Statistics/Basic
You are encouraged to solve this task according to the task description, using any language you may know.
Statistics is all about large groups of numbers. When talking about a set of sampled data, most frequently used is their mean value and standard deviation (stddev). If you have set of data where , the mean is , while the stddev is .
When examining a large quantity of data, one often uses a histogram, which shows the counts of data samples falling into a prechosen set of intervals (or bins). When plotted, often as bar graphs, it visually indicates how often each data value occurs.
Task Using your language's random number routine, generate real numbers in the range of [0, 1]. It doesn't matter if you chose to use open or closed range. Create 100 of such numbers (i.e. sample size 100) and calculate their mean and stddev. Do so for sample size of 1,000 and 10,000, maybe even higher if you feel like. Show a histogram of any of these sets. Do you notice some patterns about the standard deviation?
Extra Sometimes so much data need to be processed that it's impossible to keep all of them at once. Can you calculate the mean, stddev and histogram of a trillion numbers? (You don't really need to do a trillion numbers, just show how it can be done.)
- Hint
For a finite population with equal probabilities at all points, one can derive:
Or, more verbosely:
See also
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Ada
A plain solution for moderate sample sizes
<lang Ada>with Ada.Text_IO, Ada.Command_Line, Ada.Numerics.Float_Random,
Ada.Numerics.Generic_Elementary_Functions;
procedure Basic_Stat is
package FRG renames Ada.Numerics.Float_Random; package TIO renames Ada.Text_IO;
type Counter is range 0 .. 2**31-1; type Result_Array is array(Natural range <>) of Counter;
package FIO is new TIO.Float_IO(Float);
procedure Put_Histogram(R: Result_Array; Scale, Full: Counter) is
begin
for I in R'Range loop
FIO.Put(Float'Max(0.0, Float(I)/10.0 - 0.05),
Fore => 1, Aft => 2, Exp => 0); TIO.Put("..");
FIO.Put(Float'Min(1.0, Float(I)/10.0 + 0.05),
Fore => 1, Aft => 2, Exp => 0); TIO.Put(": ");
for J in 1 .. (R(I)* Scale)/Full loop
Ada.Text_IO.Put("X");
end loop;
Ada.Text_IO.New_Line;
end loop;
end Put_Histogram;
procedure Put_Mean_Et_Al(Sample_Size: Counter;
Val_Sum, Square_Sum: Float) is
Mean: constant Float := Val_Sum / Float(Sample_Size);
package Math is new Ada.Numerics.Generic_Elementary_Functions(Float);
begin
TIO.Put("Mean: ");
FIO.Put(Mean, Fore => 1, Aft => 5, Exp => 0);
TIO.Put(", Standard Deviation: ");
FIO.Put(Math.Sqrt(abs(Square_Sum / Float(Sample_Size)
- (Mean * Mean))), Fore => 1, Aft => 5, Exp => 0);
TIO.New_Line;
end Put_Mean_Et_Al;
N: Counter := Counter'Value(Ada.Command_Line.Argument(1)); Gen: FRG.Generator; Results: Result_Array(0 .. 10) := (others => 0); X: Float; Val_Sum, Squ_Sum: Float := 0.0;
begin
FRG.Reset(Gen);
for I in 1 .. N loop
X := FRG.Random(Gen);
Val_Sum := Val_Sum + X;
Squ_Sum := Squ_Sum + X*X;
declare
Index: Integer := Integer(X*10.0);
begin
Results(Index) := Results(Index) + 1;
end;
end loop;
TIO.Put_Line("After sampling" & Counter'Image(N) & " random numnbers: ");
Put_Histogram(Results, Scale => 600, Full => N);
TIO.New_Line;
Put_Mean_Et_Al(Sample_Size => N, Val_Sum => Val_Sum, Square_Sum => Squ_Sum);
end Basic_Stat;</lang>
- Output:
from a few sample runs
> ./basic_stat 1000 After sampling 1000 random numnbers: 0.00..0.05: XXXXXXXXXXXXXXXXXXXXXXX 0.05..0.15: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.15..0.25: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.25..0.35: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.35..0.45: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.45..0.55: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.55..0.65: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.65..0.75: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.75..0.85: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.85..0.95: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.95..1.00: XXXXXXXXXXXXXXXXXXXXXXXXXXXX Mean: 0.48727, Standard Deviation: 0.28502 > ./basic_stat 10_000 After sampling 10000 random numnbers: 0.00..0.05: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.05..0.15: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.15..0.25: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.25..0.35: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.35..0.45: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.45..0.55: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.55..0.65: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.65..0.75: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.75..0.85: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.85..0.95: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.95..1.00: XXXXXXXXXXXXXXXXXXXXXXXXXXXXX Mean: 0.50096, Standard Deviation: 0.28869 > ./basic_stat 100_000 After sampling 100000 random numnbers: 0.00..0.05: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.05..0.15: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.15..0.25: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.25..0.35: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.35..0.45: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.45..0.55: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.55..0.65: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.65..0.75: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.75..0.85: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.85..0.95: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.95..1.00: XXXXXXXXXXXXXXXXXXXXXXXXXXXXX Mean: 0.50178, Standard Deviation: 0.28805
Making the solution ready for one trillion samples
Depending on where you live, one trillion is either 10^12 or 10^18 [1]. Below, I'll assume 10^12, which implies a number of operations I can still perform on my PC.
The above program will fail with such large inputs for two reasons:
1. The type Counter cannot hold such large numbers.
2. The variables Val_Sum and Squ_Sum will numerically fail, because the type Float only provides about six decimal digits of accuracy. I.e., at some point, Val_Sum and (a little bit later) Squ_Sum are so large that adding a value below 1 has no effect, any more.
To make the program ready for sample size 10^12, we modify it as follows.
1. Change the type Counter to hold such large numbers.
2. Define a type High_Precision, that will hold (at least) 15 decimal digits. Define Val_Sum and Squ_Sum as being from that type. Include the neccessary type conversions.
3. Provide some progress report, during the running time.
This is the modified program
<lang Ada>with Ada.Text_IO, Ada.Command_Line, Ada.Numerics.Float_Random,
Ada.Numerics.Generic_Elementary_Functions;
procedure Long_Basic_Stat is
package FRG renames Ada.Numerics.Float_Random; package TIO renames Ada.Text_IO;
type Counter is range 0 .. 2**63-1; type Result_Array is array(Natural range <>) of Counter; type High_Precision is digits 15;
package FIO is new TIO.Float_IO(Float);
procedure Put_Histogram(R: Result_Array; Scale, Full: Counter) is
begin
for I in R'Range loop
FIO.Put(Float'Max(0.0, Float(I)/10.0 - 0.05),
Fore => 1, Aft => 2, Exp => 0); TIO.Put("..");
FIO.Put(Float'Min(1.0, Float(I)/10.0 + 0.05),
Fore => 1, Aft => 2, Exp => 0); TIO.Put(": ");
for J in 1 .. (R(I)* Scale)/Full loop
Ada.Text_IO.Put("X");
end loop;
Ada.Text_IO.New_Line;
end loop;
end Put_Histogram;
procedure Put_Mean_Et_Al(Sample_Size: Counter;
Val_Sum, Square_Sum: Float) is
Mean: constant Float := Val_Sum / Float(Sample_Size);
package Math is new Ada.Numerics.Generic_Elementary_Functions(Float);
begin
TIO.Put("Mean: ");
FIO.Put(Mean, Fore => 1, Aft => 5, Exp => 0);
TIO.Put(", Standard Deviation: ");
FIO.Put(Math.Sqrt(abs(Square_Sum / Float(Sample_Size)
- (Mean * Mean))), Fore => 1, Aft => 5, Exp => 0);
TIO.New_Line;
end Put_Mean_Et_Al;
N: Counter := Counter'Value(Ada.Command_Line.Argument(1)); Gen: FRG.Generator; Results: Result_Array(0 .. 10) := (others => 0); X: Float; Val_Sum, Squ_Sum: High_Precision := 0.0;
begin
FRG.Reset(Gen);
for Outer in 1 .. 1000 loop
for I in 1 .. N/1000 loop
X := FRG.Random(Gen);
Val_Sum := Val_Sum + High_Precision(X);
Squ_Sum := Squ_Sum + High_Precision(X)*High_Precision(X);
declare
Index: Integer := Integer(X*10.0);
begin
Results(Index) := Results(Index) + 1;
end;
end loop;
if Outer mod 50 = 0 then
TIO.New_Line(1);
TIO.Put_Line(Integer'Image(Outer/10) &"% done; current results:");
Put_Mean_Et_Al(Sample_Size => (Counter(Outer)*N)/1000,
Val_Sum => Float(Val_Sum),
Square_Sum => Float(Squ_Sum));
else
Ada.Text_IO.Put(".");
end if;
end loop;
TIO.New_Line(4);
TIO.Put_Line("After sampling" & Counter'Image(N) & " random numnbers: ");
Put_Histogram(Results, Scale => 600, Full => N);
TIO.New_Line;
Put_Mean_Et_Al(Sample_Size => N,
Val_Sum => Float(Val_Sum), Square_Sum => Float(Squ_Sum));
end Long_Basic_Stat;</lang>
- Output:
for sample size 10^12 took one night on my PC
................................................. 5% done; current results: Mean: 0.50000, Standard Deviation: 0.28867 ................................................. 10% done; current results: Mean: 0.50000, Standard Deviation: 0.28867 ................................................. 15% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 20% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 25% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 30% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 35% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 40% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 45% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 50% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 55% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 60% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 65% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 70% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 75% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 80% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 85% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 90% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 95% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 ................................................. 100% done; current results: Mean: 0.50000, Standard Deviation: 0.28868 After sampling 1000000000000 random numnbers: 0.00..0.05: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.05..0.15: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.15..0.25: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.25..0.35: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.35..0.45: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.45..0.55: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.55..0.65: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.65..0.75: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.75..0.85: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.85..0.95: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 0.95..1.00: XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX Mean: 0.50000, Standard Deviation: 0.28868
