Statistics/Normal distribution

From Rosetta Code
Task
Statistics/Normal distribution
You are encouraged to solve this task according to the task description, using any language you may know.

The Normal (or Gaussian) distribution is a frequently used distribution in statistics. While most programming languages provide a uniformly distributed random number generator, one can derive normally distributed random numbers from a uniform generator.


The task
  1. Take a uniform random number generator and create a large (you decide how large) set of numbers that follow a normal (Gaussian) distribution. Calculate the dataset's mean and stddev, and show the histogram here.
  2. Mention any native language support for the generation of normally distributed random numbers.


Reference



C

/*
* RosettaCode example: Statistics/Normal distribution in C
*
* The random number generator rand() of the standard C library is obsolete
* and should not be used in more demanding applications. There are plenty
* libraries with advanced features (eg. GSL) with functions to calculate
* the mean, the standard deviation, generating random numbers etc.
* However, these features are not the core of the standard C library.
*/

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <string.h>
#include <time.h>
 
 
#define NMAX 10000000
 
 
double mean(double* values, int n)
{
int i;
double s = 0;
 
for ( i = 0; i < n; i++ )
s += values[i];
return s / n;
}
 
 
double stddev(double* values, int n)
{
int i;
double average = mean(values,n);
double s = 0;
 
for ( i = 0; i < n; i++ )
s += (values[i] - average) * (values[i] - average);
return sqrt(s / (n - 1));
}
 
/*
* Normal random numbers generator - Marsaglia algorithm.
*/

double* generate(int n)
{
int i;
int m = n + n % 2;
double* values = (double*)calloc(m,sizeof(double));
double average, deviation;
 
if ( values )
{
for ( i = 0; i < m; i += 2 )
{
double x,y,rsq,f;
do {
x = 2.0 * rand() / (double)RAND_MAX - 1.0;
y = 2.0 * rand() / (double)RAND_MAX - 1.0;
rsq = x * x + y * y;
}while( rsq >= 1. || rsq == 0. );
f = sqrt( -2.0 * log(rsq) / rsq );
values[i] = x * f;
values[i+1] = y * f;
}
}
return values;
}
 
 
void printHistogram(double* values, int n)
{
const int width = 50;
int max = 0;
 
const double low = -3.05;
const double high = 3.05;
const double delta = 0.1;
 
int i,j,k;
int nbins = (int)((high - low) / delta);
int* bins = (int*)calloc(nbins,sizeof(int));
if ( bins != NULL )
{
for ( i = 0; i < n; i++ )
{
int j = (int)( (values[i] - low) / delta );
if ( 0 <= j && j < nbins )
bins[j]++;
}
 
for ( j = 0; j < nbins; j++ )
if ( max < bins[j] )
max = bins[j];
 
for ( j = 0; j < nbins; j++ )
{
printf("(%5.2f, %5.2f) |", low + j * delta, low + (j + 1) * delta );
k = (int)( (double)width * (double)bins[j] / (double)max );
while(k-- > 0) putchar('*');
printf("  %-.1f%%", bins[j] * 100.0 / (double)n);
putchar('\n');
}
 
free(bins);
}
}
 
 
int main(void)
{
double* seq;
 
srand((unsigned int)time(NULL));
 
if ( (seq = generate(NMAX)) != NULL )
{
printf("mean = %g, stddev = %g\n\n", mean(seq,NMAX), stddev(seq,NMAX));
printHistogram(seq,NMAX);
free(seq);
 
printf("\n%s\n", "press enter");
getchar();
return EXIT_SUCCESS;
}
return EXIT_FAILURE;
}
Output:
mean = 0.000477941, stddev = 0.999945

(-3.05, -2.95) |  0.1%
(-2.95, -2.85) |  0.1%
(-2.85, -2.75) |*  0.1%
(-2.75, -2.65) |*  0.1%
(-2.65, -2.55) |*  0.1%
(-2.55, -2.45) |**  0.2%
(-2.45, -2.35) |**  0.2%
(-2.35, -2.25) |***  0.3%
(-2.25, -2.15) |****  0.4%
(-2.15, -2.05) |*****  0.4%
(-2.05, -1.95) |******  0.5%
(-1.95, -1.85) |********  0.7%
(-1.85, -1.75) |*********  0.8%
(-1.75, -1.65) |***********  0.9%
(-1.65, -1.55) |*************  1.1%
(-1.55, -1.45) |****************  1.3%
(-1.45, -1.35) |******************  1.5%
(-1.35, -1.25) |*********************  1.7%
(-1.25, -1.15) |************************  1.9%
(-1.15, -1.05) |***************************  2.2%
(-1.05, -0.95) |******************************  2.4%
(-0.95, -0.85) |*********************************  2.7%
(-0.85, -0.75) |************************************  2.9%
(-0.75, -0.65) |***************************************  3.1%
(-0.65, -0.55) |*****************************************  3.3%
(-0.55, -0.45) |********************************************  3.5%
(-0.45, -0.35) |**********************************************  3.7%
(-0.35, -0.25) |***********************************************  3.8%
(-0.25, -0.15) |*************************************************  3.9%
(-0.15, -0.05) |*************************************************  4.0%
(-0.05,  0.05) |**************************************************  4.0%
( 0.05,  0.15) |*************************************************  4.0%
( 0.15,  0.25) |*************************************************  3.9%
( 0.25,  0.35) |***********************************************  3.8%
( 0.35,  0.45) |**********************************************  3.7%
( 0.45,  0.55) |********************************************  3.5%
( 0.55,  0.65) |*****************************************  3.3%
( 0.65,  0.75) |***************************************  3.1%
( 0.75,  0.85) |************************************  2.9%
( 0.85,  0.95) |*********************************  2.7%
( 0.95,  1.05) |******************************  2.4%
( 1.05,  1.15) |***************************  2.2%
( 1.15,  1.25) |************************  1.9%
( 1.25,  1.35) |*********************  1.7%
( 1.35,  1.45) |******************  1.5%
( 1.45,  1.55) |****************  1.3%
( 1.55,  1.65) |*************  1.1%
( 1.65,  1.75) |***********  0.9%
( 1.75,  1.85) |*********  0.8%
( 1.85,  1.95) |********  0.7%
( 1.95,  2.05) |******  0.5%
( 2.05,  2.15) |*****  0.4%
( 2.15,  2.25) |****  0.4%
( 2.25,  2.35) |***  0.3%
( 2.35,  2.45) |**  0.2%
( 2.45,  2.55) |**  0.2%
( 2.55,  2.65) |*  0.1%
( 2.65,  2.75) |*  0.1%
( 2.75,  2.85) |*  0.1%
( 2.85,  2.95) |  0.1%

press enter

C#

Library: Math.Net
using System;
using MathNet.Numerics.Distributions;
using MathNet.Numerics.Statistics;
 
class Program
{
static void RunNormal(int sampleSize)
{
double[] X = new double[sampleSize];
var norm = new Normal(new Random());
norm.Samples(X);
 
const int numBuckets = 10;
var histogram = new Histogram(X, numBuckets);
Console.WriteLine("Sample size: {0:N0}", sampleSize);
for (int i = 0; i < numBuckets; i++)
{
string bar = new String('#', (int)(histogram[i].Count * 360 / sampleSize));
Console.WriteLine(" {0:0.00} : {1}", histogram[i].LowerBound, bar);
}
var statistics = new DescriptiveStatistics(X);
Console.WriteLine(" Mean: " + statistics.Mean);
Console.WriteLine("StdDev: " + statistics.StandardDeviation);
Console.WriteLine();
}
static void Main(string[] args)
{
RunNormal(100);
RunNormal(1000);
RunNormal(10000);
}
}
Output:
Sample size: 100
 -2.12 : #######
 -1.66 : ############################
 -1.19 : #######################################
 -0.72 : ##############################################
 -0.26 : ###############################################################################
 0.21 : ######################################################################################
 0.68 : ################################
 1.14 : #########################
 1.61 : ###
 2.07 : ##########
  Mean: 0.0394411345297757
StdDev: 0.925286665513647

Sample size: 1,000
 -2.98 : ##
 -2.34 : ##########
 -1.69 : ##############################
 -1.05 : ################################################################
 -0.40 : ###########################################################################################
 0.24 : ########################################################################################
 0.88 : ##############################################
 1.53 : ##################
 2.17 : #####
 2.82 : ##
  Mean: 0.0868718238400114
StdDev: 0.989120264661867

Sample size: 10,000
 -3.88 :
 -3.12 : ##
 -2.35 : #################
 -1.59 : ####################################################
 -0.82 : ################################################################################################
 -0.06 : ####################################################################################################
 0.71 : ###############################################################
 1.47 : #####################
 2.23 : ####
 3.00 :
  Mean: 0.0208920122989818
StdDev: 1.00046328880424

C++

showing features of C++11 here

#include <random>
#include <map>
#include <string>
#include <iostream>
#include <cmath>
#include <iomanip>
 
int main( ) {
std::random_device myseed ;
std::mt19937 engine ( myseed( ) ) ;
std::normal_distribution<> normDistri ( 2 , 3 ) ;
std::map<int , int> normalFreq ;
int sum = 0 ; //holds the sum of the randomly created numbers
double mean = 0.0 ;
double stddev = 0.0 ;
for ( int i = 1 ; i < 10001 ; i++ )
++normalFreq[ normDistri ( engine ) ] ;
for ( auto MapIt : normalFreq ) {
sum += MapIt.first * MapIt.second ;
}
mean = sum / 10000 ;
stddev = sqrt( sum / 10000 ) ;
std::cout << "The mean of the distribution is " << mean << " , the " ;
std::cout << "standard deviation " << stddev << " !\n" ;
std::cout << "And now the histogram:\n" ;
for ( auto MapIt : normalFreq ) {
std::cout << std::left << std::setw( 4 ) << MapIt.first <<
std::string( MapIt.second / 100 , '*' ) << std::endl ;
}
return 0 ;
}

