Statistics/Normal distribution: Difference between revisions
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=={{header|jq}}== |
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'''Adapted from [[#Wren|Wren]]''' |
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{{works with|jq}} |
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'''Works with gojq, the Go implementation of jq''' (*) |
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Since jq does not have a built-in PRNG, this entry uses an external source |
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for entropy. For the sake of illustration, we will use /dev/urandom |
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as follows: |
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cat /dev/urandom | tr -cd '0-9' | fold -w 10 | |
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jq -nRr -f normal-distribution.jq |
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To save space, the function that generates the sample does not retain the observations, and for |
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simplicity, computes the sum of squared observations on the |
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assumption that overflow will not be an an issue, which is |
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reasonable as jq arithmetic uses IEEE 754 64-bit numbers. |
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(*) gojq requires an enormous amount of memory to complete the task for N=100,000, |
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and takes about 20 times longer. |
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'''Preliminaries''' |
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<lang jq># Pretty print a number to facilitate alignment of the decimal point. |
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# Input: a number without an exponent |
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# Output: a string holding the reformatted number so that there are at least `left` characters |
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# to the left of the decimal point, and exactly `right` characters to its right. |
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# Spaces are used for padding on the left, and zeros for padding on the right. |
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# No left-truncation occurs, so `left` can be specified as 0 to prevent left-padding. |
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def pp(left; right): |
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def lpad: tostring | (left - length) as $l | (" " * $l)[:$l] + .; |
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def rpad: |
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if (right > length) then . + ((right - length) * "0") |
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else .[:right] |
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end; |
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tostring as $s |
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| $s |
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| index(".") as $ix |
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| ((if $ix then $s[0:$ix] else $s end) | lpad) + "." + |
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(if $ix then $s[$ix+1:] | .[:right] else "" end | rpad) ; |
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def sigma( stream ): reduce stream as $x (0; . + $x); |
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# Input: {n, sum, ss} |
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# Output: augmented object with {mean, variance} |
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def sample_mean_and_variance: |
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.mean = (.sum/.n) |
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| .variance = ((.ss / .n) - .mean*.mean);</lang> |
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'''The Task''' |
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<lang jq># Task parameters |
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def parameters: { |
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N: 100000, |
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NUM_BINS: 12, |
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HIST_CHAR: "■", |
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HIST_CHAR_ALT: "-", |
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HIST_CHAR_SIZE: null, # null means compute dynamically |
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binSize: 0.1, |
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mu: 0.5, |
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sigma: 0.25 } |
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| .bins = [range(0; .NUM_BINS) | 0] ; |
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# input: an array of two iid rvs on [0,1] |
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# output: [z0, z1] as per the Box-Muller method -- see |
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# https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform |
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def normal(mu; sigma): |
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def pi: (1|atan) * 4; |
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. as [$u1, $u2] |
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| pi as $pi |
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| (sigma * ((-2 * ($u1|log))|sqrt)) as $mag |
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| [ $mag * ((2 * $pi * $u2)|cos) + mu, |
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$mag * ((2 * $pi * $u2)|sin) + mu ] ; |
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# Generate a random sample as specified by ., the task object (see `parameters`). |
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# Output: updated task object with sample statistics and .bins for creating a histogram. |
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# Each call to `input` should yield a string of random decimal digits |
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# such that the ensemble of ("0." + input | tonumber) can be considered to be iid rv on [0,1]. |
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def generate: |
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# uniformly distributed random variable on [0,1]: |
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def udrv: "0." + input | tonumber; |
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# Maybe compute the bucket size: |
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(.HIST_CHAR_SIZE = (.HIST_CHAR_SIZE // (.N / (.NUM_BINS * 20) | ceil))) as $p |
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| reduce range(0; $p.N/2) as $i ($p; |
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([udrv, udrv] | normal($p.mu; $p.sigma)) as $rns |
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| reduce (0,1) as $j (.; |
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$rns[$j] as $rn |
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| .n += 1 |
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| .sum += $rn |
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| .ss += ($rn*$rn) |
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| (if $rn < 0 then 0 |
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elif $rn >= 1 then ($p.NUM_BINS - 1) |
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else ($rn/.binSize)|floor + 1 |
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end ) as $bn |
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| .bins[$bn] += 1 |
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# to retain the observations: .samples[$i*2 + $j] = $rn |
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)) ; |
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# Input: an object with |
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# {NUM_BINS, HIST_CHAR_SIZE, HIST_CHAR, HIST_CHAR_ALT, binSize, bins} |
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# Output: a stream of strings |
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def histogram: |
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def tidy: pp(2;1); |
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range(0; .NUM_BINS) as $i |
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| ((.bins[$i] / .HIST_CHAR_SIZE)|floor) as $bs |
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| (if $i == 0 or $i == .NUM_BINS -1 |
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then .HIST_CHAR_ALT else .HIST_CHAR end) as $char |
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| (if $bs == 0 then "" else $char * $bs end) as $hist |
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| if $i == 0 |
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then " -∞ ..< 0.0 \($hist)" # .bins[0] |
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elif ($i < .NUM_BINS - 1) |
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then "\(.binSize * ($i-1) | tidy) ..<\(.binSize * $i|tidy) \($hist)" # .bins[$i]] |
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else " 1.0 .. +∞ \($hist)" # .bins[.NUM_BINS - 1] |
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end; |
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def task: |
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parameters |
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| generate |
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| sample_mean_and_variance as $mv |
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| (if .HIST_CHAR_SIZE == 1 then "" else "s" end) as $plural |
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| "Summary statistics for \(.N) observations from N(\(.mu), \(.sigma)):", |
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" mean: \($mv.mean | pp(2;4))", |
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" variance: \($mv.variance | pp(2;4))", |
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" unadjusted stddev: \($mv.variance | sqrt | pp(2;4))", |
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" Range Number of observations (each \(.HIST_CHAR) represents \(.HIST_CHAR_SIZE) observation\($plural))", |
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histogram ; |
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task</lang> |
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{{out}} |
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<pre> |
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Summary statistics for 100000 observations from N(0.5, 0.25): |
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mean: 0.5001 |
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variance: 0.0622 |
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unadjusted stddev: 0.2495 |
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Range Number of observations (each ■ represents 417 observations) |
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-∞ ..< 0.0 ----- |
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0.0 ..< 0.1 ■■■■■■■■ |
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0.1 ..< 0.2 ■■■■■■■■■■■■■■ |
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0.2 ..< 0.3 ■■■■■■■■■■■■■■■■■■■■■■■ |
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0.3 ..< 0.4 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ |
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0.4 ..< 0.5 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ |
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0.5 ..< 0.6 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ |
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0.6 ..< 0.7 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ |
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0.7 ..< 0.8 ■■■■■■■■■■■■■■■■■■■■■■■ |
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0.8 ..< 0.9 ■■■■■■■■■■■■■■ |
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0.9 ..< 1.0 ■■■■■■■■ |
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1.0 .. +∞ ------ |
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</pre> |
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=={{header|Julia}}== |
=={{header|Julia}}== |
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Revision as of 16:05, 3 October 2021
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
- 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 standard deviation, and show a histogram of the data.