The same program should still work fine for sample size 10^18, but I'll need my PC in the meantime. ;-)
C
Sample code. <lang C>#include <stdio.h>
- include <stdlib.h>
- include <math.h>
- include <stdint.h>
- define n_bins 10
double rand01() { return rand() / (RAND_MAX + 1.0); }
double avg(int count, double *stddev, int *hist) { double x[count]; double m = 0, s = 0;
for (int i = 0; i < n_bins; i++) hist[i] = 0; for (int i = 0; i < count; i++) { m += (x[i] = rand01()); hist[(int)(x[i] * n_bins)] ++; }
m /= count; for (int i = 0; i < count; i++) s += x[i] * x[i]; *stddev = sqrt(s / count - m * m); return m; }
void hist_plot(int *hist) { int max = 0, step = 1; double inc = 1.0 / n_bins;
for (int i = 0; i < n_bins; i++) if (hist[i] > max) max = hist[i];
/* scale if numbers are too big */ if (max >= 60) step = (max + 59) / 60;
for (int i = 0; i < n_bins; i++) { printf("[%5.2g,%5.2g]%5d ", i * inc, (i + 1) * inc, hist[i]); for (int j = 0; j < hist[i]; j += step) printf("#"); printf("\n"); } }
/* record for moving average and stddev. Values kept are sums and sum data^2
* to avoid excessive precision loss due to divisions, but some loss is inevitable */
typedef struct { uint64_t size; double sum, x2; uint64_t hist[n_bins]; } moving_rec;
void moving_avg(moving_rec *rec, double *data, int count) { double sum = 0, x2 = 0; /* not adding data directly to the sum in case both recorded sum and * count of this batch are large; slightly less likely to lose precision*/ for (int i = 0; i < count; i++) { sum += data[i]; x2 += data[i] * data[i]; rec->hist[(int)(data[i] * n_bins)]++; }
rec->sum += sum; rec->x2 += x2; rec->size += count; }
int main() { double m, stddev; int hist[n_bins], samples = 10;
while (samples <= 10000) { m = avg(samples, &stddev, hist); printf("size %5d: %g %g\n", samples, m, stddev); samples *= 10; }
printf("\nHistograph:\n"); hist_plot(hist);
printf("\nMoving average:\n N Mean Sigma\n"); moving_rec rec = { 0, 0, 0, {0} }; double data[100]; for (int i = 0; i < 10000; i++) { for (int j = 0; j < 100; j++) data[j] = rand01();
moving_avg(&rec, data, 100);
if ((i % 1000) == 999) { printf("%4lluk %f %f\n", rec.size/1000, rec.sum / rec.size, sqrt(rec.x2 * rec.size - rec.sum * rec.sum)/rec.size ); } } }</lang>
C#
<lang csharp>using System; using MathNet.Numerics.Statistics;
class Program {
static void Run(int sampleSize)
{
double[] X = new double[sampleSize];
var r = new Random();
for (int i = 0; i < sampleSize; i++)
X[i] = r.NextDouble();
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)
{
Run(100);
Run(1000);
Run(10000);
}
}</lang>
- Output:
Sample size: 100 0.00 : ################################################## 0.10 : ############################ 0.20 : ########################################### 0.30 : ############################ 0.40 : ########################################### 0.50 : ######################### 0.60 : ############################################## 0.70 : ######################### 0.80 : ######################### 0.90 : ########################################### Mean: 0.481181871658741 StdDev: 0.301957945953801 Sample size: 1,000 0.00 : ################################### 0.10 : ################################### 0.20 : ############################ 0.30 : ################################# 0.40 : ####################################### 0.50 : ######################################### 0.60 : ###################################### 0.70 : ################################# 0.80 : ################################## 0.90 : ###################################### Mean: 0.508802390412802 StdDev: 0.28593657047378 Sample size: 10,000 0.00 : ################################## 0.10 : ####################################### 0.20 : ################################# 0.30 : #################################### 0.40 : ################################### 0.50 : ##################################### 0.60 : #################################### 0.70 : ################################### 0.80 : ################################## 0.90 : ################################### Mean: 0.499069400830039 StdDev: 0.287103198996064
C++
<lang Cpp>#include <iostream>
- include <random>
- include <vector>
- include <cstdlib>
- include <algorithm>
- include <cmath>
void printStars ( int number ) {
if ( number > 0 ) {
for ( int i = 0 ; i < number + 1 ; i++ )
std::cout << '*' ;
} std::cout << '\n' ;
}
int main( int argc , char *argv[] ) {
const int numberOfRandoms = std::atoi( argv[1] ) ;
std::random_device rd ;
std::mt19937 gen( rd( ) ) ;
std::uniform_real_distribution<> distri( 0.0 , 1.0 ) ;
std::vector<double> randoms ;
for ( int i = 0 ; i < numberOfRandoms + 1 ; i++ )
randoms.push_back ( distri( gen ) ) ;
std::sort ( randoms.begin( ) , randoms.end( ) ) ;
double start = 0.0 ;
for ( int i = 0 ; i < 9 ; i++ ) {
double to = start + 0.1 ;
int howmany = std::count_if ( randoms.begin( ) , randoms.end( ),
[&start , &to] ( double c ) { return c >= start && c < to ; } ) ;
if ( start == 0.0 ) //double 0.0 output as 0
std::cout << "0.0" << " - " << to << ": " ;
else
std::cout << start << " - " << to << ": " ;
if ( howmany > 50 ) //scales big interval numbers to printable length
howmany = howmany / ( howmany / 50 ) ;
printStars ( howmany ) ;
start += 0.1 ;
}
double mean = std::accumulate( randoms.begin( ) , randoms.end( ) , 0.0 ) / randoms.size( ) ;
double sum = 0.0 ;
for ( double num : randoms )
sum += std::pow( num - mean , 2 ) ;
double stddev = std::pow( sum / randoms.size( ) , 0.5 ) ;
std::cout << "The mean is " << mean << " !" << std::endl ;
std::cout << "Standard deviation is " << stddev << " !" << std::endl ;
return 0 ;
}</lang>
- Output:
./statistics 100 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: ********* The mean is 0.493563 ! Standard deviation is 0.297152 !
CoffeeScript
<lang coffeescript> generate_statistics = (n) ->
hist = {}
update_hist = (r) -> hist[Math.floor 10*r] ||= 0 hist[Math.floor 10*r] += 1
sum = 0 sum_squares = 0.0
for i in [1..n] r = Math.random() sum += r sum_squares += r*r update_hist r mean = sum / n stddev = Math.sqrt((sum_squares / n) - mean*mean)
[n, mean, stddev, hist]
display_statistics = (n, mean, stddev, hist) ->
console.log "-- Stats for sample size #{n}"
console.log "mean: #{mean}"
console.log "sdev: #{stddev}"
for x, cnt of hist
bars = repeat "=", Math.floor(cnt*300/n)
console.log "#{x/10}: #{bars} #{cnt}"
repeat = (c, n) ->
s = s += c for i in [1..n] s
for n in [100, 1000, 10000, 1000000]
[n, mean, stddev, hist] = generate_statistics n display_statistics n, mean, stddev, hist
</lang>
- Output:
> coffee stats.coffee -- Stats for sample size 100 mean: 0.5058459933893755 sdev: 0.2752669422150894 0: ================== 6 0.1: ============================================= 15 0.2: =========================== 9 0.3: ===================== 7 0.4: ============================================= 15 0.5: ======================== 8 0.6: ================================= 11 0.7: ========================================== 14 0.8: ===================== 7 0.9: ======================== 8 -- Stats for sample size 1000 mean: 0.49664502244861797 sdev: 0.2942483939245344 0: ========================== 89 0.1: ===================================== 126 0.2: =========================== 93 0.3: ==================================== 121 0.4: =========================== 93 0.5: ====================== 75 0.6: ================================ 108 0.7: ======================== 82 0.8: ============================== 101 0.9: ================================= 112 -- Stats for sample size 10000 mean: 0.4985696110446239 sdev: 0.29007446138438986 0: ============================== 1005 0.1: ============================== 1016 0.2: ============================== 1022 0.3: ============================== 1012 0.4: ============================ 958 0.5: =============================== 1035 0.6: ============================= 974 0.7: ============================= 968 0.8: ============================= 973 0.9: =============================== 1037 -- Stats for sample size 1000000 mean: 0.5001718024678293 sdev: 0.2887130780006248 0: ============================== 100113 0.1: ============================= 99830 0.2: ============================== 100029 0.3: ============================= 99732 0.4: ============================= 99911 0.5: ============================= 99722 0.6: ============================== 100780 0.7: ============================= 99812 0.8: ============================= 99875 0.9: ============================== 100196
D
<lang d>import std.stdio, std.algorithm, std.array, std.typecons,
std.range, std.exception;
auto meanStdDev(R)(R numbers) /*nothrow*/ @safe /*@nogc*/ {
if (numbers.empty)
return tuple(0.0L, 0.0L);
real sx = 0.0, sxx = 0.0;
ulong n;
foreach (x; numbers) {
sx += x;
sxx += x ^^ 2;
n++;
}
return tuple(sx / n, (n * sxx - sx ^^ 2) ^^ 0.5L / n);
}
void showHistogram01(R)(R numbers) /*@safe*/ {
enum maxWidth = 50; // N. characters.
ulong[10] bins;
foreach (immutable x; numbers) {
immutable index = cast(size_t)(x * bins.length);
enforce(index >= 0 && index < bins.length);
bins[index]++;
}
immutable real maxFreq = bins.reduce!max;
foreach (immutable n, immutable i; bins)
writefln(" %3.1f: %s", n / real(bins.length),
replicate("*", cast(int)(i / maxFreq * maxWidth)));
writeln;
}
version (statistics_basic_main) {
void main() @safe {
import std.random;
foreach (immutable p; 1 .. 7) {
auto n = iota(10L ^^ p).map!(_ => uniform(0.0L, 1.0L));
writeln(10L ^^ p, " numbers:");
writefln(" Mean: %8.6f, SD: %8.6f", n.meanStdDev.tupleof);
n.showHistogram01;
}
}
}</lang> Compile with "-version=statistics_basic_main" to run the main function.