Output:

The mean of the distribution is 1 , the standard deviation 1 !
And now the histogram:
-10 
-9  
-8  
-7  
-6  
-5  
-4  *
-3  **
-2  ****
-1  ******
0   *********************
1   ************
2   ************
3   ***********
4   *********
5   ******
6   ****
7   **
8   *
9   
10  
11  
12  
13  

D

This uses the Box-Muller method as in the Go entry, and the module from the Statistics/Basic. A ziggurat-based normal generator for the Phobos standard library is in the works.

import std.stdio, std.random, std.math, std.range, std.algorithm,
statistics_basic;
 
struct Normals {
double mu, sigma;
double[2] state;
size_t index = state.length;
enum empty = false;
 
void popFront() pure nothrow { index++; }
 
@property double front() {
if (index >= state.length) {
immutable r = sqrt(-2 * uniform!"]["(0., 1.0).log) * sigma;
immutable x = 2 * PI * uniform01;
state = [mu + r * x.sin, mu + r * x.cos];
index = 0;
}
return state[index];
}
}
 
void main() {
const data = Normals(0.0, 0.5).take(100_000).array;
writefln("Mean: %8.6f, SD: %8.6f\n", data.meanStdDev[]);
data.map!q{ max(0.0, min(0.9999, a / 3 + 0.5)) }.showHistogram01;
}
Output:
Mean: 0.000528, SD: 0.502245

 0.0: *
 0.1: ******
 0.2: *****************
 0.3: ***********************************
 0.4: *************************************************
 0.5: **************************************************
 0.6: **********************************
 0.7: *****************
 0.8: ******
 0.9: *

Elixir

defmodule Statistics do
def normal_distribution(n, w\\5) do
{sum, sum2, hist} = generate(n, w)
mean = sum / n
stddev = :math.sqrt(sum2 / n - mean*mean)
 
IO.puts "size: #{n}"
IO.puts "mean: #{mean}"
IO.puts "stddev: #{stddev}"
{min, max} = Map.to_list(hist)
|> Enum.filter_map(fn {_k,v} -> v >= n/120/w end, fn {k,_v} -> k end)
|> Enum.min_max
Enum.each(min..max, fn i ->
bar = String.duplicate("=", trunc(120 * w * Map.get(hist, i, 0) / n))
 :io.fwrite "~4.1f: ~s~n", [i/w, bar]
end)
IO.puts ""
end
 
defp generate(n, w) do
Enum.reduce(1..n, {0, 0, %{}}, fn _,{sum, sum2, hist} ->
z = :rand.normal
{sum+z, sum2+z*z, Map.update(hist, round(w*z), 1, &(&1+1))}
end)
end
end
 
Enum.each([100,1000,10000], fn n ->
Statistics.normal_distribution(n)
end)
Output:
size:   100
mean:   0.027742416007234007
stddev: 1.0209597927405403
-2.6: ============
-2.4: 
-2.2: ============
-2.0: ======
-1.8: 
-1.6: 
-1.4: ==============================
-1.2: ======
-1.0: ==============================
-0.8: ==========================================
-0.6: ==========================================
-0.4: ================================================
-0.2: ================================================
 0.0: ==============================
 0.2: ====================================
 0.4: ==========================================
 0.6: ======================================================
 0.8: ==========================================
 1.0: ================================================
 1.2: ==============================
 1.4: ======
 1.6: ============
 1.8: ============
 2.0: 
 2.2: 
 2.4: ======
 2.6: ======

size:   1000
mean:   -0.025562168667763084
stddev: 1.0338288521306742
-3.2: =
-3.0: 
-2.8: =
-2.6: ===
-2.4: ==
-2.2: ======
-2.0: ==
-1.8: =============
-1.6: ===============
-1.4: =================
-1.2: =================
-1.0: ====================================
-0.8: ===================================
-0.6: ============================================
-0.4: ============================================
-0.2: ===============================================
 0.0: =========================================
 0.2: ===========================================
 0.4: =============================================
 0.6: =======================================
 0.8: ================================
 1.0: ============================
 1.2: ========================
 1.4: ==================
 1.6: ==========
 1.8: =====
 2.0: ========
 2.2: ====
 2.4: =====
 2.6: =
 2.8: =

size:   10000
mean:   -0.009132420943007152
stddev: 0.9979508347451509
-2.6: =
-2.4: ===
-2.2: ====
-2.0: =====
-1.8: =========
-1.6: ==============
-1.4: ================
-1.2: =======================
-1.0: ============================
-0.8: =================================
-0.6: ============================================
-0.4: ===========================================
-0.2: ==============================================
 0.0: ==================================================
 0.2: ============================================
 0.4: ===========================================
 0.6: =======================================
 0.8: =====================================
 1.0: ============================
 1.2: =======================
 1.4: ================
 1.6: ==============
 1.8: =========
 2.0: ======
 2.2: ===
 2.4: ==
 2.6: =

Fortran

Works with: Fortran version 95 and later

Using the Marsaglia polar method

program Normal_Distribution
implicit none
 
integer, parameter :: i64 = selected_int_kind(18)
integer, parameter :: r64 = selected_real_kind(15)
integer(i64), parameter :: samples = 1000000_i64
real(r64) :: mean, stddev
real(r64) :: sumn = 0, sumnsq = 0
integer(i64) :: n = 0
integer(i64) :: bin(-50:50) = 0
integer :: i, ind
real(r64) :: ur1, ur2, nr1, nr2, s
 
n = 0
do while(n <= samples)
call random_number(ur1)
call random_number(ur2)
ur1 = ur1 * 2.0 - 1.0
ur2 = ur2 * 2.0 - 1.0
 
s = ur1*ur1 + ur2*ur2
if(s >= 1.0_r64) cycle
 
nr1 = ur1 * sqrt(-2.0*log(s)/s)
ind = floor(5.0*nr1)
bin(ind) = bin(ind) + 1_i64
sumn = sumn + nr1
sumnsq = sumnsq + nr1*nr1
 
nr2 = ur2 * sqrt(-2.0*log(s)/s)
ind = floor(5.0*nr2)
bin(ind) = bin(ind) + 1_i64
sumn = sumn + nr2
sumnsq = sumnsq + nr2*nr2
n = n + 2_i64
end do
 
mean = sumn / n
stddev = sqrt(sumnsq/n - mean*mean)
 
write(*, "(a, i0)") "sample size = ", samples
write(*, "(a, f17.15)") "Mean : ", mean,
write(*, "(a, f17.15)") "Stddev : ", stddev
 
do i = -15, 15
write(*, "(f4.1, a, a)") real(i)/5.0, ": ", repeat("=", int(bin(i)*500/samples))
end do
 
end program
Output:
sample size = 1000
Mean :   0.043096320705032
Stddev : 0.981532585231540
-3.0:
-2.8:
-2.6: ==
-2.4: ==
-2.2: ====
-2.0: ======
-1.8: =======
-1.6: ============
-1.4: ================
-1.2: =====================
-1.0: ===========================
-0.8: =======================
-0.6: ==================================
-0.4: =====================================
-0.2: ==========================================
 0.0: ===============================================
 0.2: ====================================
 0.4: =================================
 0.6: ==================================
 0.8: =============================
 1.0: ====================
 1.2: ==========================
 1.4: ===========
 1.6: =========
 1.8: ====
 2.0: ======
 2.2: ===
 2.4:
 2.6:
 2.8: =
 3.0:

sample size = 1000000
Mean :   0.000166653231289
Stddev : 1.000025612171690
-3.0:
-2.8: =
-2.6: =
-2.4: ==
-2.2: ====
-2.0: ======
-1.8: =========
-1.6: ============
-1.4: =================
-1.2: =====================
-1.0: ==========================
-0.8: ===============================
-0.6: ===================================
-0.4: ======================================
-0.2: =======================================
 0.0: =======================================
 0.2: ======================================
 0.4: ==================================
 0.6: ===============================
 0.8: ==========================
 1.0: =====================
 1.2: =================
 1.4: ============
 1.6: =========
 1.8: ======
 2.0: ====
 2.2: ==
 2.4: =
 2.6: =
 2.8:
 3.0:

FreeBASIC

' FB 1.05.0 Win64
 
Const pi As Double = 3.141592653589793
Randomize
 
' Generates normally distributed random numbers with mean 0 and standard deviation 1
Function randomNormal() As Double
Return Cos(2.0 * pi * Rnd) * Sqr(-2.0 * Log(Rnd))
End Function
 
Sub normalStats(sampleSize As Integer)
If sampleSize < 1 Then Return
Dim r(1 To sampleSize) As Double
Dim h(-1 To 10) As Integer '' all zero by default
Dim sum As Double = 0.0
Dim hSum As Integer = 0
 