- Mention any native language support for the generation of normally distributed random numbers.
- Reference
- You may refer to code in Statistics/Basic if available.
C
<lang 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;
}</lang>
- 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#
<lang csharp>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);
}
}</lang>
- 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 <lang cpp>#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 ;
}</lang> 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. <lang d>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;
}</lang>
- 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
<lang 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)</lang>
- 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: =
Factor
<lang factor>USING: assocs formatting kernel math math.functions math.statistics random sequences sorting ;
2,000,000 [ 0 1 normal-random-float ] replicate ! make data set dup [ mean ] [ population-std ] bi ! calculate and show "Mean: %f\nStdev: %f\n\n" printf ! mean and stddev
[ 10 * floor 10 / ] map ! map data to buckets histogram >alist [ first ] sort-with ! create histogram sorted by bucket (key) dup values supremum ! find maximum count [
[ /f 100 * >integer ] keepd ! how big should this histogram bar be? [ [ CHAR: * ] "" replicate-as ] dip ! make the bar "% 5.2f: %s %d\n" printf ! print a line of the histogram
] curry assoc-each</lang>
- Output:
Mean: 0.000798 Stdev: 1.000549 -4.90: 2 -4.80: 1 -4.70: 1 -4.60: 3 -4.50: 3 -4.40: 6 -4.30: 15 -4.20: 13 -4.10: 16 -4.00: 42 -3.90: 62 -3.80: 68 -3.70: 98 -3.60: 145 -3.50: 205 -3.40: 311 -3.30: 379 -3.20: 580 -3.10: 739 -3.00: * 1002 -2.90: * 1349 -2.80: ** 1893 -2.70: *** 2499 -2.60: **** 3211 -2.50: ***** 4035 -2.40: ****** 5141 -2.30: ******* 6392 -2.20: ********* 7869 -2.10: ************ 9780 -2.00: ************** 11787 -1.90: ****************** 14483 -1.80: ********************* 17183 -1.70: ************************* 20387 -1.60: ****************************** 24049 -1.50: ********************************** 27555 -1.40: **************************************** 32153 -1.30: ********************************************* 36707 -1.20: *************************************************** 40921 -1.10: ********************************************************* 45928 -1.00: *************************************************************** 50707 -0.90: ********************************************************************* 55697 -0.80: *************************************************************************** 60377 -0.70: ******************************************************************************** 64358 -0.60: ************************************************************************************ 67928 -0.50: ***************************************************************************************** 71911 -0.40: ********************************************************************************************* 75054 -0.30: ************************************************************************************************ 77073 -0.20: ************************************************************************************************** 78768 -0.10: *************************************************************************************************** 79732 0.00: **************************************************************************************************** 79952 0.10: *************************************************************************************************** 79412 0.20: ************************************************************************************************ 77511 0.30: ********************************************************************************************* 74487 0.40: ****************************************************************************************** 72250 0.50: ************************************************************************************** 68789 0.60: ******************************************************************************** 64408 0.70: *************************************************************************** 60122 0.80: ********************************************************************* 55619 0.90: *************************************************************** 50869 1.00: ********************************************************* 45883 1.10: **************************************************** 41586 1.20: ********************************************** 37145 1.30: *************************************** 31715 1.40: ********************************** 27779 1.50: ****************************** 24270 1.60: ************************* 20516 1.70: ********************* 17221 1.80: ***************** 14344 1.90: ************** 11789 2.00: ************ 9796 2.10: ********* 7922 2.20: ******* 6331 2.30: ****** 5138 2.40: ***** 4044 2.50: *** 3065 2.60: ** 2397 2.70: ** 1846 2.80: * 1462 2.90: * 1001 3.00: 765 3.10: 587 3.20: 393 3.30: 299 3.40: 197 3.50: 132 3.60: 100 3.70: 74 3.80: 59 3.90: 32 4.00: 29 4.10: 12 4.20: 15 4.30: 6 4.40: 3 4.50: 4 4.60: 3 4.70: 2 4.80: 1
Fortran
Using the Marsaglia polar method <lang fortran>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</lang>
- 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
<lang 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</lang> 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. <lang go>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))
}
}</lang> Output:
mean: -0.0034970888831523488 stddev: 1.0040682925006286 * **** ********* *************** ******************* ****************** ************** ********* **** *
Haskell
<lang 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</lang>
- 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 <lang j>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.)</lang> Example Usage <lang j> DataSet=: rnorm01 1e5
(mean , stddev) DataSet
0.000781667 1.00154
require 'plot' plot (5 %~ i: 25) ([;histogram) DataSet</lang>
Java
<lang java>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());
}
}</lang>
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: **
jq
Adapted from Wren
Works with gojq, the Go implementation of jq (*)
Since jq does not have a built-in PRNG, this entry uses an external source for entropy. For the sake of illustration, we will use /dev/urandom as follows:
cat /dev/urandom | tr -cd '0-9' | fold -w 10 |
jq -nRr -f normal-distribution.jq
To save space, the function that generates the sample does not retain the observations, and for simplicity, computes the sum of squared observations on the assumption that overflow will not be an an issue, which is reasonable as jq arithmetic uses IEEE 754 64-bit numbers.