- Output:
10 numbers: Mean: 0.651336, SD: 0.220208 0.0: ************************* 0.1: ************************************************** 0.2: 0.3: ************************************************** 0.4: 0.5: ************************* 0.6: ************************* 0.7: ************************* 0.8: ************************* 0.9: ************************* 100 numbers: Mean: 0.470756, SD: 0.291080 0.0: ************************************* 0.1: ******************************************* 0.2: ******************************* 0.3: ******************************* 0.4: ****************** 0.5: ********************* 0.6: **************************** 0.7: ************************************************** 0.8: ******************************* 0.9: ****************** 1000 numbers: Mean: 0.519127, SD: 0.287775 0.0: *************************************** 0.1: ******************************************* 0.2: **************************************** 0.3: **************************************** 0.4: ************************************ 0.5: ****************************************** 0.6: ************************************************** 0.7: ************************************** 0.8: ******************************************** 0.9: ********************************** 10000 numbers: Mean: 0.503266, SD: 0.289198 0.0: ********************************************** 0.1: ********************************************** 0.2: ************************************************** 0.3: ************************************************ 0.4: *********************************************** 0.5: ********************************************* 0.6: *********************************************** 0.7: ************************************************ 0.8: ********************************************** 0.9: ********************************************** 100000 numbers: Mean: 0.500945, SD: 0.289076 0.0: ************************************************* 0.1: ************************************************* 0.2: ************************************************* 0.3: ************************************************* 0.4: ************************************************* 0.5: ************************************************* 0.6: ************************************************* 0.7: ************************************************* 0.8: ************************************************** 0.9: ************************************************* 1000000 numbers: Mean: 0.499970, SD: 0.288635 0.0: ************************************************* 0.1: ************************************************* 0.2: ************************************************* 0.3: ************************************************* 0.4: ************************************************* 0.5: ************************************************** 0.6: ************************************************* 0.7: ************************************************* 0.8: ************************************************* 0.9: *************************************************
Dart
<lang d>/* Import math library to get:
* 1) Square root function : Math.sqrt(x) * 2) Power function : Math.pow(base, exponent) * 3) Random number generator : Math.Random() */
import 'dart:math' as Math show sqrt, pow, Random;
// Returns average/mean of a list of numbers num mean(List<num> l) => l.reduce((num value,num element)=>value+element)/l.length;
// Returns standard deviation of a list of numbers num stdev(List<num> l) => Math.sqrt((1/l.length)*l.map((num x)=>x*x).reduce((num value,num element) => value+element) - Math.pow(mean(l),2));
/* CODE TO PRINT THE HISTOGRAM STARTS HERE
* * Histogram has ten fields, one for every tenth between 0 and 1 * To do this, we save the histogram as a global variable * that will hold the number of occurences of each tenth in the sample */
List<num> histogram = new List.filled(10,0);
/*
* METHOD TO CREATE A RANDOM SAMPLE OF n NUMBERS (Returns a list) * * While creating each value, this method also increments the * appropriate index of the histogram */
List<num> randomsample(num n){
List<num> l = new List<num>(n);
histogram = new List.filled(10,0);
num random = new Math.Random();
for (int i = 0; i < n; i++){
l[i] = random.nextDouble();
histogram[conv(l[i])] += 1;
}
return l;
}
/*
* METHOD TO RETURN A STRING OF n ASTERIXES (yay ASCII art) */
String stars(num n){
String s = ;
for (int i = 0; i < n; i++){
s = s + '*';
}
return s;
}
/*
* METHOD TO DRAW THE HISTOGRAM * 1) Get to total for all the values in the histogram * 2) For every field in the histogram: * a) Compute the frequency for every field in the histogram * b) Print the frequency as asterixes */
void drawhistogram(){
int total = histogram.reduce((num element,num value)=>element+value);
double freq;
for (int i = 0; i < 10; i++){
freq = histogram[i]/total;
print('${i/10} - ${(i+1)/10} : ' + stars(conv(30*freq)));
}
}
/* HELPER METHOD:
* converts values between 0-1 to integers between 0-9 inclusive * useful to figure out which random value generated * corresponds to which field in the histogram */
int conv(num i) => (10*i).floor();
/* MAIN FUNCTION
* * Create 5 histograms and print the mean and standard deviation for each: * 1) Sample Size = 100 * 2) Sample Size = 1000 * 3) Sample Size = 10000 * 4) Sample Size = 100000 * 5) Sample Size = 1000000 * */
void main(){
List<num> l;
num m;
num s;
List<int> sampleSizes = [100,1000,10000,100000,1000000];
for (int samplesize in sampleSizes){
print('--------------- Sample size $samplesize ----------------');
l = randomsample(samplesize);
m = mean(l);
s = stdev(l);
drawhistogram();
print();
print('mean: ${m.toStringAsPrecision(8)} standard deviation: ${s.toStringAsPrecision(8)}');
print();
}
}</lang>
- Output:
--------------- Sample size 100 ---------------- 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 : ************************** mean: 0.49246975 standard deviation: 0.27789056 --------------- Sample size 1000 ---------------- 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 : ******************************* mean: 0.51170283 standard deviation: 0.28170178 -------------- Sample size 10000 ---------------- 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 : ******************************* mean: 0.50517609 standard deviation: 0.28923152 -------------- Sample size 100000 ---------------- 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 : ****************************** mean: 0.49994544 standard deviation: 0.28879394 -------------- Sample size 1000000 ---------------- 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 : ****************************** mean: 0.50013331 standard deviation: 0.28864180
Elixir
<lang elixir>defmodule Statistics do
def basic(n) do
{sum, sum2, hist} = generate(n)
mean = sum / n
stddev = :math.sqrt(sum2 / n - mean*mean)
IO.puts "size: #{n}"
IO.puts "mean: #{mean}"
IO.puts "stddev: #{stddev}"
Enum.each(0..9, fn i ->
:io.fwrite "~.1f:~s~n", [0.1*i, String.duplicate("=", trunc(500 * hist[i] / n))]
end)
IO.puts ""
end
defp generate(n) do
hist = for i <- 0..9, into: %{}, do: {i,0}
Enum.reduce(1..n, {0, 0, hist}, fn _,{sum, sum2, h} ->
r = :rand.uniform
{sum+r, sum2+r*r, Map.update!(h, trunc(10*r), &(&1+1))}
end)
end
end
Enum.each([100,1000,10000], fn n ->
Statistics.basic(n)
end)</lang>
- Output:
size: 100 mean: 0.5360891830207845 stddev: 0.2934821336243825 0.0:======================================================= 0.1:========================= 0.2:============================================================ 0.3:============================================= 0.4:============================== 0.5:======================================== 0.6:=========================================================================== 0.7:======================================================= 0.8:======================================================= 0.9:============================================================ size: 1000 mean: 0.4928249370693845 stddev: 0.2877164661860377 0.0:========================================================= 0.1:============================================== 0.2:================================================ 0.3:==================================================== 0.4:================================================ 0.5:====================================================== 0.6:================================================ 0.7:================================================== 0.8:=================================================== 0.9:=========================================== size: 10000 mean: 0.4969580860984137 stddev: 0.289282008094715 0.0:================================================== 0.1:==================================================== 0.2:================================================ 0.3:================================================= 0.4:================================================ 0.5:=================================================== 0.6:================================================== 0.7:================================================ 0.8:================================================= 0.9:=================================================
Factor
<lang factor>USING: assocs formatting grouping io kernel literals math math.functions math.order math.statistics prettyprint random sequences sequences.deep sequences.repeating ; IN: rosetta-code.statistics-basic
CONSTANT: granularity
$[ 11 iota [ 10 /f ] map 2 clump ]
- mean/std ( seq -- a b )
[ mean ] [ population-std ] bi ;
- .mean/std ( seq -- )
mean/std [ "Mean: " write . ] [ "STD: " write . ] bi* ;
- count-between ( seq a b -- n )
[ between? ] 2curry count ;
- histo ( seq -- seq )
granularity [ first2 count-between ] with map ;
- bar ( n -- str )
[ dup 50 < ] [ 10 / ] until 2 * >integer "*" swap repeat ;
- (.histo) ( seq -- seq' )
[ bar ] map granularity swap zip flatten 3 group ;
- .histo ( seq -- )
(.histo) [ "%.1f - %.1f %s\n" vprintf ] each ;
- stats ( n -- )
dup "Statistics %d:\n" printf random-units [ histo .histo ] [ .mean/std nl ] bi ;
- main ( -- )
{ 100 1,000 10,000 } [ stats ] each ;
MAIN: main</lang>
- Output:
Statistics 100: 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 ****************************** Mean: 0.5125865184454739 STD: 0.3011535351273979 Statistics 1000: 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 ****************** Mean: 0.4822182628505952 STD: 0.2874411306988986 Statistics 10000: 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 ******************** Mean: 0.5030027112958179 STD: 0.2895932850375331
Fortran
This version will handle numbers as large as 1 trillion or more if you are prepared to wait long enough <lang fortran>program basic_stats
implicit none
integer, parameter :: i64 = selected_int_kind(18)
integer, parameter :: r64 = selected_real_kind(15)
integer(i64), parameter :: samples = 1000000000_i64
real(r64) :: r
real(r64) :: mean, stddev
real(r64) :: sumn = 0, sumnsq = 0
integer(i64) :: n = 0
integer(i64) :: bin(10) = 0
integer :: i, ind
call random_seed
n = 0 do while(n <= samples) call random_number(r) ind = r * 10 + 1 bin(ind) = bin(ind) + 1_i64 sumn = sumn + r sumnsq = sumnsq + r*r n = n + 1_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 = 1, 10
write(*, "(f3.1, a, a)") real(i)/10.0, ": ", repeat("=", int(bin(i)*500/samples))
end do
end program</lang>
- Output:
sample size = 100 Mean : 0.507952672404959 Stddev : 0.290452178516586 0.1: ============================================= 0.2: ============================================================ 0.3: ============================== 0.4: ================================================================= 0.5: ============================================= 0.6: ======================================================= 0.7: ================================================================= 0.8: ================================================== 0.9: ========================= 1.0: ================================================================= sample size = 1000 Mean : 0.505018948813265 Stddev : 0.287904987339785 0.1: ============================================== 0.2: ================================================ 0.3: ======================================================== 0.4: =============================================== 0.5: ================================================== 0.6: =========================================== 0.7: ======================================================== 0.8: ================================================== 0.9: =================================================== 1.0: =================================================== sample size = 10000 Mean : 0.508929669066967 Stddev : 0.287243609812712 0.1: ============================================== 0.2: ================================================ 0.3: ================================================= 0.4: ================================================== 0.5: ================================================ 0.6: =================================================== 0.7: ================================================== 0.8: ================================================== 0.9: ==================================================== 1.0: =================================================== sample size = 1000000000 Mean : 0.500005969962249 Stddev : 0.288673875345505 0.1: ================================================= 0.2: ================================================= 0.3: ================================================= 0.4: ================================================= 0.5: ================================================== 0.6: ================================================= 0.7: ================================================== 0.8: ================================================= 0.9: ================================================== 1.0: =================================================
FreeBASIC
<lang freebasic>' FB 1.05.0 Win64
Randomize
Sub basicStats(sampleSize As Integer)
If sampleSize < 1 Then Return Dim r(1 To sampleSize) As Double Dim h(0 To 9) As Integer all zero by default Dim sum As Double = 0.0 Dim hSum As Integer = 0
' Generate 'sampleSize' random numbers in the interval [0, 1) ' calculate their sum ' and in which box they will fall when drawing the histogram For i As Integer = 1 To sampleSize r(i) = Rnd sum += r(i) h(Int(r(i) * 10)) += 1 Next
For i As Integer = 0 To 9 : 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 = 0 To 9
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 > 500 then normalize histogram to 500 Dim scale As Double = 1.0 If sampleSize > 500 Then scale = 500.0 / sampleSize Print "Sample size "; sampleSize Print Print Using " Mean #.######"; mean; Print Using " SD #.######"; sd Print For i As Integer = 0 To 9 Print Using " #.## : "; i/10.0; Print Using "##### " ; h(i); numStars = Int(h(i) * scale + 0.5) Print String(numStars, "*") Next
End Sub
basicStats 100 Print basicStats 1000 Print basicStats 10000 Print basicStats 100000 Print Print "Press any key to quit" Sleep</lang>
- Output:
Sample size 100 Mean 0.485580 SD 0.269003 0.00 : 7 ******* 0.10 : 10 ********** 0.20 : 12 ************ 0.30 : 17 ***************** 0.40 : 8 ******** 0.50 : 10 ********** 0.60 : 11 *********** 0.70 : 9 ********* 0.80 : 9 ********* 0.90 : 7 ******* Sample size 1000 Mean 0.504629 SD 0.292029 0.00 : 99 ************************************************** 0.10 : 99 ************************************************** 0.20 : 93 *********************************************** 0.30 : 108 ****************************************************** 0.40 : 101 *************************************************** 0.50 : 97 ************************************************* 0.60 : 90 ********************************************* 0.70 : 110 ******************************************************* 0.80 : 102 *************************************************** 0.90 : 101 *************************************************** Sample size 10000 Mean 0.500027 SD 0.290618 0.00 : 1039 **************************************************** 0.10 : 997 ************************************************** 0.20 : 978 ************************************************* 0.30 : 988 ************************************************* 0.40 : 998 ************************************************** 0.50 : 959 ************************************************ 0.60 : 1037 **************************************************** 0.70 : 1004 ************************************************** 0.80 : 965 ************************************************ 0.90 : 1035 **************************************************** Sample size 100000 Mean 0.499503 SD 0.288730 0.00 : 10194 *************************************************** 0.10 : 9895 ************************************************* 0.20 : 9875 ************************************************* 0.30 : 9922 ************************************************** 0.40 : 10202 *************************************************** 0.50 : 9981 ************************************************** 0.60 : 10034 ************************************************** 0.70 : 10012 ************************************************** 0.80 : 9957 ************************************************** 0.90 : 9928 **************************************************
Go
<lang go>package main
import (
"fmt" "math" "math/rand" "strings"
)
func main() {
sample(100) sample(1000) sample(10000)
}
func sample(n int) {
// generate data
d := make([]float64, n)
for i := range d {
d[i] = rand.Float64()
}
// show mean, standard deviation
var sum, ssq float64
for _, s := range d {
sum += s
ssq += s * s
}
fmt.Println(n, "numbers")
m := sum / float64(n)
fmt.Println("Mean: ", m)
fmt.Println("Stddev:", math.Sqrt(ssq/float64(n)-m*m))
// show histogram
h := make([]int, 10)
for _, s := range d {
h[int(s*10)]++
}
for _, c := range h {
fmt.Println(strings.Repeat("*", c*205/int(n)))
}
fmt.Println()
}</lang>
- Output:
100 numbers Mean: 0.5231064889267764 Stddev: 0.292668237816841 **************** **************** ************************ ********************** ****************** ****************** **************** ************************** ************************ ******************** 1000 numbers Mean: 0.496026080160094 Stddev: 0.2880988956436907 ********************* ******************** ***************** *********************** ****************** ********************** ******************** ********************* ****************** ******************* 10000 numbers Mean: 0.5009091903581223 Stddev: 0.289269693719711 ******************* ******************** ******************** ******************** ********************* ******************** ******************* ******************* ******************** *********************
The usual approach to the extra problem is sampling. That is, to not do it.