' Generate 'sampleSize' normally distributed random numbers with mean 0.5 and standard deviation 0.25
' calculate their sum
' and in which box they will fall when drawing the histogram
For i As Integer = 1 To sampleSize
r(i) = 0.5 + randomNormal / 4.0
sum += r(i)
If r(i) < 0.0 Then
h(-1) += 1
ElseIf r(i) >= 1.0 Then
h(10) += 1
Else
h(Int(r(i) * 10)) += 1
End If
Next
 
For i As Integer = -1 To 10 : hSum += h(i) : Next
' adjust one of the h() values if necessary to ensure hSum = sampleSize
Dim adj As Integer = sampleSize - hSum
If adj <> 0 Then
For i As Integer = -1 To 10
h(i) += adj
If h(i) >= 0 Then Exit For
h(i) -= adj
Next
End If
 
Dim mean As Double = sum / sampleSize
 
Dim sd As Double
sum = 0.0
' Now calculate their standard deviation
For i As Integer = 1 To sampleSize
sum += (r(i) - mean) ^ 2.0
Next
sd = Sqr(sum/sampleSize)
 
' Draw a histogram of the data with interval 0.1
Dim numStars As Integer
' If sample size > 300 then normalize histogram to 300
Dim scale As Double = 1.0
If sampleSize > 300 Then scale = 300.0 / sampleSize
Print "Sample size "; sampleSize
Print
Print Using " Mean #.######"; mean;
Print Using " SD #.######"; sd
Print
For i As Integer = -1 To 10
If i = -1 Then
Print Using "< 0.00 : ";
ElseIf i = 10 Then
Print Using ">=1.00 : ";
Else
Print Using " #.## : "; i/10.0;
End If
Print Using "##### " ; h(i);
numStars = Int(h(i) * scale + 0.5)
Print String(numStars, "*")
Next
End Sub
 
normalStats 100
Print
normalStats 1000
Print
normalStats 10000
Print
normalStats 100000
Print
Print "Press any key to quit"
Sleep

Sample output:

Output:
Sample size  100

  Mean 0.486977  SD 0.244147

< 0.00 :     2 **
  0.00 :     5 *****
  0.10 :     4 ****
  0.20 :    14 **************
  0.30 :    12 ************
  0.40 :    15 ***************
  0.50 :    17 *****************
  0.60 :    11 ***********
  0.70 :     9 *********
  0.80 :     7 *******
  0.90 :     1 *
>=1.00 :     3 ***

Sample size  1000

  Mean 0.489234  SD 0.247606

< 0.00 :    18 *****
  0.00 :    32 **********
  0.10 :    73 **********************
  0.20 :   111 *********************************
  0.30 :   138 *****************************************
  0.40 :   151 *********************************************
  0.50 :   153 **********************************************
  0.60 :   114 **********************************
  0.70 :   101 ******************************
  0.80 :    51 ***************
  0.90 :    38 ***********
>=1.00 :    20 ******

Sample size  10000

  Mean 0.498239  SD 0.249235

< 0.00 :   225 *******
  0.00 :   333 **********
  0.10 :   589 ******************
  0.20 :   999 ******************************
  0.30 :  1320 ****************************************
  0.40 :  1542 **********************************************
  0.50 :  1581 ***********************************************
  0.60 :  1323 ****************************************
  0.70 :   963 *****************************
  0.80 :   591 ******************
  0.90 :   314 *********
>=1.00 :   220 *******

Sample size  100000

  Mean 0.500925  SD 0.248910

< 0.00 :  2173 *******
  0.00 :  3192 **********
  0.10 :  5938 ******************
  0.20 :  9715 *****************************
  0.30 : 13351 ****************************************
  0.40 : 15399 **********************************************
  0.50 : 15680 ***********************************************
  0.60 : 13422 ****************************************
  0.70 :  9633 *****************************
  0.80 :  5993 ******************
  0.90 :  3207 **********
>=1.00 :  2297 *******

Go

Box-Muller method shown here. Go has a normally distributed random function in the standard library, as shown in the Go Random numbers solution. It uses the ziggurat method.

package main
 
import (
"fmt"
"math"
"math/rand"
"strings"
)
 
// Box-Muller
func norm2() (s, c float64) {
r := math.Sqrt(-2 * math.Log(rand.Float64()))
s, c = math.Sincos(2 * math.Pi * rand.Float64())
return s * r, c * r
}
 
func main() {
const (
n = 10000
bins = 12
sig = 3
scale = 100
)
var sum, sumSq float64
h := make([]int, bins)
for i, accum := 0, func(v float64) {
sum += v
sumSq += v * v
b := int((v + sig) * bins / sig / 2)
if b >= 0 && b < bins {
h[b]++
}
}; i < n/2; i++ {
v1, v2 := norm2()
accum(v1)
accum(v2)
}
m := sum / n
fmt.Println("mean:", m)
fmt.Println("stddev:", math.Sqrt(sumSq/float64(n)-m*m))
for _, p := range h {
fmt.Println(strings.Repeat("*", p/scale))
}
}

Output:

mean: -0.0034970888831523488
stddev: 1.0040682925006286

*
****
*********
***************
*******************
******************
**************
*********
****
*

Haskell

import Data.Map (Map, empty, insert, findWithDefault, toList)
import Data.Maybe (fromMaybe)
import Text.Printf (printf)
import Data.Function (on)
import Data.List (sort, maximumBy, minimumBy)
import Control.Monad.Random (RandomGen, Rand, evalRandIO, getRandomR)
import Control.Monad (replicateM)
 
-- Box-Muller
getNorm :: RandomGen g => Rand g Double
getNorm = do
u0 <- getRandomR (0.0, 1.0)
u1 <- getRandomR (0.0, 1.0)
let r = sqrt $ (-2.0) * log u0
theta = 2.0 * pi * u1
return $ r * sin theta
 
putInBin :: Double -> Map Int Int -> Double -> Map Int Int
putInBin binWidth t v =
let bin = round (v / binWidth)
count = findWithDefault 0 bin t
in insert bin (count+1) t
 
runTest :: Int -> IO ()
runTest n = do
rs <- evalRandIO $ replicateM n getNorm
let binWidth = 0.1
 
tally v (sv, sv2, t) = (sv+v, sv2 + v*v, putInBin binWidth t v)
 
(sum, sum2, tallies) = foldr tally (0.0, 0.0, empty) rs
 
tallyList = sort $ toList tallies
 
printStars tallies binWidth maxCount selection =
let count = findWithDefault 0 selection tallies
bin = binWidth * fromIntegral selection
maxStars = 100
starCount = if maxCount <= maxStars
then count
else maxStars * count `div` maxCount
stars = replicate starCount '*'
in printf "%5.2f: %s  %d\n" bin stars count
 
mean = sum / fromIntegral n
stddev = sqrt (sum2/fromIntegral n - mean*mean)
 
printf "\n"
printf "sample count: %d\n" n
printf "mean:  %9.7f\n" mean
printf "stddev:  %9.7f\n" stddev
 
let maxCount = snd $ maximumBy (compare `on` snd) tallyList
maxBin = fst $ maximumBy (compare `on` fst) tallyList
minBin = fst $ minimumBy (compare `on` fst) tallyList
 
mapM_ (printStars tallies binWidth maxCount) [minBin..maxBin]
 
main = do
runTest 1000
runTest 2000000
Output:
sample count: 1000
mean:         -0.0269949
stddev:       0.9795285
-3.10: **  2
-3.00:   0
-2.90:   0
-2.80: **  2
-2.70: *  1
-2.60: ****  4
-2.50: **  2
-2.40: **  2
-2.30:   0
-2.20: ***  3
-2.10: *****  5
-2.00: ******  6
-1.90: ******  6
-1.80: ***********  11
-1.70: ************  12
-1.60: *******  7
-1.50: *************  13
-1.40: *****************  17
-1.30: ********************  20
-1.20: ****************  16
-1.10: *****************  17
-1.00: **********************  22
-0.90: ***************************  27
-0.80: **********************  22
-0.70: ********************************  32
-0.60: *********************************  33
-0.50: ******************************************  42
-0.40: ***********************************************  47
-0.30: ************************************************  48
-0.20: ***************************  27
-0.10: *****************************  29
 0.00: ***********************************************  47
 0.10: ***************************************************  51
 0.20: ******************************************  42
 0.30: ********************************  32
 0.40: *********************************  33
 0.50: *****************************************  41
 0.60: ******************************************  42
 0.70: ****************************  28
 0.80: **********************  22
 0.90: ***************************  27
 1.00: *******************  19
 1.10: **********************  22
 1.20: ************************  24
 1.30: ********************  20
 1.40: *****************  17
 1.50: **********  10
 1.60: *************  13
 1.70: ****  4
 1.80: ***  3
 1.90: *******  7
 2.00: ******  6
 2.10: *  1
 2.20: *  1
 2.30: *******  7
 2.40: ***  3
 2.50:   0
 2.60: *  1
 2.70:   0
 2.80:   0
 2.90:   0
 3.00: *  1
 3.10:   0
 3.20:   0
 3.30: *  1