(*) gojq requires an enormous amount of memory to complete the task for N=100,000, and takes about 20 times longer.
Preliminaries <lang jq># Pretty print a number to facilitate alignment of the decimal point.
- Input: a number without an exponent
- Output: a string holding the reformatted number so that there are at least `left` characters
- to the left of the decimal point, and exactly `right` characters to its right.
- Spaces are used for padding on the left, and zeros for padding on the right.
- No left-truncation occurs, so `left` can be specified as 0 to prevent left-padding.
def pp(left; right):
def lpad: tostring | (left - length) as $l | (" " * $l)[:$l] + .;
def rpad:
if (right > length) then . + ((right - length) * "0")
else .[:right]
end;
tostring as $s
| $s
| index(".") as $ix
| ((if $ix then $s[0:$ix] else $s end) | lpad) + "." +
(if $ix then $s[$ix+1:] | .[:right] else "" end | rpad) ;
def sigma( stream ): reduce stream as $x (0; . + $x);
- Input: {n, sum, ss}
- Output: augmented object with {mean, variance}
def sample_mean_and_variance:
.mean = (.sum/.n) | .variance = ((.ss / .n) - .mean*.mean);</lang>
The Task <lang jq># Task parameters def parameters: {
N: 100000, NUM_BINS: 12, HIST_CHAR: "■", HIST_CHAR_ALT: "-", HIST_CHAR_SIZE: null, # null means compute dynamically binSize: 0.1, mu: 0.5, sigma: 0.25 } | .bins = [range(0; .NUM_BINS) | 0] ;
- input: an array of two iid rvs on [0,1]
- output: [z0, z1] as per the Box-Muller method -- see
- https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform
def normal(mu; sigma):
def pi: (1|atan) * 4;
. as [$u1, $u2]
| pi as $pi
| (sigma * ((-2 * ($u1|log))|sqrt)) as $mag
| [ $mag * ((2 * $pi * $u2)|cos) + mu,
$mag * ((2 * $pi * $u2)|sin) + mu ] ;
- Generate a random sample as specified by ., the task object (see `parameters`).
- Output: updated task object with sample statistics and .bins for creating a histogram.
- Each call to `input` should yield a string of random decimal digits
- such that the ensemble of ("0." + input | tonumber) can be considered to be iid rv on [0,1].
def generate:
# uniformly distributed random variable on [0,1]:
def udrv: "0." + input | tonumber;
# Maybe compute the bucket size:
(.HIST_CHAR_SIZE = (.HIST_CHAR_SIZE // (.N / (.NUM_BINS * 20) | ceil))) as $p
| reduce range(0; $p.N/2) as $i ($p;
([udrv, udrv] | normal($p.mu; $p.sigma)) as $rns
| reduce (0,1) as $j (.;
$rns[$j] as $rn
| .n += 1 | .sum += $rn | .ss += ($rn*$rn)
| (if $rn < 0 then 0
elif $rn >= 1 then ($p.NUM_BINS - 1)
else ($rn/.binSize)|floor + 1
end ) as $bn
| .bins[$bn] += 1
# to retain the observations: .samples[$i*2 + $j] = $rn
)) ;
- Input: an object with
- {NUM_BINS, HIST_CHAR_SIZE, HIST_CHAR, HIST_CHAR_ALT, binSize, bins}
- Output: a stream of strings
def histogram:
def tidy: pp(2;1);
range(0; .NUM_BINS) as $i
| ((.bins[$i] / .HIST_CHAR_SIZE)|floor) as $bs
| (if $i == 0 or $i == .NUM_BINS -1
then .HIST_CHAR_ALT else .HIST_CHAR end) as $char
| (if $bs == 0 then "" else $char * $bs end) as $hist
| if $i == 0
then " -∞ ..< 0.0 \($hist)" # .bins[0]
elif ($i < .NUM_BINS - 1)
then "\(.binSize * ($i-1) | tidy) ..<\(.binSize * $i|tidy) \($hist)" # .bins[$i]]
else " 1.0 .. +∞ \($hist)" # .bins[.NUM_BINS - 1]
end;
def task:
parameters
| generate
| sample_mean_and_variance as $mv
| (if .HIST_CHAR_SIZE == 1 then "" else "s" end) as $plural
| "Summary statistics for \(.N) observations from N(\(.mu), \(.sigma)):",
" mean: \($mv.mean | pp(2;4))",
" variance: \($mv.variance | pp(2;4))",
" unadjusted stddev: \($mv.variance | sqrt | pp(2;4))",
" Range Number of observations (each \(.HIST_CHAR) represents \(.HIST_CHAR_SIZE) observation\($plural))",
histogram ;
task</lang>
- Output:
Summary statistics for 100000 observations from N(0.5, 0.25):
mean: 0.5001
variance: 0.0622
unadjusted stddev: 0.2495
Range Number of observations (each ■ represents 417 observations)
-∞ ..< 0.0 -----
0.0 ..< 0.1 ■■■■■■■■
0.1 ..< 0.2 ■■■■■■■■■■■■■■
0.2 ..< 0.3 ■■■■■■■■■■■■■■■■■■■■■■■
0.3 ..< 0.4 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
0.4 ..< 0.5 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
0.5 ..< 0.6 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
0.6 ..< 0.7 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
0.7 ..< 0.8 ■■■■■■■■■■■■■■■■■■■■■■■
0.8 ..< 0.9 ■■■■■■■■■■■■■■
0.9 ..< 1.0 ■■■■■■■■
1.0 .. +∞ ------
Julia
Julia has the builtin package "Distributions" to generate random numbers from a standard distribution (Normal, Chisq etc.). <lang julia>using Printf, Distributions, Gadfly
data = rand(Normal(0, 1), 1000) @printf("N = %i\n", length(data)) @printf("μ = %2.2f\tσ = %2.2f\n", mean(data), std(data)) @printf("range = (%2.2f, %2.2f\n)", minimum(data), maximum(data)) h = plot(x=data, Geom.histogram) draw(PNG("norm_hist.png", 10cm, 10cm), h)</lang>
- Output:
N = 1000 μ = 0.02 σ = 0.97 range = (-2.76, 3.42)
Kotlin
<lang scala>// 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)
}</lang>
- 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
<lang 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'
^}</lang>
- 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 <lang lb>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</lang>
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. <lang Lua>function gaussian (mean, variance)
return math.sqrt(-2 * variance * math.log(math.random())) *
math.cos(2 * 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)</lang>
- 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: <lang maple>with(Statistics): n := 100000: X := Sample( Normal(0,1), n ); Mean( X ); StandardDeviation( X ); Histogram( X );</lang>
Mathematica/Wolfram Language
<lang 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
</lang>
MATLAB / Octave
<lang Matlab> N = 100000;
x = randn(N,1); mean(x) std(x) [nn,xx] = hist(x,100); bar(xx,nn);</lang>
Nim
In module “random”, Nim provides two procedures named gauss to generate random values following normal distribution and following Gauss distribution with given mean and standard deviation.