To show really show how computations could be done a trillion numbers however, here is an outline of a map reduce strategy. The main task indicated that numbers should be generated before doing any computations on them. Consistent with that, The function getSegment returns data based on a starting and ending index, as if it were accessing some large data store.
The following runs comfortably on a simulated data size of 10 million. To scale to a trillion, and to use real data, you would want to use a technique like Distributed_programming#Go to distribute work across multiple computers, and on each computer, use a technique like Parallel_calculations#Go to distribute work across multiple cores within each computer. You would tune parameters like the constant threshold in the code below to optimize cache performance. <lang go>package main
import (
"fmt" "math" "math/rand" "strings"
)
func main() {
bigSample(1e7)
}
func bigSample(n int64) {
sum, ssq, h := reduce(0, n)
// compute final statistics and output as above
fmt.Println(n, "numbers")
m := sum / float64(n)
fmt.Println("Mean: ", m)
fmt.Println("Stddev:", math.Sqrt(ssq/float64(n)-m*m))
for _, c := range h {
fmt.Println(strings.Repeat("*", c*205/int(n)))
}
fmt.Println()
}
const threshold = 1e6
func reduce(start, end int64) (sum, ssq float64, h []int) {
n := end - start
if n < threshold {
d := getSegment(start, end)
return computeSegment(d)
}
// map to two sub problems
half := (start + end) / 2
sum1, ssq1, h1 := reduce(start, half)
sum2, ssq2, h2 := reduce(half, end)
// combine results
for i, c := range h2 {
h1[i] += c
}
return sum1 + sum2, ssq1 + ssq2, h1
}
func getSegment(start, end int64) []float64 {
d := make([]float64, end-start)
for i := range d {
d[i] = rand.Float64()
}
return d
}
func computeSegment(d []float64) (sum, ssq float64, h []int) {
for _, s := range d {
sum += s
ssq += s * s
}
h = make([]int, 10)
for _, s := range d {
h[int(s*10)]++
}
return
}</lang>
- Output:
10000000 numbers Mean: 0.4999673191148989 Stddev: 0.2886663876567514 ******************** ******************** ******************** ******************** ******************** ******************** ******************** ******************** ******************** ********************
Haskell
<lang Haskell>import Data.Foldable (foldl') --' import System.Random (randomRs, newStdGen) import Control.Monad (zipWithM_) import System.Environment (getArgs)
intervals :: [(Double, Double)] intervals = map conv [0 .. 9]
where
xs = [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]
conv s =
let [h, l] = take 2 $ drop s xs
in (h, l)
count :: [Double] -> [Int] count rands = map (\iv -> foldl (loop iv) 0 rands) intervals
where
loop :: (Double, Double) -> Int -> Double -> Int
loop (lo, hi) n x
| lo <= x && x < hi = n + 1
| otherwise = n
-- ^ fuses length and filter within (lo,hi) data Pair a b =
Pair !a
!b
-- accumulate sum and length in one fold sumLen :: [Double] -> Pair Double Double sumLen = fion2 . foldl (\(Pair s l) x -> Pair (s + x) (l + 1)) (Pair 0.0 0)
where fion2 :: Pair Double Int -> Pair Double Double fion2 (Pair s l) = Pair s (fromIntegral l)
-- safe division on pairs divl :: Pair Double Double -> Double divl (Pair _ 0.0) = 0.0 divl (Pair s l) = s / l
-- sumLen and divl are separate for stddev below mean :: [Double] -> Double mean = divl . sumLen
stddev :: [Double] -> Double stddev xs = sqrt $ foldl (\s x -> s + (x - m) ^ 2) 0 xs / l
where p@(Pair s l) = sumLen xs m = divl p
main = do
nr <- read . head <$> getArgs
-- or in code, e.g. let nr = 1000
rands <- take nr . randomRs (0.0, 1.0) <$> newStdGen
putStrLn $ "The mean is " ++ show (mean rands) ++ " !"
putStrLn $ "The standard deviation is " ++ show (stddev rands) ++ " !"
zipWithM_
(\iv fq -> putStrLn $ ivstr iv ++ ": " ++ fqstr fq)
intervals
(count rands)
where
fqstr i =
replicate
(if i > 50
then div i (div i 50)
else i)
'*'
ivstr (lo, hi) = show lo ++ " - " ++ show hi
-- To avoid Wiki formatting issue foldl = foldl'</lang>
- Output:
./Statistics 100 The mean is 0.5007604927009823 ! The standard deviation is 0.2933668702954616 ! 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: ************* ./Statistics 10000 The mean is 0.49399049116152155 ! The standard deviation is 0.28782134281196275 ! 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: ***************************************************
Hy
<lang lisp>(import
[numpy.random [random]] [numpy [mean std]] [matplotlib.pyplot :as plt])
(for [n [100 1000 10000]]
(setv v (random n)) (print "Mean:" (mean v) "SD:" (std v)))
(plt.hist (random 1000)) (plt.show)</lang>
Icon and Unicon
The following uses the stddev procedure from the Standard_deviation task. In this example,
<lang Icon>procedure main(A)
W := 50 # avg width for histogram bar B := 10 # histogram bins if *A = 0 then put(A,100) # 100 if none specified
while N := get(A) do { # once per argument
write("\nN=",N)
N := 0 < integer(N) | next # skip if invalid
stddev() # reset
m := 0.
H := list(B,0) # Histogram of
every i := 1 to N do { # calc running ...
s := stddev(r := ?0) # ... std dev
m +:= r/N # ... mean
H[integer(*H*r)+1] +:= 1 # ... histogram
}
write("mean=",m)
write("stddev=",s)
every i := 1 to *H do # show histogram
write(right(real(i)/*H,5)," : ",repl("*",integer(*H*50./N*H[i])))
}
end</lang>
- Output:
N=100 mean=0.4941076275054806 stddev=0.2812938788216594 0.1 : **************************************** 0.2 : ******************************************************* 0.3 : ******************************************************* 0.4 : ********************************************************************** 0.5 : **************************************** 0.6 : ********************************************* 0.7 : **************************************** 0.8 : ***************************************************************** 0.9 : **************************************** 1.0 : ************************************************** N=10000 mean=0.4935428224375008 stddev=0.2884171825227816 0.1 : *************************************************** 0.2 : *************************************************** 0.3 : *************************************************** 0.4 : ************************************************** 0.5 : **************************************************** 0.6 : ************************************************* 0.7 : *********************************************** 0.8 : ************************************************ 0.9 : ************************************************** 1.0 : *********************************************** N=1000000 mean=0.4997503773607869 stddev=0.2886322440610256 0.1 : ************************************************* 0.2 : ************************************************** 0.3 : ************************************************** 0.4 : ************************************************** 0.5 : ************************************************* 0.6 : ************************************************** 0.7 : ************************************************* 0.8 : ************************************************* 0.9 : ************************************************** 1.0 : *************************************************
J
J has library routines to compute mean and standard deviation: <lang j> require 'stats'
(mean,stddev) 1000 ?@$ 0
0.484669 0.287482
(mean,stddev) 10000 ?@$ 0
0.503642 0.290777
(mean,stddev) 100000 ?@$ 0
0.499677 0.288726</lang>
And, for a histogram:
<lang j>histogram=: <: @ (#/.~) @ (i.@#@[ , I.) require'plot' plot ((% * 1 + i.)100) ([;histogram) 10000 ?@$ 0</lang>
but these are not quite what is being asked for here.
Instead:
<lang j>histogram=: <: @ (#/.~) @ (i.@#@[ , I.)
meanstddevP=: 3 :0
NB. compute mean and std dev of y random numbers NB. picked from even distribution between 0 and 1 NB. and display a normalized ascii histogram for this sample NB. note: uses population mean (0.5), not sample mean, for stddev NB. given the equation specified for this task. h=.s=.t=. 0 chunk=. 1e6 bins=. (%~ 1 + i.) 10 for. i. <.y%chunk do. data=. chunk ?@$ 0 h=. h+ bins histogram data s=. s+ +/ data t=. t+ +/ *: data-0.5 end. data=. (chunk|y) ?@$ 0 h=. h+ bins histogram data s=. s+ +/ data t=. t+ +/ *: data - 0.5 smoutput (<.300*h%y) #"0 '#' (s%y) , %:t%y
)</lang>
Example use:
<lang j> meanstddevP 1000
0.488441 0.289744
meanstddevP 10000
0.49697 0.289433
meanstddevP 100000
0.500872 0.288241</lang>
(That said, note that these numbers are random, so reported standard deviation will vary with the random sample being tested.)
This could handle a trillion random numbers on a bog-standard computer, but I am not inclined to wait that long.