sample count: 2000000
mean:         0.0001017
stddev:       0.9994329
-4.60:   3
-4.50:   2
-4.40:   3
-4.30:   9
-4.20:   18
-4.10:   19
-4.00:   20
-3.90:   41
-3.80:   77
-3.70:   84
-3.60:   105
-3.50:   189
-3.40:   245
-3.30:   350
-3.20:   460
-3.10:   619
-3.00: *  838
-2.90: *  1234
-2.80: *  1586
-2.70: **  2063
-2.60: ***  2716
-2.50: ****  3503
-2.40: *****  4345
-2.30: *******  5678
-2.20: ********  7160
-2.10: ***********  8856
-2.00: *************  10915
-1.90: ****************  13299
-1.80: *******************  15599
-1.70: ***********************  19004
-1.60: ***************************  22321
-1.50: ********************************  25940
-1.40: *************************************  29622
-1.30: ******************************************  34213
-1.20: ************************************************  38922
-1.10: ******************************************************  43415
-1.00: ************************************************************  48250
-0.90: ******************************************************************  53210
-0.80: ************************************************************************  58127
-0.70: ******************************************************************************  62438
-0.60: ***********************************************************************************  66650
-0.50: ****************************************************************************************  70298
-0.40: ********************************************************************************************  73739
-0.30: ***********************************************************************************************  75831
-0.20: **************************************************************************************************  78222
-0.10: ***************************************************************************************************  79412
 0.00: ****************************************************************************************************  79801
 0.10: ***************************************************************************************************  79255
 0.20: *************************************************************************************************  78163
 0.30: ************************************************************************************************  76667
 0.40: ********************************************************************************************  73554
 0.50: ****************************************************************************************  70391
 0.60: ***********************************************************************************  66566
 0.70: ******************************************************************************  62857
 0.80: ************************************************************************  57962
 0.90: ******************************************************************  53407
 1.00: ************************************************************  48565
 1.10: ******************************************************  43496
 1.20: ************************************************  38799
 1.30: ******************************************  34156
 1.40: *************************************  29713
 1.50: ********************************  25946
 1.60: ***************************  22264
 1.70: ***********************  18843
 1.80: *******************  15780
 1.90: ****************  13151
 2.00: *************  10905
 2.10: **********  8690
 2.20: ********  7102
 2.30: *******  5693
 2.40: *****  4459
 2.50: ****  3550
 2.60: ***  2603
 2.70: **  2155
 2.80: **  1619
 2.90: *  1121
 3.00: *  914
 3.10:   607
 3.20:   478
 3.30:   349
 3.40:   216
 3.50:   170
 3.60:   113
 3.70:   79
 3.80:   58
 3.90:   48
 4.00:   33
 4.10:   20
 4.20:   9
 4.30:   8
 4.40:   7
 4.50:   3
 4.60:   3
 4.70:   0
 4.80:   1
 4.90:   1

J

Solution

runif01=: ?@$ 0:                                           NB. random uniform number generator
rnorm01=. (2 o. 2p1 * runif01) * [: %: _2 * ^.@runif01 NB. random normal number generator (Box-Muller)
 
mean=: +/ % # NB. mean
stddev=: (<:@# %~ +/)&.:*:@(- mean) NB. standard deviation
histogram=: <:@(#/.~)@(i.@#@[ , I.)

Example Usage

   DataSet=: rnorm01 1e5
(mean , stddev) DataSet
0.000781667 1.00154
require 'plot'
plot (5 %~ i: 25) ([;histogram) DataSet

Java

Translation of: D
Works with: Java version 8
import static java.lang.Math.*;
import static java.util.Arrays.stream;
import java.util.Locale;
import java.util.function.DoubleSupplier;
import static java.util.stream.Collectors.joining;
import java.util.stream.DoubleStream;
import static java.util.stream.IntStream.range;
 
public class Test implements DoubleSupplier {
 
private double mu, sigma;
private double[] state = new double[2];
private int index = state.length;
 
Test(double m, double s) {
mu = m;
sigma = s;
}
 
static double[] meanStdDev(double[] numbers) {
if (numbers.length == 0)
return new double[]{0.0, 0.0};
 
double sx = 0.0, sxx = 0.0;
long n = 0;
for (double x : numbers) {
sx += x;
sxx += pow(x, 2);
n++;
}
 
return new double[]{sx / n, pow((n * sxx - pow(sx, 2)), 0.5) / n};
}
 
static String replicate(int n, String s) {
return range(0, n + 1).mapToObj(i -> s).collect(joining());
}
 
static void showHistogram01(double[] numbers) {
final int maxWidth = 50;
long[] bins = new long[10];
 
for (double x : numbers)
bins[(int) (x * bins.length)]++;
 
double maxFreq = stream(bins).max().getAsLong();
 
for (int i = 0; i < bins.length; i++)
System.out.printf(" %3.1f: %s%n", i / (double) bins.length,
replicate((int) (bins[i] / maxFreq * maxWidth), "*"));
System.out.println();
}
 
@Override
public double getAsDouble() {
index++;
if (index >= state.length) {
double r = sqrt(-2 * log(random())) * sigma;
double x = 2 * PI * random();
state = new double[]{mu + r * sin(x), mu + r * cos(x)};
index = 0;
}
return state[index];
 
}
 
public static void main(String[] args) {
Locale.setDefault(Locale.US);
double[] data = DoubleStream.generate(new Test(0.0, 0.5)).limit(100_000)
.toArray();
 
double[] res = meanStdDev(data);
System.out.printf("Mean: %8.6f, SD: %8.6f%n", res[0], res[1]);
 
showHistogram01(stream(data).map(a -> max(0.0, min(0.9999, a / 3 + 0.5)))
.toArray());
}
}
Mean: -0.001870, SD: 0.500539
 0.0: **
 0.1: *******
 0.2: ******************
 0.3: ************************************
 0.4: ***************************************************
 0.5: **************************************************
 0.6: ***********************************
 0.7: ******************
 0.8: *******
 0.9: **

Kotlin

Translation of: FreeBASIC
// version 1.1.2
 
val rand = java.util.Random()
 
fun normalStats(sampleSize: Int) {
if (sampleSize < 1) return
val r = DoubleArray(sampleSize)
val h = IntArray(12) // all zero by default
/*
Generate 'sampleSize' normally distributed random numbers with mean 0.5 and SD 0.25
and calculate in which box they will fall when drawing the histogram
*/

for (i in 0 until sampleSize) {
r[i] = 0.5 + rand.nextGaussian() / 4.0
when {
r[i] < 0.0 -> h[0]++
r[i] >= 1.0 -> h[11]++
else -> h[1 + (r[i] * 10).toInt()]++
}
}
 
// adjust one of the h[] values if necessary to ensure they sum to sampleSize
val adj = sampleSize - h.sum()
if (adj != 0) {
for (i in 0..11) {
h[i] += adj
if (h[i] >= 0) break
h[i] -= adj
}
}
 
val mean = r.average()
val sd = Math.sqrt(r.map { (it - mean) * (it - mean) }.average())
 
// Draw a histogram of the data with interval 0.1
var numStars: Int
// If sample size > 300 then normalize histogram to 300
val scale = if (sampleSize <= 300) 1.0 else 300.0 / sampleSize
println("Sample size $sampleSize\n")
println(" Mean ${"%1.6f".format(mean)} SD ${"%1.6f".format(sd)}\n")
for (i in 0..11) {
when (i) {
0 -> print("< 0.00 : ")
11 -> print(">=1.00 : ")
else -> print("  %1.2f : ".format(i / 10.0))
}
print("%5d ".format(h[i]))
numStars = (h[i] * scale + 0.5).toInt()
println("*".repeat(numStars))
}
println()
}
 
fun main(args: Array<String>) {
val sampleSizes = intArrayOf(100, 1_000, 10_000, 100_000)
for (sampleSize in sampleSizes) normalStats(sampleSize)
}
Output:
Sample size 100

  Mean 0.525211  SD 0.266316

< 0.00 :     3 ***
  0.10 :     1 *
  0.20 :     3 ***
  0.30 :    11 ***********
  0.40 :    14 **************
  0.50 :    13 *************
  0.60 :    15 ***************
  0.70 :    13 *************
  0.80 :    10 **********
  0.90 :    11 ***********
  1.00 :     4 ****
>=1.00 :     2 **

Sample size 1000

  Mean 0.500948  SD 0.255757

< 0.00 :    29 *********
  0.10 :    35 ***********
  0.20 :    70 *********************
  0.30 :    71 *********************
  0.40 :   138 *****************************************
  0.50 :   139 ******************************************
  0.60 :   168 **************************************************
  0.70 :   123 *************************************
  0.80 :   110 *********************************
  0.90 :    62 *******************
  1.00 :    32 **********
>=1.00 :    23 *******

Sample size 10000

  Mean 0.501376  SD 0.248317

< 0.00 :   240 *******
  0.10 :   305 *********
  0.20 :   617 *******************
  0.30 :   927 ****************************
  0.40 :  1291 ***************************************
  0.50 :  1554 ***********************************************
  0.60 :  1609 ************************************************
  0.70 :  1319 ****************************************
  0.80 :   983 *****************************
  0.90 :   609 ******************
  1.00 :   324 **********
>=1.00 :   222 *******

Sample size 100000

  Mean 0.499427  SD 0.250533

< 0.00 :  2341 *******
  0.10 :  3246 **********
  0.20 :  6005 ******************
  0.30 :  9718 *****************************
  0.40 : 13247 ****************************************
  0.50 : 15595 ***********************************************
  0.60 : 15271 **********************************************
  0.70 : 13405 ****************************************
  0.80 :  9653 *****************************
  0.90 :  5990 ******************
  1.00 :  3257 **********
>=1.00 :  2272 *******

