > 2e-8) && y < x) y = y + 0.4 if (y > x) y = x return 1 - integrate.call(0, y, (y/h).truncate, f0) / Math.gamma(a) } var chi2UniformDistance = Fn.new { |ds| var expected = Nums.mean(ds) var sum = Nums.sum(ds.map { |d| (d - expected).pow(2) }.toList) return sum / expected } var chi2Probability = Fn.new { |dof, dist| gammaIncomplete.call(0.5*dof, 0.5*dist) } var chiIsUniform = Fn.new { |ds, significance| var dof = ds.count - 1 var dist = chi2UniformDistance.call(ds) return chi2Probability.call(dof, dist) > significance } var dsets = [ [199809, 200665, 199607, 200270, 199649], [522573, 244456, 139979, 71531, 21461] ] for (ds in dsets) { System.print("Dataset: %(ds)") var dist = chi2UniformDistance.call(ds) var dof = ds.count - 1 Fmt.write("DOF:$d  Distance: $.4f", dof, dist) var prob = chi2Probability.call(dof, dist) Fmt.write(" Probability:$.6f", prob)
var uniform = chiIsUniform.call(ds, 0.05) ? "Yes" : "No"
System.print("  Uniform? %(uniform)\n")
}
Output:
Dataset: [199809, 200665, 199607, 200270, 199649]
DOF: 4  Distance: 4.1463  Probability: 0.386571  Uniform? Yes

Dataset: [522573, 244456, 139979, 71531, 21461]
DOF: 4  Distance: 790063.2759  Probability: 0.000000  Uniform? No


zkl

Translation of: C
Translation of: D
/* Numerical integration method */
fcn Simpson3_8(f,a,b,N){  // fcn,double,double,Int --> double
h,h1:=(b - a)/N, h/3.0;
h*[1..3*N - 1].reduce('wrap(sum,j){
l1:=(if(j%3) 3.0 else 2.0);
sum + l1*f(a + h1*j);
},f(a) + f(b))/8.0;
}

const A=12;
fcn Gamma_Spouge(z){  // double --> double
var coefs=fcn{  // this runs only once, at construction time
a,coefs:=A.toFloat(),(A).pump(List(),0.0);
k1_factrl:=1.0;
coefs[0]=(2.0*(0.0).pi).sqrt();
foreach k in ([1.0..A-1]){
coefs[k]=(a - k).exp() * (a - k).pow(k - 0.5) / k1_factrl;
k1_factrl*=-k;
}
coefs
}();

( [1..A-1].reduce('wrap(accum,k){ accum + coefs[k]/(z + k) },coefs[0])
* (-(z + A)).exp()*(z + A).pow(z + 0.5) )
/ z;
}

fcn f0(t,aa1){ t.pow(aa1)*(-t).exp() }

fcn GammaIncomplete_Q(a,x){  // double,double --> double
h:=1.5e-2;  /* approximate integration step size */
/* this cuts off the tail of the integration to speed things up */
y:=a - 1; f:=f0.fp1(y);
while((f(y)*(x - y)>2.0e-8) and (y<x)){ y+=0.4; }
if(y>x) y=x;
1.0 - Simpson3_8(f,0.0,y,(y/h).toInt())/Gamma_Spouge(a);
}
fcn chi2UniformDistance(ds){ // --> double
dslen   :=ds.len();
expected:=dslen.reduce('wrap(sum,k){ sum + ds[k] },0.0)/dslen;
sum     := dslen.reduce('wrap(sum,k){ x:=ds[k] - expected; sum + x*x },0.0);
sum/expected
}

fcn chi2Probability(dof,distance){ GammaIncomplete_Q(0.5*dof, 0.5*distance) }

fcn chiIsUniform(dset,significance=0.05){
significance < chi2Probability(-1.0 + dset.len(),chi2UniformDistance(dset))
}
datasets:=T( T(199809.0, 200665.0, 199607.0, 200270.0, 199649.0),
T(522573.0, 244456.0, 139979.0,  71531.0,  21461.0) );
println(" %4s %12s  %12s %8s   %s".fmt(
"dof", "distance", "probability", "Uniform?", "dataset"));
foreach ds in (datasets){
dof :=ds.len() - 1;
dist:=chi2UniformDistance(ds);
prob:=chi2Probability(dof,dist);
println("%4d %12.3f  %12.8f    %5s    %6s".fmt(
dof, dist, prob, chiIsUniform(ds) and "YES" or "NO",
ds.concat(",")));
}
Output:
  dof     distance   probability Uniform?   dataset
4        4.146    0.38657083      YES    199809,200665,199607,200270,199649
4   790063.276    0.00000000       NO    522573,244456,139979,71531,21461