Java
<lang java>import static java.lang.Math.pow; import static java.util.Arrays.stream; import static java.util.stream.Collectors.joining; import static java.util.stream.IntStream.range;
public class Test {
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();
}
public static void main(String[] a) {
Locale.setDefault(Locale.US);
for (int p = 1; p < 7; p++) {
double[] n = range(0, (int) pow(10, p))
.mapToDouble(i -> Math.random()).toArray();
System.out.println((int)pow(10, p) + " numbers:");
double[] res = meanStdDev(n);
System.out.printf(" Mean: %8.6f, SD: %8.6f%n", res[0], res[1]);
showHistogram01(n);
}
}
}</lang>
10 numbers: Mean: 0.564409, SD: 0.249601 0.0: * 0.1: ***************** 0.2: ***************** 0.3: ***************** 0.4: ***************** 0.5: ***************** 0.6: * 0.7: *************************************************** 0.8: ********************************** 0.9: * 100 numbers: Mean: 0.487440, SD: 0.283866 0.0: ************************************ 0.1: ************************************ 0.2: ********************** 0.3: *************************************************** 0.4: *************************************************** 0.5: ***************************** 0.6: ************************************ 0.7: ************************************ 0.8: ************************************ 0.9: ***************************** 1000 numbers: Mean: 0.500521, SD: 0.285790 0.0: ********************************************** 0.1: ******************************************** 0.2: ****************************************** 0.3: **************************************** 0.4: ************************************************** 0.5: *************************************************** 0.6: ************************************************ 0.7: ************************************************ 0.8: **************************************** 0.9: ******************************************* 10000 numbers: Mean: 0.499363, SD: 0.288427 0.0: ************************************************* 0.1: ************************************************* 0.2: ************************************************ 0.3: ************************************************* 0.4: *************************************************** 0.5: ************************************************ 0.6: *************************************************** 0.7: ************************************************ 0.8: ************************************************ 0.9: ************************************************ 100000 numbers: Mean: 0.500154, SD: 0.287981 0.0: ************************************************* 0.1: ************************************************** 0.2: ************************************************** 0.3: ************************************************** 0.4: ************************************************** 0.5: *************************************************** 0.6: ************************************************** 0.7: ************************************************** 0.8: ************************************************* 0.9: ************************************************** 1000000 numbers: Mean: 0.500189, SD: 0.288560 0.0: ************************************************** 0.1: ************************************************** 0.2: ************************************************** 0.3: *************************************************** 0.4: ************************************************** 0.5: ************************************************** 0.6: ************************************************** 0.7: ************************************************** 0.8: ************************************************** 0.9: **************************************************
Jsish
<lang javascript>#!/usr/bin/env jsish "use strict";
function statisticsBasic(args:array|string=void, conf:object=void) {
var options = { // Rosetta Code, Statistics/Basic
rootdir :, // Root directory.
samples : 0 // Set sample size from options
};
var self = { };
parseOpts(self, options, conf);
function generateStats(n:number):object {
var i, sum = 0, sum2 = 0;
var hist = new Array(10);
hist.fill(0);
for (i = 0; i < n; i++) {
var r = Math.random();
sum += r;
sum2 += r*r;
hist[Math.floor((r*10))] += 1;
}
var mean = sum/n;
var stddev = Math.sqrt((sum2 / n) - mean*mean);
var obj = {n:n, sum:sum, mean:mean, stddev:stddev};
return {n:n, sum:sum, mean:mean, stddev:stddev, hist:hist};
}
function reportStats(summary:object):void {
printf("Samples: %d, mean: %f, stddev: %f\n", summary.n, summary.mean, summary.stddev);
var max = Math.max.apply(summary, summary.hist);
for (var i = 0; i < 10; i++) {
printf("%3.1f+ %-70s %5d\n", i * 0.1, 'X'.repeat(70 * summary.hist[i] / max), summary.hist[i]);
}
return;
}
function main() {
LogTest('Starting', args);
switch (typeof(args)) {
case 'string': args = [args]; break;
case 'array': break;
default: args = [];
}
if (self.rootdir === )
self.rootdir=Info.scriptDir();
Math.srand(0);
if (self.samples > 0) reportStats(generateStats(self.samples));
else if (args[0] && parseInt(args[0])) reportStats(generateStats(parseInt(args[0])));
else for (var n of [100, 1000, 10000]) reportStats(generateStats(n));
debugger;
LogDebug('Done');
return 0;
}
return main();
}
provide(statisticsBasic, 1);
if (isMain()) {
if (!Interp.conf('unitTest'))
return runModule(statisticsBasic);
- ' statisticsBasic unit-test';
- statisticsBasic();
}
/*
!EXPECTSTART!
' statisticsBasic unit-test' statisticsBasic() ==> Samples: 100, mean: 0.534517, stddev: 0.287124 0.0+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 8 0.1+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 11 0.2+ XXXXXXXXXXXXXXXXXXXXXXXXXX 6 0.3+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 10 0.4+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 10 0.5+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 11 0.6+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 8 0.7+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 16 0.8+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 7 0.9+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 13 Samples: 1000, mean: 0.490335, stddev: 0.286562 0.0+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 98 0.1+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 122 0.2+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 85 0.3+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 106 0.4+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 105 0.5+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 101 0.6+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 93 0.7+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 106 0.8+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 98 0.9+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 86 Samples: 10000, mean: 0.499492, stddev: 0.287689 0.0+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 969 0.1+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 992 0.2+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 1067 0.3+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 1011 0.4+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 973 0.5+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 1031 0.6+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 971 0.7+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 999 0.8+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 991 0.9+ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 996 0
!EXPECTEND!
- /</lang>
- Output:
prompt$ jsish -u statisticsBasic.jsi [PASS] statisticsBasic.jsi
Julia
<lang julia>using Printf
function hist(numbers)
maxwidth = 50
h = fill(0, 10)
for n in numbers
h[ceil(Int, 10n)] += 1
end
mx = maximum(h)
for (n, i) in enumerate(h)
@printf("%3.1f: %s\n", n / 10, "+" ^ floor(Int, i / mx * maxwidth))
end
end
for i in 1:6
n = rand(10 ^ i)
println("\n##\n## $(10 ^ i) numbers")
@printf("μ: %8.6f; σ: %8.6f\n", mean(n), std(n))
hist(n)
end</lang>
- Output:
## ## 10 numbers μ: 0.513345; σ: 0.261532 0.1: 0.2: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.3: 0.4: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.5: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.6: 0.7: +++++++++++++++++++++++++ 0.8: +++++++++++++++++++++++++ 0.9: ++++++++++++++++++++++++++++++++++++++++++++++++++ 1.0: ## ## 100 numbers μ: 0.483039; σ: 0.289858 0.1: ++++++++++++++++++++++++++++++++++++++++++ 0.2: ++++++++++++++++++++++++++++++++++++++++++ 0.3: ++++++++++++++++++++++++++++++++++++++++++ 0.4: ++++++++++++++++++++++++++++++ 0.5: ++++++++++++++++++++++++++++++++++++++++++++++ 0.6: ++++++++++++++++++++++++++++++ 0.7: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.8: +++++++++++++++++++ 0.9: ++++++++++++++++++++++++++++++++++++++++++++++ 1.0: ++++++++++++++++++++++++++++++++++ ## ## 1000 numbers μ: 0.482115; σ: 0.288932 0.1: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.2: ++++++++++++++++++++++++++++++++++++++++ 0.3: ++++++++++++++++++++++++++++++++++++++++ 0.4: ++++++++++++++++++++++++++++++++++++++++++ 0.5: ++++++++++++++++++++++++++++++++++++ 0.6: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.7: +++++++++++++++++++++++++++++++++++++++ 0.8: ++++++++++++++++++++++++++++++++++++++ 0.9: ++++++++++++++++++++++++++++++++++++++++ 1.0: +++++++++++++++++++++++++++++++++++ ## ## 10000 numbers μ: 0.502500; σ: 0.288759 0.1: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.2: ++++++++++++++++++++++++++++++++++++++++++++++ 0.3: ++++++++++++++++++++++++++++++++++++++++++++++ 0.4: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.5: +++++++++++++++++++++++++++++++++++++++++++++++ 0.6: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.7: +++++++++++++++++++++++++++++++++++++++++++++++ 0.8: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.9: ++++++++++++++++++++++++++++++++++++++++++++++++ 1.0: +++++++++++++++++++++++++++++++++++++++++++++++++ ## ## 100000 numbers μ: 0.499489; σ: 0.288911 0.1: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.2: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.3: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.4: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.5: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.6: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.7: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.8: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.9: +++++++++++++++++++++++++++++++++++++++++++++++++ 1.0: ++++++++++++++++++++++++++++++++++++++++++++++++ ## ## 1000000 numbers μ: 0.500268; σ: 0.288622 0.1: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.2: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.3: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.4: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.5: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.6: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.7: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.8: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.9: +++++++++++++++++++++++++++++++++++++++++++++++++ 1.0: +++++++++++++++++++++++++++++++++++++++++++++++++
Klong
Using the "mu" (mean) and "sd" (standard deviation) functions from the Klong statistics library: <lang K> .l("nstat.kg") bar::{x{x;.d("*")}:*0;.p("")} hist10::{[s];#'=s@<s::_x*10} plot::{[s];.p("");.p("n = ",$x);
(!10){.d(x%10);.d(" ");bar(y)}'_(100%x)*(hist10(s::{x;.rn()}'!x));
.p("mean = ",$mu(s));.p("sd = ",$sd(s))}
plot(100) plot(1000) plot(10000) </lang>
- Output:
n = 100 0.0 ***************** 0.1 ****** 0.2 ******** 0.3 ********** 0.4 *********** 0.5 ********* 0.6 *********** 0.7 ********** 0.8 ****** 0.9 ************ mean = 0.482634518758 sd = 0.300804579739938409 n = 1000 0.0 ******* 0.1 ******** 0.2 *********** 0.3 *********** 0.4 ********* 0.5 *********** 0.6 ******** 0.7 ************ 0.8 ********** 0.9 ******** mean = 0.510119356421 sd = 0.277396945925369919 n = 10000 0.0 ********** 0.1 ********* 0.2 ********* 0.3 ********** 0.4 ********* 0.5 ********** 0.6 ********* 0.7 ********* 0.8 ********** 0.9 ********** mean = 0.49854591894824 sd = 0.290375399458904972
Kotlin
<lang scala>// version 1.1.2
val rand = java.util.Random()
fun basicStats(sampleSize: Int) {
if (sampleSize < 1) return
val r = DoubleArray(sampleSize)
val h = IntArray(10) // all zero by default
/*
Generate 'sampleSize' random numbers in the interval [0, 1)
and calculate in which box they will fall when drawing the histogram
*/
for (i in 0 until sampleSize) {
r[i] = rand.nextDouble()
h[(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..9) {
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 > 500 then normalize histogram to 500
val scale = if (sampleSize <= 500) 1.0 else 500.0 / sampleSize
println("Sample size $sampleSize\n")
println(" Mean ${"%1.6f".format(mean)} SD ${"%1.6f".format(sd)}\n")
for (i in 0..9) {
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) basicStats(sampleSize)
}</lang> Sample run:
- Output:
Sample size 100 Mean 0.489679 SD 0.286151 0.00 : 12 ************ 0.10 : 7 ******* 0.20 : 13 ************* 0.30 : 9 ********* 0.40 : 10 ********** 0.50 : 8 ******** 0.60 : 14 ************** 0.70 : 10 ********** 0.80 : 8 ******** 0.90 : 9 ********* Sample size 1000 Mean 0.497003 SD 0.290002 0.00 : 104 **************************************************** 0.10 : 92 ********************************************** 0.20 : 107 ****************************************************** 0.30 : 109 ******************************************************* 0.40 : 96 ************************************************ 0.50 : 111 ******************************************************** 0.60 : 87 ******************************************** 0.70 : 79 **************************************** 0.80 : 117 *********************************************************** 0.90 : 98 ************************************************* Sample size 10000 Mean 0.505243 SD 0.288944 0.00 : 991 ************************************************** 0.10 : 938 *********************************************** 0.20 : 1034 **************************************************** 0.30 : 958 ************************************************ 0.40 : 963 ************************************************ 0.50 : 1003 ************************************************** 0.60 : 1081 ****************************************************** 0.70 : 995 ************************************************** 0.80 : 1001 ************************************************** 0.90 : 1036 **************************************************** Sample size 100000 Mean 0.500501 SD 0.288766 0.00 : 10015 ************************************************** 0.10 : 9844 ************************************************* 0.20 : 10012 ************************************************** 0.30 : 10160 *************************************************** 0.40 : 10051 ************************************************** 0.50 : 9938 ************************************************** 0.60 : 9934 ************************************************** 0.70 : 9914 ************************************************** 0.80 : 10057 ************************************************** 0.90 : 10075 **************************************************
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 => { #h->get(integer(#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 }
with scale in array(100,1000,10000,100000) do => {^ local(n = array) loop(#scale) => { #n->insert(decimal_random) } 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: 0.291640, mean: 0.549633 Second method: sd: 0.291640, mean: 0.549633 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: 0.288696, mean: 0.500533 Second method: sd: 0.288696, mean: 0.500533 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: 0.289180, mean: 0.496726 Second method: sd: 0.289180, mean: 0.496726 0.0: ++++++++++++++++++++++++ 0.1: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.2: ++++++++++++++++++++++++++++++++++++++++++++++ 0.3: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.4: +++++++++++++++++++++++++++++++++++++++++++++++ 0.5: +++++++++++++++++++++++++++++++++++++++++++++++ 0.6: +++++++++++++++++++++++++++++++++++++++++++++++ 0.7: +++++++++++++++++++++++++++++++++++++++++++++++ 0.8: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.9: +++++++++++++++++++++++++++++++++++++++++++++++ 1.0: +++++++++++++++++++++++ 100000 numbers: Naive method: sd: 0.288785, mean: 0.500985 Second method: sd: 0.288785, mean: 0.500985 0.0: +++++++++++++++++++++++++ 0.1: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.2: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.3: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.4: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.5: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.6: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.7: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.8: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.9: ++++++++++++++++++++++++++++++++++++++++++++++++++ 1.0: ++++++++++++++++++++++++
Liberty BASIC
Be aware that the PRNG in LB has a SLIGHT bias. <lang lb> call sample 100 call sample 1000 call sample 10000
end
sub sample n
dim dat( n)
for i =1 to n
dat( i) =rnd( 1)
next i
'// show mean, standard deviation
sum =0
sSq =0
for i =1 to n
sum =sum +dat( i)
sSq =sSq +dat( i)^2
next i
print n; " data terms used."