Lasso

define stat1(a) => {
if(#a->size) => {
local(mean = (with n in #a sum #n) / #a->size)
local(sdev = math_pow(((with n in #a sum Math_Pow((#n - #mean),2)) / #a->size),0.5))
return (:#sdev, #mean)
else
return (:0,0)
}
}
define stat2(a) => {
if(#a->size) => {
local(sx = 0, sxx = 0)
with x in #a do => {
#sx += #x
#sxx += #x*#x
}
local(sdev = math_pow((#a->size * #sxx - #sx * #sx),0.5) / #a->size)
return (:#sdev, #sx / #a->size)
else
return (:0,0)
}
}
define histogram(a) => {
local(
out = '\r',
h = array(0,0,0,0,0,0,0,0,0,0,0),
maxwidth = 50,
sc = 0
)
with n in #a do => {
if((#n * 10) <= 0) => {
#h->get(1) += 1
else((#n * 10) >= 10)
#h->get(#h->size) += 1
else
#h->get(integer(decimal(#n)*10)+1) += 1
}
 
}
local(mx = decimal(with n in #h max #n))
with i in #h do => {
#out->append((#sc/10.0)->asString(-precision=1)+': '+('+' * integer(#i / #mx * #maxwidth))+'\r')
#sc++
}
return #out
}
define normalDist(mean,sdev) => {
// Uses Box-Muller transform
return ((-2 * decimal_random->log)->sqrt * (2 * pi * decimal_random)->cos) * #sdev + #mean
}
 
with scale in array(100,1000,10000) do => {^
local(n = array)
loop(#scale) => { #n->insert(normalDist(0.5, 0.2)) }
local(sdev1,mean1) = stat1(#n)
local(sdev2,mean2) = stat2(#n)
#scale' numbers:\r'
'Naive method: sd: '+#sdev1+', mean: '+#mean1+'\r'
'Second method: sd: '+#sdev2+', mean: '+#mean2+'\r'
histogram(#n)
'\r\r'
^}
Output:
100 numbers:
Naive  method: sd: 0.199518, mean: 0.506059
Second  method: sd: 0.199518, mean: 0.506059

0.0: ++
0.1: ++++
0.2: +++++++++++++++++
0.3: ++++++++++++++++++++++
0.4: ++++++++++++++++++++++++++++++++++++++++++++++++++
0.5: +++++++++++++++++++++++++++++++++++++++
0.6: +++++++++++++++++++++++++++++++++
0.7: ++++++++++++++++++++++++
0.8: ++++++++++++++++++++
0.9: ++++
1.0: ++


1000 numbers:
Naive  method: sd: 0.199653, mean: 0.504046
Second  method: sd: 0.199653, mean: 0.504046

0.0: +++
0.1: ++++++
0.2: ++++++++++++++++
0.3: ++++++++++++++++++++++++++++++
0.4: +++++++++++++++++++++++++++++++++++++++++++++++
0.5: ++++++++++++++++++++++++++++++++++++++++++++++++++
0.6: ++++++++++++++++++++++++++++++++++++++++++++++
0.7: +++++++++++++++++++++++++
0.8: +++++++++++++++++++
0.9: +++++++
1.0: ++++


10000 numbers:
Naive  method: sd: 0.202354, mean: 0.502519
Second  method: sd: 0.202354, mean: 0.502519

0.0: +++
0.1: +++++++
0.2: +++++++++++++++
0.3: +++++++++++++++++++++++++++++
0.4: ++++++++++++++++++++++++++++++++++++++++++
0.5: ++++++++++++++++++++++++++++++++++++++++++++++++++
0.6: +++++++++++++++++++++++++++++++++++++++++++
0.7: ++++++++++++++++++++++++++++++
0.8: ++++++++++++++++
0.9: +++++++
1.0: ++++

Liberty BASIC

Uses LB Statistics/Basic

call sample 100000
 
end
 
sub sample n
dim dat( n)
for i =1 to n
dat( i) =normalDist( 1, 0.2)
next i
 
'// show mean, standard deviation. Find max, min.
mx =-1000
mn = 1000
sum =0
sSq =0
for i =1 to n
d =dat( i)
mx =max( mx, d)
mn =min( mn, d)
sum =sum +d
sSq =sSq +d^2
next i
print n; " data terms used."
 
mean =sum / n
print "Largest term was "; mx; " & smallest was "; mn
range =mx -mn
print "Mean ="; mean
 
print "Stddev ="; ( sSq /n -mean^2)^0.5
 
'// show histogram
nBins =50
dim bins( nBins)
for i =1 to n
z =int( ( dat( i) -mn) /range *nBins)
bins( z) =bins( z) +1
next i
for b =0 to nBins -1
for j =1 to int( nBins *bins( b)) /n *30)
print "#";
next j
print
next b
print
end sub
 
function normalDist( m, s) ' Box Muller method
u =rnd( 1)
v =rnd( 1)
normalDist =( -2 *log( u))^0.5 *cos( 2 *3.14159265 *v)
end function
100000 data terms used.
Largest term was 4.12950792 & smallest was -4.37934139
Mean =-0.26785425e-2
Stddev =1.00097669


#
##
###
#####
########
############
################
########################
##############################
######################################
##############################################
########################################################
###################################################################
##############################################################################
#######################################################################################
################################################################################################
####################################################################################################
########################################################################################################
#####################################################################################################
##############################################################################################
#########################################################################################
##################################################################################
#########################################################################
##############################################################
####################################################
##########################################
#################################
##########################
##################
#############
#########
######
####
##
#
#

Lua

Lua provides math.random() to generate uniformly distributed random numbers. The function gaussian() shown here uses math.random() to generate normally distributed random numbers with given mean and variance.

function gaussian (mean, variance)
return math.sqrt(-2 * variance * math.log(math.random())) *
math.cos(2 * variance * math.pi * math.random()) + mean
end
 
function mean (t)
local sum = 0
for k, v in pairs(t) do
sum = sum + v
end
return sum / #t
end
 
function std (t)
local squares, avg = 0, mean(t)
for k, v in pairs(t) do
squares = squares + ((avg - v) ^ 2)
end
local variance = squares / #t
return math.sqrt(variance)
end
 
function showHistogram (t)
local lo = math.ceil(math.min(unpack(t)))
local hi = math.floor(math.max(unpack(t)))
local hist, barScale = {}, 200
for i = lo, hi do
hist[i] = 0
for k, v in pairs(t) do
if math.ceil(v - 0.5) == i then
hist[i] = hist[i] + 1
end
end
io.write(i .. "\t" .. string.rep('=', hist[i] / #t * barScale))
print(" " .. hist[i])
end
end
 
math.randomseed(os.time())
local t, average, variance = {}, 50, 10
for i = 1, 1000 do
table.insert(t, gaussian(average, variance))
end
print("Mean:", mean(t) .. ", expected " .. average)
print("StdDev:", std(t) .. ", expected " .. math.sqrt(variance) .. "\n")
showHistogram(t)
Output:
Mean:   50.008328894275, expected 50
StdDev: 3.2374717425824, expected 3.1622776601684

41       3
42      = 8
43      == 11
44      ==== 22
45      ======= 38
46      ============ 60
47      ============== 73
48      ================== 92
49      ======================= 118
50      =========================== 136
51      ========================= 128
52      ================= 89
53      ================= 89
54      =========== 56
55      ======= 37
56      === 18
57      = 7
58      = 5
59      = 6
60       2

Maple

Maple has a built-in for sampling directly from Normal distributions:

with(Statistics):
n := 100000:
X := Sample( Normal(0,1), n );
Mean( X );
StandardDeviation( X );
Histogram( X );

Mathematica

x:= RandomReal[1]
SampleNormal[n_] := (Print[#//Length, " numbers, Mean : ", #//Mean, ", StandardDeviation : ", #//StandardDeviation];
Histogram[#, BarOrigin -> Left,Axes -> False])& [(Table[(-2*Log[x])^0.5*Cos[2*Pi*x], {n} ]]
 
Invocation:
SampleNormal[ 10000 ]
->10000 numbers, Mean : -0.0122308, StandardDeviation : 1.00646
 

Mma NormalDistribution.png

MATLAB / Octave

  N = 100000;	
x = randn(N,1);
mean(x)
std(x)
[nn,xx] = hist(x,100);
bar(xx,nn);

PARI/GP

Works with: PARI/GP version 2.4.3 and above
rnormal()={
my(u1=random(1.),u2=random(1.);
sqrt(-2*log(u1))*cos(2*Pi*u1)
\\ Could easily be extended with a second normal at very little cost.
};
mean(v)={
sum(i=1,#v,v[i])/#v
};
stdev(v,mu="")={
if(mu=="",mu=mean(v));
sqrt(sum(i=1,#v,(v[i]-mu)^2))/#v
};
histogram(v,bins=16,low=0,high=1)={
my(u=vector(bins),width=(high-low)/bins);
for(i=1,#v,u[(v[i]-low)\width+1]++);
u
};
show(n)={
my(v=vector(n,i,rnormal()),m=mean(v),s=stdev(v,m),h,sz=ceil(n/300));
h=histogram(v,,vecmin(v)-.1,vecmax(v)+.1);
for(i=1,#h,for(j=1,h[i]\sz,print1("#"));print());
};
show(10^4)

For versions before 2.4.3, define

rreal()={
my(pr=32*ceil(default(realprecision)*log(10)/log(4294967296))); \\ Current precision
random(2^pr)*1.>>pr
};

and use rreal() in place of random(1.).