Here is a way to generate random values following normal distribution from random values following uniform distribution. It uses the Basic form of the Box-Muller transform.
<lang Nim>import math, random, sequtils, stats, strformat, strutils
proc drawHistogram(ns: seq[float]) =
# Distribute values in bins. const NBins = 50 var minval = min(ns) var maxval = max(ns) var h = newSeq[int](NBins + 1) for n in ns: let pos = ((n - minval) * NBins / (maxval - minval)).toInt inc h[pos]
# Eliminate extremes values. const MaxWidth = 50 let mx = max(h) var first = 0 while (h[first] / mx * MaxWidth).toInt == 0: inc first var last = h.high while (h[last] / mx * MaxWidth).toInt == 0: dec last
# Draw the histogram.
echo ""
for n in first..last:
echo repeat('+', (h[n] / mx * MaxWidth).toInt)
echo ""
const N = 100_000
randomize()
let u1, u2 = newSeqWith(N, rand(1.0))
var z = newSeq[float](N) for i in 0..<N:
z[i] = sqrt(-2 * ln(u1[i])) * cos(2 * PI * u2[i])
echo &"μ = {z.mean:.12f} σ = {z.standardDeviation:.12f}" z.drawHistogram()</lang>
- Output:
μ = -0.001105836229 σ = 0.999906544722 + + ++ +++ +++++ +++++++ +++++++++ ++++++++++++ ++++++++++++++++ +++++++++++++++++++++ ++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++ +++++++++++++++++++++++++ ++++++++++++++++++++ ++++++++++++++++ ++++++++++++ +++++++++ ++++++ +++++ +++ ++ + +
PARI/GP
<lang parigp>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)</lang>
For versions before 2.4.3, define <lang parigp>rreal()={
my(pr=32*ceil(default(realprecision)*log(10)/log(4294967296))); \\ Current precision random(2^pr)*1.>>pr
};</lang>
and use rreal() in place of random(1.).
A PARI implementation: <lang C>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;
}</lang>
Use mpsincos and caching to generate two values at nearly the same cost.
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 <lang pascal>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.</lang>
- 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
<lang perl>use constant pi => 3.14159265; use List::Util qw(sum reduce min max);
sub normdist {
my($m, $sigma) = @_; my $r = sqrt -2 * log rand; my $theta = 2 * pi * rand; $r * cos($theta) * $sigma + $m;
}
$size = 100000; $mean = 50; $stddev = 4;
push @dataset, normdist($mean,$stddev) for 1..$size;
my $m = sum(@dataset) / $size; print "m = $m\n";
my $sigma = sqrt( (reduce { $a + $b **2 } 0,@dataset) / $size - $m**2 ); print "sigma = $sigma\n";
$hash{int $_}++ for @dataset;
my $scale = 180 * $stddev / $size;
my @subbar = < ⎸ ▏ ▎ ▍ ▌ ▋ ▊ ▉ █ >;
for $i (min(@dataset)..max(@dataset)) {
my $x = ($hash{$i} // 0) * $scale;
my $full = int $x;
my $part = 8 * ($x - $full);
my $t1 = '█' x $full;
my $t2 = $subbar[$part];
print "$i\t$t1$t2\n";
}
</lang>
- Output:
32 ⎸ 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 ⎸
Phix
<lang Phix>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)</lang>
- Output:
100000 data terms used. Largest term was 4.30779 & smallest was -4.11902 Mean = -0.00252597 Stddev = 1.00067 # ## #### ###### ########## ############# ################## ######################## ################################# ######################################## #################################################### ############################################################# ###################################################################### ############################################################################### ####################################################################################### ############################################################################################### ################################################################################################# ##################################################################################################### ################################################################################################### ################################################################################################ ######################################################################################## ############################################################################### ####################################################################### ############################################################ ################################################# ####################################### ############################## ######################### ################ ############ ######### ###### #### ## #
<lang Phix>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)</lang>
- 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
<lang 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</lang> 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. <lang python>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)</lang>
- Output:
Sample mean = 49.9822; Stddev = 4.00938; max = 66.8091; min = 33.5283 for 100000 values
R
Generate normal random numbers from uniform random numbers, using the Box-Muller transform. Both x and y are normally distributed, and independent. <lang r>n <- 100000 u <- sqrt(-2*log(runif(n))) v <- 2*pi*runif(n) x <- u*cos(v) y <- v*sin(v) hist(x)</lang>
R can generate random normal distributed numbers using the rnorm function:
<lang r>n <- 100000
x <- rnorm(n, mean=0, sd=1)
mean(x)
sd(x)
hist(x)</lang>
Racket
This shows how one would generate samples from a normal distribution, compute statistics and plot a histogram.