mean =sum / n print "Mean ="; mean
print "Stddev ="; ( sSq /n -mean^2)^0.5
'// show histogram
nBins =10
dim bins( nBins)
for i =1 to n
z =int( nBins *dat( i))
bins( z) =bins( z) +1
next i
for b =0 to nBins -1
for j =1 to int( nBins *bins( b)) /n *70)
print "#";
next j
print
next b
print
end sub </lang>
100000 data terms used. Mean =0.49870232 Stddev =0.28926563 ###################################################################### ###################################################################### ###################################################################### ###################################################################### ##################################################################### ##################################################################### ##################################################################### ##################################################################### ###################################################################### #####################################################################
Lua
The standard deviation seems to converge to around 0.28. I expect there's a good reason for this, though it's entirely beyond me. <lang lua> math.randomseed(os.time())
function randList (n) -- Build table of size n local numbers = {} for i = 1, n do table.insert(numbers, math.random()) -- range correct by default end return numbers end
function mean (t) -- Find mean average of values in table t local sum = 0 for k, v in pairs(t) do sum = sum + v end return sum / #t end
function stdDev (t) -- Find population standard deviation of table 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) -- Draw histogram of given table to stdout local histBars, compVal = {} for range = 0, 9 do histBars[range] = 0 for k, v in pairs(t) do compVal = tonumber(string.format("%0.1f", v - 0.05)) if compVal == range / 10 then histBars[range] = histBars[range] + 1 end end end for k, v in pairs(histBars) do io.write("0." .. k .. " " .. string.rep('=', v / #t * 200)) print(" " .. v) end print() end
function showStats (tabSize) -- Create and display statistics info local numList = randList(tabSize) print("Table of size " .. #numList) print("Mean average: " .. mean(numList)) print("Standard dev: " .. stdDev(numList)) showHistogram(numList) end
for power = 2, 5 do -- Start of main procedure showStats(10 ^ power) end </lang>
Maple
The following samples 100 uniformly distributed numbers between 0 and 1: <lang maple>with(Statistics): X_100 := Sample( Uniform(0,1), 100 ); Mean( X_100 ); StandardDeviation( X_100 ); Histogram( X_100 );</lang> It is also possible to make a procedure that outputs the mean, standard deviation, and a histogram for a given number of random uniformly distributed numbers: <lang maple>sample := proc( n )
local data;
data := Sample( Uniform(0,1), n );
printf( "Mean: %.4f\nStandard Deviation: %.4f",
Statistics:-Mean( data ),
Statistics:-StandardDeviation( data ) );
return Statistics:-Histogram( data );
end proc: sample( 1000 );</lang>
Mathematica
<lang Mathematica>Sample[n_]:= (Print[#//Length," numbers, Mean : ",#//Mean,", StandardDeviation : ",#//StandardDeviation ];
BarChart[BinCounts[#,{0,1,.1}], Axes->False, BarOrigin->Left])&[(RandomReal[1,#])&[ n ]]
Sample/@{100,1 000,10 000,1 000 000} </lang>
- Output:
100 numbers, Mean : 0.478899, StandardDeviation : 0.322265 1000 numbers, Mean : 0.503383, StandardDeviation : 0.278352 10000 numbers, Mean : 0.498278, StandardDeviation : 0.28925 1000000 numbers, Mean : 0.500248, StandardDeviation : 0.288713
MATLAB / Octave
<lang Matlab> % Initialize
N = 0; S=0; S2 = 0; binlist = 0:.1:1; h = zeros(1,length(binlist)); % initialize histogram
% read data and perform computation while (1)
% read next sample x
if (no_data_available) break; end;
N = N + 1;
S = S + x;
S2= S2+ x*x;
ix= sum(x < binlist);
h(ix) = h(ix)+1;
end
% generate output m = S/N; % mean sd = sqrt(S2/N-mean*mean); % standard deviation bar(binlist,h)</lang>
Nim
<lang nim>import math, strutils randomize()
proc sd(ns): auto =
var sx, sxx = 0.0
for x in ns:
sx += x
sxx += x * x
let sd = if ns.len > 0: sqrt(float(ns.len) * sxx - sx * sx) / float(ns.len)
else: 0
(sd, sx / float(ns.len))
proc histogram(ns) =
var h = newSeq[int](10) for n in ns: let pos = int(n * 10) inc h[pos]
const maxWidth = 50 let mx = max(h) echo "" for n, i in h: echo n/10,": ",repeatChar(int(i / mx * maxWidth), '+') echo ""
for i in [10, 100, 1_000, 10_000, 100_000]:
var n = newSeq[float](i) for x in 0..n.high: n[x] = random(1.0) echo "\n##\n## ",i," numbers\n##" let (sd, mean) = sd(n) echo "sd: ",sd,", mean: ",mean histogram(n)</lang>
- Output:
## ## 10 numbers ## sd: 0.2738118959385979, mean: 0.4717111448227304 0.0: +++++++++++++++++++++++++ 0.1: +++++++++++++++++++++++++ 0.2: 0.3: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.4: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.5: 0.6: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.7: +++++++++++++++++++++++++ 0.8: 0.9: +++++++++++++++++++++++++ [...] ## ## 100000 numbers ## sd: 0.2884329643843962, mean: 0.4997598571602153 0.0: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.1: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.2: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.3: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.4: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.5: ++++++++++++++++++++++++++++++++++++++++++++++++ 0.6: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.7: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.8: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.9: ++++++++++++++++++++++++++++++++++++++++++++++++
Oforth
<lang Oforth>: main(n) | l m std i nb |
// Create list and calculate avg and stddev ListBuffer init(n, #[ Float rand ]) dup ->l avg ->m 0 l apply(#[ sq +]) n / m sq - sqrt ->std System.Out "n = " << n << ", avg = " << m << ", std = " << std << cr
// Histo
0.0 0.9 0.1 step: i [
l count(#[ between(i, i 0.1 +) ]) 400 * n / asInteger ->nb
System.Out i <<wjp(3, JUSTIFY_RIGHT, 2) " - " <<
i 0.1 + <<wjp(3, JUSTIFY_RIGHT, 2) " - " <<
StringBuffer new "*" <<n(nb) << cr
] ;</lang>
- Output:
>100 main n = 100, avg = 0.483425493606762, std = 0.280986417046947 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 - ************************ ok >main(1000) n = 1000, avg = 0.514985138392994, std = 0.288119541786792 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 - ********************************************* ok >main(10000) n = 10000, avg = 0.501457911440693, std = 0.289120988428389 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 - **************************************** ok >main(100000) n = 100000, avg = 0.499807481461133, std = 0.28907281580804 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 - **************************************** ok >main(1000000) n = 1000000, avg = 0.500078448259022, std = 0.288580229525348 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 - *************************************** ok >
PARI/GP
<lang parigp>mean(v)={
vecsum(v)/#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,random(1.)),mu=mean(v),s=stdev(v,mu),h=histogram(v),sz=ceil(n/50/16));
for(i=1,16,for(j=1,h[i]\sz,print1("#"));print());
print("Mean: "mu);
print("Stdev: "s);
}; show(100); show(1000); show(10000);</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.).