A PARI implementation:

GEN
rnormal(long prec)
{
pari_sp ltop = avma;
GEN u1, u2, left, right, ret;
u1 = randomr(prec);
u2 = randomr(prec);
left = sqrtr_abs(shiftr(mplog(u1), 1));
right = mpcos(mulrr(shiftr(mppi(prec), 1), u2));
 
ret = mulrr(left, right);
ret = gerepileupto(ltop, ret);
return ret;
}

Use mpsincos and caching to generate two values at nearly the same cost.

Pascal

Works with: free Pascal

//not neccessary include unit math if using function rnorm

got some trouble with math.randg needs this call randg(cMean,cMean*cStdDiv), whereas randg(cMean,cStdDiv) to get the same results??

From Free Pascal Docs unit math

Program Example40;
{$IFDEF FPC}
{$MOde objFPC}
{$ENDIF}
{ Program to demonstrate the randg function. }
Uses Math;
 
type
tTestData = extended;//because of math.randg
ttstfunc = function (mean, sd: tTestData): tTestData;
tExArray = Array of tTestData;
tSolution = record
SolExArr : tExArray;
SollowVal,
SolHighVal,
SolMean,
SolStdDiv : tTestData;
SolSmpCnt : LongInt;
end;
 
function getSol(genFunc:ttstfunc;Mean,StdDiv: tTestData;smpCnt: LongInt): tSolution;
var
GenValue,
sumValue,
sumsqrVal : extended;
Begin
with result do
Begin
SolSmpCnt := smpCnt;
SolMean := 0;
SolStdDiv := 0;
SolLowVal := Mean+50* StdDiv;
SolHighVal := Mean-50* StdDiv;
setlength(SolExArr,smpCnt);
if smpCnt <= 0 then
EXIT;
sumValue := 0;
sumsqrVal := 0;
repeat
GenValue := genFunc(Mean,StdDiv);
sumValue := sumvalue+GenValue;
sumsqrVal := sumsqrVal+sqr(GenValue);
IF GenValue < SollowVal then
SollowVal:= GenValue
else
IF GenValue > SolHighVal then
SolHighVal := GenValue;
dec(smpCnt);
SolExArr[smpCnt] := GenValue;
until smpCnt<= 0;
SolMean := sumValue/SolSmpCnt;
SolStdDiv := sqrt(sumsqrVal/SolSmpCnt-sqr(SolMean));
end;
end;
 
//http://wiki.freepascal.org/Generating_Random_Numbers#Normal_.28Gaussian.29_Distribution
function rnorm (mean, sd: tTestData): tTestData;
{Calculates Gaussian random numbers according to the Box-Müller approach}
var
u1, u2: extended;
begin
u1 := random;
u2 := random;
rnorm := mean * abs(1 + sqrt(-2 * (ln(u1))) * cos(2 * pi * u2) * sd);
end;
 
procedure Histo(const sol:TSolution;Colcnt,ColLen :LongInt);
var
CntHisto : array of integer;
LoLmt,HiLmt,span : tTestData;
i, j,cnt,maxCnt: LongInt;
sCross : Ansistring;
Begin
setlength(CntHisto,Colcnt);
with Sol do
Begin
span := solHighVal-solLowVal;
LoLmt := solLowVal;
writeln('Count: ',SolSmpCnt:10,' Mean ',SolMean:10:6,' StdDiv ',SolStdDIv:10:6);
writeln('span : ',span:10:5,' Low ',solLowVal:10:6,' high ',solHighVal:10:6);
 
end;
maxCnt := 0;
For j := 0 to Colcnt-1 do
Begin
HiLmt:= LoLmt+span/Colcnt;
cnt := 0;
with sol do
For i := 0 to High(SolExArr) do
IF (HiLmt > SolExArr[i]) AND (SolExArr[i]>= LoLmt) then
inc(cnt);
CntHisto[j] := cnt;
IF maxCnt < cnt then
maxCnt := cnt;
LoLmt:= HiLmt;
end;
inc(CntHisto[Colcnt]); // for HiLmt itself
writeln;
LoLmt := sol.solLowVal;
For i := 0 to Colcnt-1 do
Begin
Writeln(LoLmt:8:4,': ');
cnt:= Round(CntHisto[i]*ColLen/maxCnt);
setlength(sCross,cnt+3);
fillChar(sCross[1],3,' ');
fillChar(sCross[4],cnt,'#');
writeln(CntHisto[i]:10,sCross);
LoLmt := LoLmt+span/Colcnt;
end;
Writeln(sol.solHighVal:8:4,': ');
end;
 
const
cHistCnt = 11;
cColLen = 65;
 
cStdDiv = 0.25;
cMean = 20*cStdDiv;
var
mySol : tSolution;
begin
Randomize;
// test of randg of unit math
Writeln('function randg');
mySol := getSol(@randg,cMean,cMean*cStdDiv,100000);
Histo(mySol,cHistCnt,cColLen);
writeln;
// test of rnorm from wiki
Writeln('function rnorm');
mySol := getSol(@rnorm,cMean,cStdDiv,1000000);
Histo(mySol,cHistCnt,cColLen);
end.
Output:

function randg Count: 100000 Mean 5.000326 StdDiv 1.250027 span : 10.65123 Low -0.333310 high 10.317922

-0.3333:
       25
 0.6350:
      287   #
 1.6033:
     2291   #####
 2.5716:
     9531   #####################
 3.5399:
    22608   #################################################
 4.5082:
    29953   #################################################################
 5.4765:
    22917   ##################################################
 6.4447:
     9716   #####################
 7.4130:
     2352   #####
 8.3813:
      295   #
 9.3496:
       24
10.3179:

function rnorm Count: 1000000 Mean 4.998391 StdDiv 1.251103 span : 11.08994 Low 0.001521 high 11.091461

 0.0015:
      704
 1.0097:
     7797   ##
 2.0179:
    49235   ###########
 3.0261:
   162761   ####################################
 4.0342:
   293242   #################################################################
 5.0424:
   285818   ###############################################################
 6.0506:
   150781   #################################
 7.0588:
    42641   #########
 8.0669:
     6467   #
 9.0751:
      528
10.0833:
       25
11.0915:

Perl 6

constant τ = 2 * pi;
 
sub normdist ($m,) {
my $r = sqrt -2 * log rand;
my= τ * rand;
$r * cos() *+ $m;
}
 
sub MAIN ($size = 100000, $mean = 50, $stddev = 4) {
my @dataset = normdist($mean,$stddev) xx $size;
 
my $m = [+](@dataset) / $size;
say (:$m);
 
my= sqrt [+](@dataset X** 2) / $size - $m**2;
say (:);
 
(my %hash){.round}++ for @dataset;
my $scale = 180 * $stddev / $size;
constant @subbar = < ⎸ ▏ ▎ ▍ ▌ ▋ ▊ ▉ █ >;
for %hash.keys».Int.minmax(+*) -> $i {
my $x = (%hash{$i} // 0) * $scale;
my $full = floor $x;
my $part = 8 * ($x - $full);
say $i, "\t", '█' x $full, @subbar[$part];
}
}
Output:
"m" => 50.006107405837142e0
"σ" => 4.0814435639885254e0
33	⎸
34	⎸
35	⎸
36	▏
37	▎
38	▊
39	█▋
40	███⎸
41	█████▊
42	██████████⎸
43	███████████████▋
44	███████████████████████▏
45	████████████████████████████████▌
46	███████████████████████████████████████████▍
47	██████████████████████████████████████████████████████▏
48	███████████████████████████████████████████████████████████████▏
49	█████████████████████████████████████████████████████████████████████▋
50	███████████████████████████████████████████████████████████████████████▊
51	█████████████████████████████████████████████████████████████████████▌
52	███████████████████████████████████████████████████████████████⎸
53	██████████████████████████████████████████████████████▎
54	███████████████████████████████████████████⎸
55	████████████████████████████████▌
56	███████████████████████▍
57	███████████████▉
58	█████████▉
59	█████▍
60	███▍
61	█▋
62	▊
63	▍
64	▏
65	⎸
66	⎸
67	⎸

Phix

Translation of: Liberty_BASIC
procedure sample(integer n)
-- show mean, standard deviation. Find max, min.
sequence dat = repeat(0,n)
for i=1 to n do
dat[i] = sqrt(-2*log(rnd()))*cos(2*PI*rnd())
end for
printf(1,"%d data terms used.\n",{n})
 
atom mean = sum(dat)/n,
mx = max(dat),
mn = min(dat),
range = mx-mn
printf(1,"Largest term was %g & smallest was %g\n",{mx,mn})
printf(1,"Mean = %g\n",{mean})
printf(1,"Stddev = %g\n",sqrt(sum(sq_mul(dat,dat))/n-mean*mean))
 
-- show histogram
integer nBins = 50
sequence bins = repeat(0,nBins+1)
for i=1 to n do
bins[floor((dat[i]-mn)/range*nBins)+1] += 1
end for
for b=1 to nBins do
puts(1,repeat('#',floor(nBins*bins[b]/n*30))&"\n")
end for
end procedure
 
sample(100000)
Output:
100000 data terms used.
Largest term was 4.30779 & smallest was -4.11902
Mean = -0.00252597
Stddev = 1.00067