<lang racket>
- 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") </lang>
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>
- 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))])))))))
</lang>
Raku
(formerly Perl 6)
<lang perl6>sub normdist ($m, $σ) {
my $r = sqrt -2 * log rand; my $Θ = τ * rand; $r * cos($Θ) * $σ + $m;
}
sub MAIN ($size = 100000, $mean = 50, $stddev = 4) {
my @dataset = normdist($mean,$stddev) xx $size;
my $m = [+](@dataset) / $size; say (:$m);
my $σ = sqrt [+](@dataset X** 2) / $size - $m**2; say (:$σ);
(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];
}
}</lang>
- 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 ⎸
REXX
The REXX language doesn't have any "higher math" BIF functions like SIN, COS, LN, LOG, SQRT, EXP, POW, etc,
so we hoi polloi programmers have to roll our own.
<lang rexx>/*REXX program generates 10,000 normally distributed numbers (Gaussian distribution).*/
numeric digits 20 /*use twenty decimal digs for accuracy.*/
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; #.g= sqrt( -2 * ln( rand() ) ) * cos( 2 * pi * rand() )
end /*g*/ /* [↑] uniform random number ───► #.g */
s= 0 mn= #.1; mx= mn; noise= n * .0005 /*calculate the noise: 1/20th % of N.*/ ss= 0
do j=1 for n; _= #.j /*the sum, and the sum of squares. */
s= s + _; ss= ss + _ * _ /*the sum, and the sum of squares. */
mn= min(mn, _); mx= max(mx, _) /*find the minimum and the maximum. */
end /*j*/
!.= 0 say 'number of data points = ' fmt(n ) say ' minimum = ' fmt(mn ) say ' maximum = ' fmt(mx ) say ' arithmetic mean = ' fmt(s/n) say ' standard deviation = ' fmt(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 (usable) 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, !.?) /*find the minimum. */
?mx= max(?mx, !.?) /* " " 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. */ /*───────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/ fmt: parse arg @; return left(, (@>=0) + 2 * datatype(@, 'W'))@ /*prepend a blank if #>=0, add 2 blanks if whole.*/ e: e = 2.7182818284590452353602874713526624977572470936999595749669676277240766303535; return e pi: pi= 3.1415926535897932384626433832795028841971693993751058209749445923078164062862; return pi r2r: return arg(1) // (pi() * 2) /*normalize the given angle (in radians) to ±2pi.*/ rand: return random(1, 1e5) / 1e5 /*REXX generates uniform random postive integers.*/ /*───────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/ ln: procedure; parse arg x,f; call e; ig= x>1.5; is= 1 -2*(ig\==1); ii= 0; xx= x; 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; z= 1; _= 1
x= x*x; p= z; do k=2 by 2; _= -_ * x / (k*(k-1)); z= z + _; if z=p then leave; p= z; end; return z
/*───────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/ 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</lang>
This REXX program makes use of scrsize REXX program (or BIF) which is used to determine the screen size of the terminal (console); this is to aid in maximizing the width of the horizontal histogram.
The SCRSIZE.REX REXX program is included here ──► SCRSIZE.REX.
- output when using the default input:
(The output shown when the screen size is 62x140.)
The graph is shown at 3/4 size.
number of data points = 10000
minimum = -3.8181072371544448250
maximum = 3.5445917138265268562
arithmetic mean = -0.01406470979976873427
standard deviation = 0.99486092191249231518
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Run BASIC
<lang runbasic> 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</lang>
- Output:
Samples :100000 Largest :4.61187177 Smallest :-4.21695424 Range :8.82882601 Mean :-9.25042513e-4 Stand Dev :1.00680067 = == === ===== ======== ============= ================= ======================= ============================== ======================================= =============================================== ========================================================= =================================================================== =========================================================================== =================================================================================== ======================================================================================= ========================================================================================== ======================================================================================== ====================================================================================== ================================================================================= ============================================================================ ================================================================== ======================================================== ============================================== ===================================== ============================ ===================== =============== ========== ======= ===== === = =
Rust
<lang rust>//! Rust rosetta example for normal distribution use math::{histogram::Histogram, traits::ToIterator}; use rand; use rand_distr::{Distribution, Normal};
/// Returns the mean of the provided samples /// /// ## Arguments /// * data -- reference to float32 array fn mean(data: &[f32]) -> Option<f32> {
let sum: f32 = data.iter().sum(); Some(sum / data.len() as f32)
}
/// Returns standard deviation of the provided samples /// /// ## Arguments /// * data -- reference to float32 array fn standard_deviation(data: &[f32]) -> Option<f32> {