Perl
<lang perl>my @histogram = (0) x 10; my $sum = 0; my $sum_squares = 0; my $n = $ARGV[0];
for (1..$n) {
my $current = rand(); $sum+= $current; $sum_squares+= $current ** 2; $histogram[$current * @histogram]+= 1;
}
my $mean = $sum / $n;
print "$n numbers\n",
"Mean: $mean\n",
"Stddev: ", sqrt(($sum_squares / $n) - ($mean ** 2)), "\n";
for my $i (0..$#histogram) {
printf "%.1f - %.1f : ", $i/@histogram, (1 + $i)/@histogram;
print "*" x (30 * $histogram[$i] * @histogram/$n); # 30 stars expected per row print "\n";
}</lang>
Usage:
perl rand_statistics.pl (number of values)
$ perl rand_statistics.pl 100 100 numbers Mean: 0.531591369804339 Stddev: 0.28440375340793 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 : ************************************ $ perl rand_statistics.pl 1000 1000 numbers Mean: 0.51011452684812 Stddev: 0.29490201218115 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 : ********************************* $ perl rand_statistics.pl 10000 10000 numbers Mean: 0.495329167703333 Stddev: 0.285944419431566 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 : ****************************** $ perl rand_statistics.pl 10000000 10000000 numbers Mean: 0.499973935749229 Stddev: 0.2887231680817 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 : *******************************
Perl 6
<lang perl6>for 100, 1_000, 10_000 -> $N {
say "size: $N";
my @data = rand xx $N;
printf "mean: %f\n", my $mean = $N R/ [+] @data;
printf "stddev: %f\n", sqrt
$mean**2 R- $N R/ [+] @data »**» 2;
printf "%.1f %s\n", .key, '=' x (500 * .value.elems / $N)
for sort @data.classify: (10 * *).Int / 10;
say ;
}</lang>
- Output:
size: 100 mean: 0.52518699464629726 stddev: 0.28484207464779548 0.0 ============================== 0.1 ====================================================================== 0.2 =================================== 0.3 ================================================== 0.4 ============================================================ 0.5 ============================================= 0.6 ==================== 0.7 =========================================================================== 0.8 ====================================================================== 0.9 ============================================= size: 1000 mean: 0.51043974182914975 stddev: 0.29146336553431618 0.0 ============================================== 0.1 ================================================== 0.2 =========================================== 0.3 ======================================================== 0.4 =================================================== 0.5 ======================================= 0.6 =========================================================== 0.7 ==================================================== 0.8 ============================================== 0.9 ======================================================== size: 10000 mean: 0.50371817503544458 stddev: 0.2900716333092252 0.0 =================================================== 0.1 ================================================= 0.2 ============================================= 0.3 ==================================================== 0.4 ============================================== 0.5 ==================================================== 0.6 ================================================ 0.7 =================================================== 0.8 ==================================================== 0.9 ==================================================
Phix
To do a trillion samples, I would change the existing generate loop into an inner 100_000_000 loop that still uses the fast native types, with everything outside that changed to bigatom, and of course add an outer loop which sums into them. <lang Phix>function generate_statistics(integer n) sequence hist = repeat(0,10) atom sum_r = 0,
sum_squares = 0.0
for i=1 to n do
atom r = rnd()
sum_r += r
sum_squares += r*r
hist[floor(10*r)+1] += 1
end for
atom mean = sum_r / n
atom stddev = sqrt((sum_squares / n) - mean*mean)
return {n, mean, stddev, hist}
end function
procedure display_statistics(sequence x) atom n, mean, stddev sequence hist
{n, mean, stddev, hist} = x
printf(1,"-- Stats for sample size %d\n",{n})
printf(1,"mean: %g\n",{mean})
printf(1,"sdev: %g\n",{stddev})
for i=1 to length(hist) do
integer cnt = hist[i]
string bars = repeat('=',floor(cnt*300/n))
printf(1,"%.1f: %s %d\n",{i/10,bars,cnt})
end for
end procedure
for n=2 to 5 do
display_statistics(generate_statistics(power(10,n+(n=5))))
end for</lang>
- Output:
-- Stats for sample size 100 mean: 0.530925 sdev: 0.303564 0.1: ======================== 8 0.2: ======================================= 13 0.3: ============================== 10 0.4: ================== 6 0.5: ===================== 7 0.6: ================================= 11 0.7: ================================= 11 0.8: ===================== 7 0.9: ======================================= 13 1.0: ========================================== 14
-- Stats for sample size 1000 mean: 0.50576 sdev: 0.288862 0.1: ============================ 95 0.2: ============================== 103 0.3: ============================= 98 0.4: =========================== 93 0.5: ============================== 101 0.6: ============================= 99 0.7: =============================== 105 0.8: ============================= 97 0.9: ================================ 108 1.0: ============================== 101
-- Stats for sample size 10000 mean: 0.498831 sdev: 0.28841 0.1: ============================= 987 0.2: =============================== 1060 0.3: ============================ 953 0.4: ============================= 980 0.5: ============================== 1013 0.6: ============================= 997 0.7: ================================ 1089 0.8: ============================ 948 0.9: ============================= 974 1.0: ============================= 999
-- Stats for sample size 1000000 mean: 0.499937 sdev: 0.288898 0.1: ============================== 100071 0.2: ============================== 100943 0.3: ============================= 99594 0.4: ============================= 99436 0.5: ============================= 99806 0.6: ============================= 99723 0.7: ============================== 100040 0.8: ============================== 100280 0.9: ============================== 100264 1.0: ============================= 99843
PicoLisp
The following has no limit on the number of samples. The 'statistics' function accepts an executable body 'Prg', which it calls repeatedly to get the samples. <lang PicoLisp> (seed (time))
(scl 8)
(de statistics (Cnt . Prg)
(prinl Cnt " numbers")
(let (Sum 0 Sqr 0 Hist (need 10 NIL 0))
(do Cnt
(let N (run Prg 1) # Get next number
(inc 'Sum N)
(inc 'Sqr (*/ N N 1.0))
(inc (nth Hist (inc (/ N 0.1)))) ) )
(let M (*/ Sum Cnt)
(prinl "Mean: " (round M))
(prinl "StdDev: "
(round
(sqrt
(- (*/ Sqr Cnt) (*/ M M 1.0))
1.0 ) ) ) )
(for (I . H) Hist
(prin (format I 1) " ")
(do (*/ H 400 Cnt) (prin '=))
(prinl) ) ) )
(for I (2 4 6)
(statistics (** 10 I)
(rand 0 (dec 1.0)) )
(prinl) )
</lang>
- Output:
100 numbers Mean: 0.501 StdDev: 0.284 0.1 ======================================== 0.2 ==================================== 0.3 ==================================================== 0.4 ======================== 0.5 ======================== 0.6 ================================================================ 0.7 ======================================================== 0.8 ==================================== 0.9 ======================== 1.0 ============================================ 10000 numbers Mean: 0.501 StdDev: 0.288 0.1 ======================================= 0.2 ======================================== 0.3 ======================================= 0.4 ========================================= 0.5 ========================================= 0.6 ======================================== 0.7 ========================================= 0.8 ======================================== 0.9 ======================================== 1.0 ======================================== 1000000 numbers Mean: 0.500 StdDev: 0.289 0.1 ======================================== 0.2 ======================================== 0.3 ======================================== 0.4 ======================================== 0.5 ======================================== 0.6 ======================================== 0.7 ======================================== 0.8 ======================================== 0.9 ======================================== 1.0 ========================================
PL/I
<lang pli> stat: procedure options (main); /* 21 May 2014 */
stats: procedure (values, mean, standard_deviation);
declare (values(*), mean, standard_deviation) float; declare n fixed binary (31) initial ( (hbound(values,1)) );
mean = sum(values)/n;
standard_deviation = sqrt( sum(values - mean)**2 / n);
end stats;
declare values (*) float controlled; declare (mean, stddev) float; declare bin(0:9) fixed; declare (i, n) fixed binary (31);
do n = 100, 1000, 10000, 100000;
allocate values(n);
values = random();
call stats (values, mean, stddev);
if n = 100 then
do;
bin = 0;
do i = 1 to 100;
bin(10*values(i)) += 1;
end;
put skip list ('Histogram for 100 values:');
do i = 0 to 9; /* display histogram */
put skip list (repeat('.', bin(i)) );
end;
end;
put skip list (n || ' values: mean=' || mean, 'stddev=' || stddev);
free values;
end;
end stat;</lang>
- Output:
Histogram for 100 values:
.......
..............
..............
...........
...............
........
...........
.........
.......
..............
100 values: mean= 4.89708E-0001 stddev= 1.64285E-0007
1000 values: mean= 4.97079E-0001 stddev= 1.07871E-0005
10000 values: mean= 4.99119E-0001 stddev= 8.35870E-0005
100000 values: mean= 5.00280E-0001 stddev= 7.88976E-0004
PureBasic
Changes were made from the Liberty BASIC version to normalize the histogram as well as implement a random float function. <lang purebasic>Procedure.f randomf()
#RNG_max_resolution = 2147483647 ProcedureReturn Random(#RNG_max_resolution) / #RNG_max_resolution
EndProcedure
Procedure sample(n)
Protected i, nBins, binNumber, tickMarks, maxBinValue
Protected.f sum, sumSq, mean
Dim dat.f(n)
For i = 1 To n
dat(i) = randomf()
Next
;show mean, standard deviation
For i = 1 To n
sum + dat(i)
sumSq + dat(i) * dat(i)
Next i
PrintN(Str(n) + " data terms used.")
mean = sum / n
PrintN("Mean =" + StrF(mean))
PrintN("Stddev =" + StrF((sumSq / n) - Sqr(mean * mean)))
;show histogram
nBins = 10
Dim bins(nBins)
For i = 1 To n
binNumber = Int(nBins * dat(i))
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 = Int(bins(binNumber) * #normalizedMaxValue / maxBinValue)
PrintN(ReplaceString(Space(tickMarks), " ", "#"))
Next
PrintN("")
EndProcedure
If OpenConsole()
sample(100) sample(1000) sample(10000) Print(#CRLF$ + #CRLF$ + "Press ENTER to exit"): Input() CloseConsole()
EndIf</lang>
- Output:
100 data terms used. Mean =0.4349198639 Stddev =-0.1744846404 ######################################################### ######################################### ################################ ################################################################# ################################ ##################################################### ###################################################################### ################ ######################## ################ 1000 data terms used. Mean =0.4960154891 Stddev =-0.1691310555 ############################################################### ####################################################### ############################################################# ###################################################################### ########################################################## ############################################################## #################################################################### ############################################################### ############################################################# ##################################################### 10000 data terms used. Mean =0.5042046309 Stddev =-0.1668083966 ################################################################## ################################################################ ################################################################## #################################################################### ################################################################ ###################################################################### #################################################################### ################################################################### #################################################################### ####################################################################
Python
The second function, sd2 only needs to go once through the numbers and so can more efficiently handle large streams of numbers. <lang python>def sd1(numbers):
if numbers:
mean = sum(numbers) / len(numbers)
sd = (sum((n - mean)**2 for n in numbers) / len(numbers))**0.5
return sd, mean
else:
return 0, 0
def sd2(numbers):
if numbers:
sx = sxx = n = 0
for x in numbers:
sx += x
sxx += x*x
n += 1
sd = (n * sxx - sx*sx)**0.5 / n
return sd, sx / n
else:
return 0, 0
def histogram(numbers):
h = [0] * 10
maxwidth = 50 # characters
for n in numbers:
h[int(n*10)] += 1
mx = max(h)
print()
for n, i in enumerate(h):
print('%3.1f: %s' % (n / 10, '+' * int(i / mx * maxwidth)))
print()
if __name__ == '__main__':
import random
for i in range(1, 6):
n = [random.random() for j in range(10**i)]
print("\n##\n## %i numbers\n##" % 10**i)
print(' Naive method: sd: %8.6f, mean: %8.6f' % sd1(n))
print(' Second method: sd: %8.6f, mean: %8.6f' % sd2(n))
histogram(n)</lang>
- Output:
for larger sets of random numbers, the distribution of numbers between the bins of the histogram evens out.