#
##
####
######
##########
#############
##################
########################
#################################
########################################
####################################################
#############################################################
######################################################################
###############################################################################
#######################################################################################
###############################################################################################
#################################################################################################
#####################################################################################################
###################################################################################################
################################################################################################
########################################################################################
###############################################################################
#######################################################################
############################################################
#################################################
#######################################
##############################
#########################
################
############
#########
######
####
##
#
Translation of: Lua
function gaussian(atom mean, atom variance)
return sqrt(-2 * variance * log(rnd())) *
cos(2 * variance * PI * rnd()) + mean
end function
 
function mean(sequence t)
return sum(t)/length(t)
end function
 
function std(sequence t)
atom squares = 0,
avg = mean(t)
for i=1 to length(t) do
squares += power(avg-t[i],2)
end for
atom variance = squares/length(t)
return sqrt(variance)
end function
 
procedure showHistogram(sequence t)
for i=ceil(min(t)) to floor(max(t)) do
integer n = 0
for k=1 to length(t) do
n += ceil(t[k]-0.5)=i
end for
integer l = floor(n/length(t)*200)
printf(1,"%d %s %d\n",{i,repeat('=',l),n})
end for
end procedure
 
sequence t = repeat(0,100000)
integer avg = 50, variance = 10
for i=1 to length(t) do
t[i] = gaussian(avg, variance)
end for
printf(1,"Mean: %g, expected %g\n",{mean(t),avg})
printf(1,"StdDev: %g, expected %g\n",{std(t),sqrt(variance)})
showHistogram(t)
Output:
Mean: 50.0041, expected 50
StdDev: 3.1673, expected 3.16228
37  2
38  7
39  30
40  92
41  220
42 = 523
43 == 1098
44 ==== 2140
45 ======= 3690
46 =========== 5753
47 =============== 7906
48 ==================== 10299
49 ======================= 11813
50 ========================= 12555
51 ======================= 11934
52 ==================== 10327
53 ================ 8099
54 =========== 5733
55 ======= 3684
56 ==== 2126
57 == 1098
58  487
59  226
60  106
61  36
62  9
63  7

PureBasic

Procedure.f randomf(resolution = 2147483647)
ProcedureReturn Random(resolution) / resolution
EndProcedure
 
Procedure.f normalDist() ;Box Muller method
ProcedureReturn Sqr(-2 * Log(randomf())) * Cos(2 * #PI * randomf())
EndProcedure
 
Procedure sample(n, nBins = 50)
Protected i, maxBinValue, binNumber
Protected.f d, mean, sum, sumSq, mx, mn, range
 
Dim dat.f(n)
For i = 1 To n
dat(i) = normalDist()
Next
 
;show mean, standard deviation, find max & min.
mx = -1000
mn = 1000
sum = 0
sumSq = 0
For i = 1 To n
d = dat(i)
If d > mx: mx = d: EndIf
If d < mn: mn = d: EndIf
sum + d
sumSq + d * d
Next
 
PrintN(Str(n) + " data terms used.")
PrintN("Largest term was " + StrF(mx) + " & smallest was " + StrF(mn))
mean = sum / n
PrintN("Mean = " + StrF(mean))
PrintN("Stddev = " + StrF((sumSq / n) - Sqr(mean * mean)))
 
;show histogram
range = mx - mn
Dim bins(nBins)
For i = 1 To n
binNumber = Int(nBins * (dat(i) - mn) / range)
bins(binNumber) + 1
Next
 
maxBinValue = 1
For i = 0 To nBins
If bins(i) > maxBinValue
maxBinValue = bins(i)
EndIf
Next
 
#normalizedMaxValue = 70
For binNumber = 0 To nBins
tickMarks = Round(bins(binNumber) * #normalizedMaxValue / maxBinValue, #PB_Round_Nearest)
PrintN(ReplaceString(Space(tickMarks), " ", "#"))
Next
PrintN("")
EndProcedure
 
If OpenConsole()
sample(100000)
 
Print(#CRLF$ + #CRLF$ + "Press ENTER to exit"): Input()
CloseConsole()
EndIf

Sample output:

100000 data terms used.
Largest term was 4.5352029800 & smallest was -4.5405135155
Mean = 0.0012346541
Stddev = 0.9959455132





#
###
######
##########
##################
############################
#########################################
#####################################################
################################################################
######################################################################
######################################################################
################################################################
#####################################################
#########################################
#############################
##################
##########
######
###
#




Python

This uses the external matplotlib package as well as the built-in standardlib function random.gauss.

from __future__ import division
import matplotlib.pyplot as plt
import random
 
mean, stddev, size = 50, 4, 100000
data = [random.gauss(mean, stddev) for c in range(size)]
 
mn = sum(data) / size
sd = (sum(x*x for x in data) / size
- (sum(data) / size) ** 2) ** 0.5
 
print("Sample mean = %g; Stddev = %g; max = %g; min = %g for %i values"
 % (mn, sd, max(data), min(data), size))
 
plt.hist(data,bins=50)
Output:
Sample mean = 49.9822; Stddev = 4.00938; max = 66.8091; min = 33.5283 for 100000 values

Normal distribution py.svg

R

R can generate random normal distributed numbers using the rnorm command:

n = 100000;
X = rnorm(n, mean = 0, sd = 1);
mean( X );
sd( X );
hist( X );

Racket

This shows how one would generate samples from a normal distribution, compute statistics and plot a histogram.

Histogram-racket.png
 
#lang racket
(require math (planet williams/science/histogram-with-graphics))
 
(define data (sample (normal-dist 50 4) 100000))
 
(displayln (~a "Mean:\t" (mean data)))
(displayln (~a "Stddev:\t" (stddev data)))
(displayln (~a "Max:\t" (apply max data)))
(displayln (~a "Min:\t" (apply min data)))
 
(define h (make-histogram-with-ranges-uniform 40 30 70))
(for ([x data]) (histogram-increment! h x))
(histogram-plot h "Normal distribution μ=50 σ=4")
 

The other part of the task was to produce normal distributed numbers from a unit distribution. The following code is an implementation of the polar method. It is a slightly modified version of code originally written by Sebastian Egner.

 
#lang racket
(require math)
 
(define random-normal
(let ([unit (uniform-dist)]
[next #f])
(λ (μ σ)
(if next
(begin0
(+ μ (* σ next))
(set! next #f))
(let loop ()
(let* ([v1 (- (* 2.0 (sample unit)) 1.0)]
[v2 (- (* 2.0 (sample unit)) 1.0)]
[s (+ (sqr v1) (sqr v2))])
(cond [(>= s 1) (loop)]
[else (define scale (sqrt (/ (* -2.0 (log s)) s)))
(set! next (* scale v2))
(+ μ (* σ scale v1))])))))))
 

REXX

The REXX language doesn't have any "higher math" BIF functions like SIN/COS/LN/LOG/SQRT/POW/etc,
so we hoi polloi programmers have to roll our own.

/*REXX program generates  10,000  normally distributed numbers  (Gaussian distribution).*/
parse arg n seed . /*obtain optional arguments from the CL*/
if n=='' | n=="," then n=10000 /*Not specified? Then use the default.*/
if datatype(seed,'W') then call random ,,seed /*seed is for repeatable RANDOM numbers*/
call pi /*call subroutine to define pi constant*/
do g=1 for n /*generate N uniform random numbers. */
#.g=sqrt(-2*ln(rand()))*cos(2*pi*rand()) /*assign a uniform random number to #. */
end /*g*/
 
mn=#.1; mx=mn; s=0; ss=0; noise=n*.0005 /*calculate the noise: 1/20th % of N.*/
 
do j=1 for n; _=#.j; s=s+_; ss=ss+_*_ /*the sum, and the sum of squares. */
mn=min(mn,#.j); mx=max(mx,#.j) /*find the minimum and the maximum. */
end /*j*/
!.=0
say 'number of data points = ' aa(n )
say ' minimum = ' aa(mn )
say ' maximum = ' aa(mx )
say ' arithmetic mean = ' aa(s/n)
say ' standard deviation = ' aa(sqrt(ss/n - (s/n)**2) )
?mn=!.1;  ?mx=?mn /*define minimum & maximum value so far*/
parse value scrSize() with sd sw . /*obtain the (true) screen size of term*/ /*◄──not all REXXes have this BIF*/
sdE=sd-4 /*the effective (useable) screen depth.*/
swE=sw-1 /* " " " " width.*/
$=1 / max(1,mx-mn) * sdE /*$ is used for scaling depth of histo*/
do i=1 for n;  ?=trunc((#.i-mn) * $) /*calculate the relative line.*/
 !.?=!.?+1 /*bump the counter. */
 ?mn=min(?mn, !.?);  ?mx=max(?mx, !.?) /*find the minimum and maximum*/
end /*i*/
f=swE/?mx /*limit graph to 1 full screen*/
do h=0 for sdE; _=!.h /*obtain a data point. */
if _>noise then say copies('─', trunc(_*f)) /*display a bar of histogram. */
end /*h*/ /*[↑] use a hyphen for histo.*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/
aa: parse arg a; return left('',(a>=0)+2*datatype(a,'W'))a /*prepend a blank if #>=0, add two blanks if its whole*/
e: e =2.7182818284590452353602874713526624977572470936999595749669676277240766303535; return e
pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862; return pi
rand: return random(, 1e5) / 1e5 /*REXX generates uniform random postive integers.*/
r2r: return arg(1) // (pi()*2) /*normalize the given angle (in radians) to ±2pi.*/
.sincos: parse arg z,_,i; x=x*x; p=z; do k=2 by 2; _=-_*x/(k*(k+i)); z=z+_; if z=p then leave; p=z; end; return z
ln: procedure; parse arg x,f; call e; ig=x>1.5; is=1-2*(ig\==1); ii=0; xx=x; return .ln()
.ln: do while ig&xx>1.5|\ig&xx<.5;_=e; do k=-1;iz=xx*_**-is;if k>=0&(ig&iz<1|\ig&iz>.5) then leave; _=_*_;izz=iz; end
xx=izz;ii=ii+is*2**k;end; x=x*e**-ii-1;z=0;_=-1;p=z; do k=1;_=-_*x;z=z+_/k;if z=p then leave;p=z;end; return z+ii
 
cos: procedure; parse arg x; x=r2r(x); a=abs(x); hpi=pi*.5; numeric fuzz min(6,digits()-3); if a=pi() then return -1
if a=hpi|a=hpi*3 then return 0; if a=pi()/3 then return .5; if a=pi()*2/3 then return -.5; return .sinCos(1,1,-1)
 