let mean = mean(data).expect("invalid mean");
let sum = data.iter().fold(0.0, |acc, &x| acc + (x - mean).powi(2));
Some((sum / data.len() as f32).sqrt())
}
/// Prints a histogram in the shell /// /// ## Arguments /// * data -- reference to float32 array /// * maxwidth -- the maxwidth of the histogram in # of characters /// * bincount -- number of bins in the histogram /// * ch -- character used to plot the graph fn print_histogram(data: &[f32], maxwidth: usize, bincount: usize, ch: char) {
let min_val = data.iter().cloned().fold(f32::NAN, f32::min);
let max_val = data.iter().cloned().fold(f32::NAN, f32::max);
let histogram = Histogram::new(Some(&data.to_vec()), bincount, min_val, max_val).unwrap();
let max_bin_value = histogram.get_counters().iter().max().unwrap();
println!();
for x in histogram.to_iter() {
let (bin_min, bin_max, freq) = x;
let bar_width = (((freq as f64) / (*max_bin_value as f64)) * (maxwidth as f64)) as u32;
let bar_as_string = (1..bar_width).fold(String::new(), |b, _| b + &ch.to_string());
println!(
"({:>6},{:>6}) |{} {:.2}%",
format!("{:.2}", bin_min),
format!("{:.2}", bin_max),
bar_as_string,
(freq as f64) * 100.0 / (data.len() as f64)
);
}
println!();
}
/// Runs the demo to generate normal distribution of three different sample sizes fn main() {
let expected_mean: f32 = 0.0; let expected_std_deviation: f32 = 4.0; let normal = Normal::new(expected_mean, expected_std_deviation).unwrap();
let mut rng = rand::thread_rng();
for &number_of_samples in &[1000, 10_000, 1_000_000] {
let data: Vec<f32> = normal
.sample_iter(&mut rng)
.take(number_of_samples)
.collect();
println!("Statistics for sample size {}:", number_of_samples);
println!("\tMean: {:?}", mean(&data).expect("invalid mean"));
println!(
"\tStandard deviation: {:?}",
standard_deviation(&data).expect("invalid standard deviation")
);
print_histogram(&data, 80, 40, '-');
}
}</lang>
- Output:
Statistics for sample size 1000: Mean: -0.11356559 Standard deviation: 4.0244555 (-13.81,-13.12) | 0.10% (-13.12,-12.44) | 0.00% (-12.44,-11.75) | 0.10% (-11.75,-11.06) | 0.20% (-11.06,-10.38) |- 0.30% (-10.38, -9.69) | 0.10% ( -9.69, -9.01) |--- 0.50% ( -9.01, -8.32) |---- 0.60% ( -8.32, -7.64) |------ 0.80% ( -7.64, -6.95) |-------------- 1.60% ( -6.95, -6.27) |----------------- 1.90% ( -6.27, -5.58) |------------------------ 2.60% ( -5.58, -4.90) |----------------------- 2.50% ( -4.90, -4.21) |---------------------------------------- 4.20% ( -4.21, -3.53) |------------------------------------- 3.90% ( -3.53, -2.84) |------------------------------------------------- 5.10% ( -2.84, -2.15) |---------------------------------------------- 4.80% ( -2.15, -1.47) |------------------------------------------------------------------------ 7.40% ( -1.47, -0.78) |---------------------------------------------------------- 6.00% ( -0.78, -0.10) |----------------------------------------------------------------------- 7.30% ( -0.10, 0.59) |------------------------------------------------------------------------------- 8.10% ( 0.59, 1.27) |----------------------------------------------------------------------- 7.30% ( 1.27, 1.96) |------------------------------------------------- 5.10% ( 1.96, 2.64) |------------------------------------------------------------ 6.20% ( 2.64, 3.33) |----------------------------------------- 4.30% ( 3.33, 4.01) |----------------------------- 3.10% ( 4.01, 4.70) |------------------------------------- 3.90% ( 4.70, 5.39) |-------------------------- 2.80% ( 5.39, 6.07) |---------------------- 2.40% ( 6.07, 6.76) |---------------- 1.80% ( 6.76, 7.44) |---------------- 1.80% ( 7.44, 8.13) |--------- 1.10% ( 8.13, 8.81) |---------- 1.20% ( 8.81, 9.50) | 0.20% ( 9.50, 10.18) | 0.00% ( 10.18, 10.87) | 0.10% ( 10.87, 11.55) |- 0.30% ( 11.55, 12.24) | 0.10% ( 12.24, 12.92) | 0.10% ( 12.92, 13.61) | 0.10% Statistics for sample size 10000: Mean: 0.02012564 Standard deviation: 3.9697735 (-15.80,-14.99) | 0.02% (-14.99,-14.18) | 0.04% (-14.18,-13.37) | 0.04% (-13.37,-12.56) | 0.04% (-12.56,-11.75) | 0.09% (-11.75,-10.94) | 0.08% (-10.94,-10.13) |- 0.25% (-10.13, -9.32) |--- 0.42% ( -9.32, -8.51) |----- 0.67% ( -8.51, -7.70) |--------- 1.10% ( -7.70, -6.89) |------------- 1.48% ( -6.89, -6.08) |------------------ 1.98% ( -6.08, -5.27) |-------------------------- 2.82% ( -5.27, -4.45) |------------------------------------ 3.80% ( -4.45, -3.64) |--------------------------------------------- 4.66% ( -3.64, -2.83) |------------------------------------------------------- 5.72% ( -2.83, -2.02) |------------------------------------------------------------------ 6.85% ( -2.02, -1.21) |---------------------------------------------------------------------------- 7.80% ( -1.21, -0.40) |---------------------------------------------------------------------------- 7.79% ( -0.40, 0.41) |------------------------------------------------------------------------------- 8.09% ( 0.41, 1.22) |----------------------------------------------------------------------------- 7.89% ( 1.22, 2.03) |--------------------------------------------------------------------------- 7.76% ( 2.03, 2.84) |-------------------------------------------------------------------- 7.06% ( 2.84, 3.65) |------------------------------------------------------- 5.74% ( 3.65, 4.46) |-------------------------------------------- 4.64% ( 4.46, 5.27) |-------------------------------------- 4.00% ( 5.27, 6.08) |---------------------------- 3.03% ( 6.08, 6.89) |------------------- 2.07% ( 6.89, 7.71) |-------------- 1.54% ( 7.71, 8.52) |-------- 0.97% ( 8.52, 9.33) |----- 0.61% ( 9.33, 10.14) |-- 0.36% ( 10.14, 10.95) |- 0.25% ( 10.95, 11.76) | 0.19% ( 11.76, 12.57) | 0.08% ( 12.57, 13.38) | 0.02% ( 13.38, 14.19) | 0.01% ( 14.19, 15.00) | 0.03% ( 15.00, 15.81) | 0.00% ( 15.81, 16.62) | 0.01% Statistics for sample size 1000000: Mean: -0.004743685 Standard deviation: 4.0006065 (-20.00,-19.02) | 0.00% (-19.02,-18.04) | 0.00% (-18.04,-17.06) | 0.00% (-17.06,-16.07) | 0.00% (-16.07,-15.09) | 0.00% (-15.09,-14.11) | 0.01% (-14.11,-13.13) | 0.03% (-13.13,-12.15) | 0.06% (-12.15,-11.16) | 0.14% (-11.16,-10.18) |- 0.28% (-10.18, -9.20) |--- 0.53% ( -9.20, -8.22) |------ 0.95% ( -8.22, -7.24) |----------- 1.51% ( -7.24, -6.25) |------------------ 2.40% ( -6.25, -5.27) |--------------------------- 3.48% ( -5.27, -4.29) |-------------------------------------- 4.82% ( -4.29, -3.31) |-------------------------------------------------- 6.27% ( -3.31, -2.32) |------------------------------------------------------------- 7.62% ( -2.32, -1.34) |----------------------------------------------------------------------- 8.77% ( -1.34, -0.36) |----------------------------------------------------------------------------- 9.58% ( -0.36, 0.62) |------------------------------------------------------------------------------- 9.74% ( 0.62, 1.60) |---------------------------------------------------------------------------- 9.39% ( 1.60, 2.59) |-------------------------------------------------------------------- 8.49% ( 2.59, 3.57) |---------------------------------------------------------- 7.30% ( 3.57, 4.55) |----------------------------------------------- 5.86% ( 4.55, 5.53) |----------------------------------- 4.45% ( 5.53, 6.51) |------------------------ 3.16% ( 6.51, 7.50) |---------------- 2.09% ( 7.50, 8.48) |--------- 1.34% ( 8.48, 9.46) |----- 0.81% ( 9.46, 10.44) |-- 0.46% ( 10.44, 11.42) | 0.23% ( 11.42, 12.41) | 0.11% ( 12.41, 13.39) | 0.06% ( 13.39, 14.37) | 0.02% ( 14.37, 15.35) | 0.01% ( 15.35, 16.34) | 0.00% ( 16.34, 17.32) | 0.00% ( 17.32, 18.30) | 0.00% ( 18.30, 19.28) | 0.00%
SAS
<lang 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
- /</lang>
Sidef
<lang ruby>define τ = Num.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 / size) say ("m: #{m}")
var σ = sqrt(dataset »**» 2 -> sum / size - m**2) say ("s: #{σ}")
var hash = Hash() dataset.each { |n| hash{ n.round } := 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])
}</lang>
- 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 █████████████████████████████████████████████████████████████████████▊ 52 ██████████████████████████████████████████████████████████████▏ 53 ████████████████████████████████████████████████████▉ 54 ███████████████████████████████████████████▉ 55 █████████████████████████████████▎ 56 ███████████████████████⎸ 57 ███████████████▋ 58 █████████▋ 59 █████▍ 60 ███▍ 61 █▊ 62 ▋ 63 ▍ 64 ▏ 65 ⎸ 66 ⎸
Stata
Pairs of normal numbers are generated from pairs of uniform numbers using the Box-Muller method. A normal density is added to the histogram for comparison. See histogram in Stata help. A Q-Q plot is also drawn.
<lang stata>clear all set obs 100000 gen u=runiform() gen v=runiform() gen r=sqrt(-2*log(u)) gen x=r*cos(2*_pi*v) gen y=r*sin(2*_pi*v) gen z=rnormal() sum x y z
Variable | Obs Mean Std. Dev. Min Max
+---------------------------------------------------------
x | 100,000 .0025861 1.002346 -4.508192 4.164336
y | 100,000 .0017389 1.001586 -4.631144 4.460274
z | 100,000 .005054 .9998861 -5.134265 4.449522
hist x, normal hist y, normal hist z, normal qqplot x z, msize(tiny)</lang>
Tcl
<lang 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</lang> Sample output:
100 numbers Mean: 0.49355955990390254 StdDev: 0.19651396178121985 *** ******* ************** *********************************** ******************************************************** ****************************************************************** ************************************************************************* ****************************************** ************************************** ************** 1000 numbers Mean: 0.5066940614105869 StdDev: 0.2016794788065389 * ***** ************** **************************** ********************************************************** **************************************************************** ************************************************************* ****************************************************** *********************************** ************ ********* * 10000 numbers Mean: 0.49980964730768285 StdDev: 0.1968441612522318 * ***** *************** ******************************* ***************************************************** ****************************************************************** ******************************************************************* **************************************************** ********************************* *************** ***** * 100000 numbers Mean: 0.49960438950922254 StdDev: 0.20060211160998606 * ** *** ****** ********* ************** ****************** *********************** ***************************** ******************************** ********************************** ********************************** ******************************** **************************** *********************** ****************** ************* ********* ****** *** ** *
The blank lines in the output are where the number of samples is too small to even merit a single unit on the histogram.