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); m.=9; numeric digits; numeric form; h=d+6
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*/; numeric digits d; return g/1

This REXX program makes use of   scrsize   REXX program (or BIF) which is used to determine the screen size of the terminal (console).
The   SCRSIZE.REX   REXX program is included here ──►   SCRSIZE.REX.

output   when using the default input:
(The output shown when the screen size is 50x80.)

number of data points =     10000
              minimum =  -3.41571894
              maximum =   3.96752904
      arithmetic mean =  -0.0150910306
   standard deviation =   0.99056458


─
─
───
─────
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────────────
─────────────────
─────────────────────
─────────────────────────────
────────────────────────────────────
───────────────────────────────────────────
───────────────────────────────────────────────
──────────────────────────────────────────────────────
────────────────────────────────────────────────────────────────────
──────────────────────────────────────────────────────────────────
────────────────────────────────────────────────────────────────────────
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───────────────────────────────────────
───────────────────────────────
─────────────────────────────
──────────────────
─────────────
────────
───────
────
───
─
─

output   when using the default input:
(The output shown when the screen size is 60x130.)

number of data points =     10000
              minimum =  -3.83073183
              maximum =   3.61051026
      arithmetic mean =   0.00421997333
   standard deviation =   0.981924955
──
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Run BASIC

 
s = 100000
h$ = "============================================================="
h$ = h$ + h$
dim ndis(s)
' mean and standard deviation.
mx = -9999
mn = 9999
sum = 0
sumSqr = 0
for i = 1 to s ' find minimum and maximum
ms = rnd(1)
ss = rnd(1)
nd = (-2 * log(ms))^0.5 * cos(2 *3.14159265 * ss) ' normal distribution
ndis(i) = nd
mx = max(mx, nd)
mn = min(mn, nd)
sum = sum + nd
sumSqr = sumSqr + nd ^ 2
next i
 
mean = sum / s
range = mx - mn
 
print "Samples  :"; s
print "Largest  :"; mx
print "Smallest  :"; mn
print "Range  :"; range
print "Mean  :"; mean
print "Stand Dev :"; (sumSqr /s -mean^2)^0.5
 
'Show chart of histogram
nBins = 50
dim bins(nBins)
for i = 1 to s
z = int((ndis(i) -mn) /range *nBins)
bins(z) = bins(z) + 1
mb = max(bins(z),mb)
next i
for b = 0 to nBins -1
print using("##",b);" ";using("#####",bins(b));" ";left$(h$,(bins(b) / mb) * 90)
next b
END
Output:
Samples   :100000
Largest   :4.61187177
Smallest  :-4.21695424
Range     :8.82882601
Mean      :-9.25042513e-4
Stand Dev :1.00680067

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SAS

data test;
n=100000;
twopi=2*constant('pi');
do i=1 to n;
u=ranuni(0);
v=ranuni(0);
r=sqrt(-2*log(u));
x=r*cos(twopi*v);
y=r*sin(twopi*v);
z=rannor(0);
output;
end;
keep x y z;
 
proc means mean stddev;
 
proc univariate;
histogram /normal;
 
run;
 
/*
Variable Mean Std Dev
----------------------------------------
x -0.0052720 0.9988467
y 0.000023995 1.0019996
z 0.0012857 1.0056536
*/

Sidef

Translation of: Perl 6
define τ = Number.tau
 
func normdist (m, σ) {
var r = sqrt(-2 * 1.rand.log)
var Θ = (τ * 1.rand)
r * Θ.cos * σ + m
}
 
var size = 100_000
var mean = 50
var stddev = 4
 
var dataset = size.of { normdist(mean, stddev) }
var m = (dataset.sum(0) / size)
say ("m: #{m}")
 
var σ = sqrt(dataset »**» 2 -> sum(0) / size - m**2)
say ("s: #{σ}")
 
var hash = Hash()
dataset.each { |n| hash{ n.round(0) } := 0 ++ }
 
var scale = (180 * stddev / size)
const subbar = < ⎸ ▏ ▎ ▍ ▌ ▋ ▊ ▉ █ >
 
for i in (hash.keys.map{.to_i}.sort) {
var x = (hash{i} * scale)
var full = x.int
var part = (8 * (x - full))
say (i, "\t", '█' * full, subbar[part])
}
Output:
m: 49.99538275618550306540055142077589
s: 4.00295544816687358837821680496471
33	⎸
34	⎸
35	⎸
36	▏
37	▎
38	▊
39	█▋
40	███▏
41	██████▏
42	█████████▍
43	███████████████▌
44	███████████████████████▋
45	████████████████████████████████▍
46	████████████████████████████████████████████▎
47	█████████████████████████████████████████████████████▍
48	███████████████████████████████████████████████████████████████▍
49	█████████████████████████████████████████████████████████████████████▌
50	████████████████████████████████████████████████████████████████████████▋
51	█████████████████████████████████████████████████████████████████████▊
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53	████████████████████████████████████████████████████▉
54	███████████████████████████████████████████▉
55	█████████████████████████████████▎
56	███████████████████████⎸
57	███████████████▋
58	█████████▋
59	█████▍
60	███▍
61	█▊
62	▋
63	▍
64	▏
65	⎸
66	⎸

Tcl

package require Tcl 8.5
# Uses the Box-Muller transform to compute a pair of normal random numbers
proc tcl::mathfunc::nrand {mean stddev} {
variable savednormalrandom
if {[info exists savednormalrandom]} {
return [expr {$savednormalrandom*$stddev + $mean}][unset savednormalrandom]
}
set r [expr {sqrt(-2*log(rand()))}]
set theta [expr {2*3.1415927*rand()}]
set savednormalrandom [expr {$r*sin($theta)}]
expr {$r*cos($theta)*$stddev + $mean}
}
proc stats {size {slotfactor 10}} {
set sum 0.0
set sum2 0.0
for {set i 0} {$i < $size} {incr i} {
set r [expr { nrand(0.5, 0.2) }]
 
incr histo([expr {int(floor($r*$slotfactor))}])
set sum [expr {$sum + $r}]
set sum2 [expr {$sum2 + $r**2}]
}
set mean [expr {$sum / $size}]
set stddev [expr {sqrt($sum2/$size - $mean**2)}]
puts "$size numbers"
puts "Mean: $mean"
puts "StdDev: $stddev"
foreach i [lsort -integer [array names histo]] {
puts [string repeat "*" [expr {$histo($i)*350/int($size)}]]
}
}
 
stats 100
puts ""
stats 1000
puts ""
stats 10000
puts ""
stats 100000 20

Sample output:

100 numbers
Mean:   0.49355955990390254
StdDev: 0.19651396178121985
***
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1000 numbers
Mean:   0.5066940614105869
StdDev: 0.2016794788065389


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10000 numbers
Mean:   0.49980964730768285
StdDev: 0.1968441612522318

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100000 numbers
Mean:   0.49960438950922254
StdDev: 0.20060211160998606





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The blank lines in the output are where the number of samples is too small to even merit a single unit on the histogram.

zkl

Translation of: Go
fcn norm2{   // Box-Muller
const PI2=(0.0).pi*2;;
rnd:=(0.0).random.fp(1); // random number in [0,1), using partial application
r,a:=(-2.0*rnd().log()).sqrt(), PI2*rnd();
return(r*a.cos(), r*a.sin()); // z0,z1
}
const N=100000, BINS=12, SIG=3, SCALE=500;
var sum=0.0,sumSq=0.0, h=BINS.pump(List(),0); // (0,0,0,...)
fcn accum(v){
sum+=v;
sumSq+=v*v;
b:=(v + SIG)*BINS/SIG/2;
if(0<=b<BINS) h[b]+=1;
};

Partial application: rnd() --> (0.0).random(1). Basically, the fp method fixes the call parameters, which are then used when the partial thing is run.

foreach i in (N/2){ v1,v2:=norm2(); accum(v1); accum(v2); }
println("Samples: %,d".fmt(N));
println("Mean: ", m:=sum/N);
println("Stddev: ", (sumSq/N - m*m).sqrt());
foreach p in (h){ println("*"*(p/SCALE)) }
Output:
Samples: 100,000
Mean:    0.0005999
Stddev:  1.003
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