VBA
<lang vb>Public Sub standard_normal()
Dim s() As Variant, bins(71) As Single
ReDim s(20000)
For i = 1 To 20000
s(i) = WorksheetFunction.Norm_S_Inv(Rnd())
Next i
For i = -35 To 35
bins(i + 36) = i / 10
Next i
Debug.Print "sample size"; UBound(s), "mean"; mean(s), "standard deviation"; standard_deviation(s)
t = WorksheetFunction.Frequency(s, bins)
For i = -35 To 35
Debug.Print Format((i - 1) / 10, "0.00");
Debug.Print "-"; Format(i / 10, "0.00"),
Debug.Print String$(t(i + 36, 1) / 10, "X");
Debug.Print
Next i
End Sub</lang>
- Output:
sample size 20000 mean-5,26306310478751E-03 standard deviation 1,00355037427319 -3,60--3,50 -3,50--3,40 -3,40--3,30 -3,30--3,20 -3,20--3,10 -3,10--3,00 -3,00--2,90 XX -2,90--2,80 X -2,80--2,70 XX -2,70--2,60 XX -2,60--2,50 XXX -2,50--2,40 XXXX -2,40--2,30 XXXXX -2,30--2,20 XXXXXXXX -2,20--2,10 XXXXXXXX -2,10--2,00 XXXXXXXXXXX -2,00--1,90 XXXXXXXXXXXXX -1,90--1,80 XXXXXXXXXXXXXXX -1,80--1,70 XXXXXXXXXXXXXXXX -1,70--1,60 XXXXXXXXXXXXXXXXXXXX -1,60--1,50 XXXXXXXXXXXXXXXXXXXXXXXX -1,50--1,40 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -1,40--1,30 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -1,30--1,20 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -1,20--1,10 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -1,10--1,00 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -1,00--0,90 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -0,90--0,80 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -0,80--0,70 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -0,70--0,60 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -0,60--0,50 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -0,50--0,40 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -0,40--0,30 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -0,30--0,20 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -0,20--0,10 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX -0,10-0,00 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0,00-0,10 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0,10-0,20 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0,20-0,30 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0,30-0,40 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0,40-0,50 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0,50-0,60 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0,60-0,70 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0,70-0,80 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0,80-0,90 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0,90-1,00 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 1,00-1,10 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 1,10-1,20 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 1,20-1,30 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 1,30-1,40 XXXXXXXXXXXXXXXXXXXXXXXXXXXXX 1,40-1,50 XXXXXXXXXXXXXXXXXXXXXXXXXX 1,50-1,60 XXXXXXXXXXXXXXXXXXXXXXXXX 1,60-1,70 XXXXXXXXXXXXXXXXXXXXXX 1,70-1,80 XXXXXXXXXXXXXXXXXX 1,80-1,90 XXXXXXXXXXXXXXX 1,90-2,00 XXXXXXXXXXX 2,00-2,10 XXXXXXXXXXXX 2,10-2,20 XXXXXXX 2,20-2,30 XXXXXX 2,30-2,40 XXXXX 2,40-2,50 XXX 2,50-2,60 XXXX 2,60-2,70 XX 2,70-2,80 XX 2,80-2,90 X 2,90-3,00 X 3,00-3,10 X 3,10-3,20 X 3,20-3,30 3,30-3,40 3,40-3,50
Wren
<lang ecmascript>import "random" for Random import "/fmt" for Fmt import "/math" for Nums
var rgen = Random.new()
// Box-Muller method from Wikipedia var normal = Fn.new { |mu, sigma|
var u1 = rgen.float() var u2 = rgen.float() var mag = sigma * (-2 * u1.log).sqrt var z0 = mag * (2 * Num.pi * u2).cos + mu var z1 = mag * (2 * Num.pi * u2).sin + mu return [z0, z1]
}
var N = 100000 var NUM_BINS = 12 var HIST_CHAR = "■" var HIST_CHAR_SIZE = 250 var bins = List.filled(NUM_BINS, 0) var binSize = 0.1 var samples = List.filled(N, 0) var mu = 0.5 var sigma = 0.25 for (i in 0...N/2) {
var rns = normal.call(mu, sigma)
for (j in 0..1) {
var rn = rns[j]
var bn
if (rn < 0) {
bn = 0
} else if (rn >= 1) {
bn = 11
} else {
bn = (rn/binSize).floor + 1
}
bins[bn] = bins[bn] + 1
samples[i*2 + j] = rn
}
}
Fmt.print("Normal distribution with mean $0.2f and S/D $0.2f for $,d samples:\n", mu, sigma, N) System.print(" Range Number of samples within that range") for (i in 0...NUM_BINS) {
var hist = HIST_CHAR * (bins[i] / HIST_CHAR_SIZE).round
if (i == 0) {
Fmt.print(" -∞ ..< 0.00 $s $,d", hist, bins[0])
} else if (i < NUM_BINS - 1) {
Fmt.print("$4.2f ..< $4.2f $s $,d", binSize * (i-1), binSize * i, hist, bins[i])
} else {
Fmt.print("1.00 ... +∞ $s $,d", hist, bins[NUM_BINS - 1])
}
} Fmt.print("\nActual mean for these samples : $0.5f", Nums.mean(samples)) Fmt.print("Actual S/D for these samples : $0.5f", Nums.stdDev(samples))</lang>
- Output:
Specimen run:
Normal distribution with mean 0.50 and S/D 0.25 for 100,000 samples:
Range Number of samples within that range
-∞ ..< 0.00 ■■■■■■■■■ 2,243
0.00 ..< 0.10 ■■■■■■■■■■■■■ 3,250
0.10 ..< 0.20 ■■■■■■■■■■■■■■■■■■■■■■■■ 5,977
0.20 ..< 0.30 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 9,723
0.30 ..< 0.40 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 13,104
0.40 ..< 0.50 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 15,601
0.50 ..< 0.60 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 15,469
0.60 ..< 0.70 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 13,334
0.70 ..< 0.80 ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ 9,659
0.80 ..< 0.90 ■■■■■■■■■■■■■■■■■■■■■■■■ 6,119
0.90 ..< 1.00 ■■■■■■■■■■■■■ 3,173
1.00 ... +∞ ■■■■■■■■■ 2,348
Actual mean for these samples : 0.50099
Actual S/D for these samples : 0.25051
zkl
<lang zkl>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;
};</lang> 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. <lang zkl>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)) }</lang>
- Output:
Samples: 100,000 Mean: 0.0005999 Stddev: 1.003 * *** ******** ****************** ***************************** ************************************** ************************************** ***************************** ****************** ******** *** *
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