... ## ## 100 numbers ## Naive method: sd: 0.288911, mean: 0.508686 Second method: sd: 0.288911, mean: 0.508686 0.0: +++++++++++++++++++++++++++++++ 0.1: ++++++++++++++++++++++++++++ 0.2: +++++++++++++++++++++++++ 0.3: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.4: ++++++++++++++++++ 0.5: +++++++++++++++++++++++++++++++ 0.6: ++++++++++++++++++ 0.7: +++++++++++++++++++++++++++++++++++++ 0.8: ++++++++++++++++++++++++++++++++++++++++ 0.9: +++++++++++++++++++++++++++++++ ... ## ## 10000000 numbers ## Naive method: sd: 0.288750, mean: 0.499839 Second method: sd: 0.288750, mean: 0.499839 0.0: ++++++++++++++++++++++++++++++++++++++++++++++++++ 0.1: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.2: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.3: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.4: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.5: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.6: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.7: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.8: +++++++++++++++++++++++++++++++++++++++++++++++++ 0.9: +++++++++++++++++++++++++++++++++++++++++++++++++
R
The challenge of processing a trillion numbers is generating them in the first place. As the errors below show, allocating 7.5 TB for such a vector is simply impractical. The workaround is to generate them, process individual data points and then discard them. The downside in this case is the time. <lang R>
- Generate the sets
a = runif(10,min=0,max=1) b = runif(100,min=0,max=1) c = runif(1000,min=0,max=1) d = runif(10000,min=0,max=1)
- Print out the set of 10 values
cat("a = ",a)
- Print out the Mean and Standard Deviations of each of the sets
cat("Mean of a : ",mean(a)) cat("Standard Deviation of a : ", sd(a)) cat("Mean of b : ",mean(b)) cat("Standard Deviation of b : ", sd(b)) cat("Mean of c : ",mean(c)) cat("Standard Deviation of c : ", sd(c)) cat("Mean of d : ",mean(d)) cat("Standard Deviation of d : ", sd(d))
- Plotting the histogram of d
hist(d)
- Following lines error out due to insufficient memory
cat("Mean of a trillion random values in the range [0,1] : ",mean(runif(10^12,min=0,max=1))) cat("Standard Deviation of a trillion random values in the range [0,1] : ", sd(runif(10^12,min=0,max=1))) </lang> Output
a = 0.3884718 0.6324655 0.9288667 0.1948398 0.5636742 0.2746207 0.4712035 0.2624648 0.45492 0.3328236> Mean of a : 0.4504351 Standard Deviation of a : 0.2171919 Mean of b : 0.5240795 Standard Deviation of b : 0.2654211 Mean of c : 0.5000978 Standard Deviation of c : 0.2882098 Mean of d : 0.4991501 Standard Deviation of d : 0.2911486 Error: cannot allocate vector of size 7450.6 Gb Error: cannot allocate vector of size 7450.6 Gb
Racket
<lang racket>
- lang racket
(require math (only-in srfi/27 random-real))
(define (histogram n xs Δx)
(define (r x) (~r x #:precision 1 #:min-width 3))
(define (len count) (exact-floor (/ (* count 200) n)))
(for ([b (bin-samples (range 0 1 Δx) <= xs)])
(displayln (~a (r (sample-bin-min b)) "-" (r (sample-bin-max b)) ": "
(make-string (len (length (sample-bin-values b))) #\*)))))
(define (task n)
(define xs (for/list ([_ n]) (random-real))) (displayln (~a "Number of samples: " n)) (displayln (~a "Mean: " (mean xs))) (displayln (~a "Standard deviance: " (stddev xs))) (histogram n xs 0.1) (newline))
(task 100) (task 1000) (task 10000) </lang>
- Output:
Number of samples: 100 Mean: 0.5466640451797568 Standard deviance: 0.29309099509716496 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: ************************ Number of samples: 1000 Mean: 0.48116201801707503 Standard deviance: 0.2873408579602762 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: ****************** Number of samples: 10000 Mean: 0.4988839808467469 Standard deviance: 0.2892924816935072 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: *******************
REXX
Twenty decimal digits are used for the calculations, but only half that (ten digits) are displayed in the output. <lang rexx>/*REXX program generates some random numbers, shows bin histogram, finds mean & stdDev. */ numeric digits 20 /*use twenty decimal digits precision, */ showDigs=digits()%2 /* ··· but only show ten decimal digits*/ parse arg size seed . /*allow specification: size, and seed.*/ if size== | size=="," then size=100 /*Not specified? Then use the default.*/ if datatype(seed,'W') then call random ,,seed /*allow a seed for the RANDOM BIF. */
- .=0 /*count of the numbers in each bin. */
do j=1 for size /*generate some random numbers. */
@.j=random(, 99999) / 100000 /*express random number as a fraction. */
_=substr(@.j'00', 3, 1) /*determine which bin the number is in,*/
#._=#._ + 1 /* ··· and bump its count. */
end /*j*/
do k=0 for 10; kp=k + 1 /*show a histogram of the bins. */
lr='0.'k ; if k==0 then lr= "0 " /*adjust for the low range. */
hr='0.'kp ; if k==9 then hr= "1 " /* " " " high range. */
barPC=right( strip( left( format( 100*#.k / size, , 2), 5)), 5) /*compute the %. */
say lr"──►"hr' ' barPC copies("─", barPC * 2 % 1 ) /*show histogram.*/
end /*k*/
say say 'sample size = ' size; say avg= mean(size) ; say ' mean = ' format(avg, , showDigs) std=stdDev(size) ; say ' stdDev = ' format(std, , showDigs) exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ mean: arg N; $=0; do m=1 for N; $=$ + @.m; end; return $/N stdDev: arg N; $=0; do s=1 for N; $=$ + (@.s-avg)**2; end; return sqrt($/N) /1 /*──────────────────────────────────────────────────────────────────────────────────────*/ sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); m.=9; numeric form; h=d+6
numeric digits; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g*.5'e'_ % 2
do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/; return g</lang>
- output when using the default input of: 100
0 ──►0.1 12.00 ────────────────────────
0.1──►0.2 12.00 ────────────────────────
0.2──►0.3 10.00 ────────────────────
0.3──►0.4 8.00 ────────────────
0.4──►0.5 12.00 ────────────────────────
0.5──►0.6 8.00 ────────────────
0.6──►0.7 11.00 ──────────────────────
0.7──►0.8 11.00 ──────────────────────
0.8──►0.9 6.00 ────────────
0.9──►1 10.00 ────────────────────
sample size = 100
mean = 0.4711358000
stdDev = 0.2920169478
- output when using the default input of: 1000
0 ──►0.1 9.50 ───────────────────
0.1──►0.2 9.90 ───────────────────
0.2──►0.3 11.70 ───────────────────────
0.3──►0.4 8.80 ─────────────────
0.4──►0.5 8.40 ────────────────
0.5──►0.6 10.20 ────────────────────
0.6──►0.7 10.30 ────────────────────
0.7──►0.8 11.40 ──────────────────────
0.8──►0.9 9.10 ──────────────────
0.9──►1 10.70 ─────────────────────
sample size = 1000
mean = 0.5037752500
stdDev = 0.2886365539
- output when using the default input of: 10000
0 ──►0.1 9.61 ───────────────────
0.1──►0.2 10.45 ────────────────────
0.2──►0.3 9.96 ───────────────────
0.3──►0.4 10.56 ─────────────────────
0.4──►0.5 9.91 ───────────────────
0.5──►0.6 10.13 ────────────────────
0.6──►0.7 10.12 ────────────────────
0.7──►0.8 9.84 ───────────────────
0.8──►0.9 9.61 ───────────────────
0.9──►1 9.81 ───────────────────
sample size = 10000
mean = 0.4968579550
stdDev = 0.2863756713
- output when using the default input of: 100000
0 ──►0.1 10.13 ────────────────────
0.1──►0.2 9.84 ───────────────────
0.2──►0.3 9.91 ───────────────────
0.3──►0.4 9.94 ───────────────────
0.4──►0.5 10.19 ────────────────────
0.5──►0.6 10.08 ────────────────────
0.6──►0.7 10.12 ────────────────────
0.7──►0.8 9.78 ───────────────────
0.8──►0.9 10.07 ────────────────────
0.9──►1 9.95 ───────────────────
sample size = 100000
mean = 0.4999883642
stdDev = 0.2884109515
- output when using the default input of: 1000000
0 ──►0.1 9.94 ───────────────────
0.1──►0.2 10.03 ────────────────────
0.2──►0.3 10.03 ────────────────────
0.3──►0.4 9.98 ───────────────────
0.4──►0.5 10.00 ────────────────────
0.5──►0.6 10.03 ────────────────────
0.6──►0.7 9.99 ───────────────────
0.7──►0.8 10.03 ────────────────────
0.8──►0.9 9.97 ───────────────────
0.9──►1 9.99 ───────────────────
sample size = 1000000
mean = 0.5000687045
stdDev = 0.2885125537
Ring
<lang ring>
- Project : Statistics/Basic
decimals(9) sample(100) sample(1000) sample(10000)
func sample(n)
samp = list(n)
for i =1 to n
samp[i] =random(9)/10
next
sum = 0
sumSq = 0
for i = 1 to n
sum = sum + samp[i]
sumSq = sumSq +pow(samp[i],2)
next
see n + " Samples used." + nl
mean = sum / n
see "Mean = " + mean + nl
see "Std Dev = " + pow((sumSq /n -pow(mean,2)),0.5) + nl
bins2 = 10
bins = list(bins2)
for i = 1 to n
z = floor(bins2 * samp[i])
if z != 0
bins[z] = bins[z] +1
ok
next
for b = 1 to bins2
see b + " " + nl
for j = 1 to floor(bins2 *bins[b]) /n *70
see "*"
next
see nl
next
see nl
</lang> Output:
100 Mean = 0.482000000 Std Dev = 0.276904316 1 *************************************************************** 2 ******************************************************** 3 ******************************************************** 4 ***************************************************************************** 5 ********************************************************************** 6 ***************************************************************************** 7 *********************************************************************************************************************** 8 ******************************************************** 9 ********************************************************************** 1000 Mean = 0.436600000 Std Dev = 0.284605762 1 ******************************************************************************* 2 ********************************************************************** 3 *********************************************************************************** 4 ************************************************************************ 5 *********************************************************************** 6 ****************************************************************** 7 ******************************************************* 8 **************************************************************** 9 ********************************************************************** 10000 Mean = 0.451940000 Std Dev = 0.287183280 1 ******************************************************************** 2 *********************************************************************** 3 ******************************************************************* 4 ********************************************************************* 5 ********************************************************************* 6 *********************************************************************** 7 ********************************************************************* 8 ************************************************************************ 9 *********************************************************************
Ruby
<lang ruby>def generate_statistics(n)
sum = sum2 = 0.0
hist = Array.new(10, 0)
n.times do
r = rand
sum += r
sum2 += r**2
hist[(10*r).to_i] += 1
end
mean = sum / n
stddev = Math::sqrt((sum2 / n) - mean**2)
puts "size: #{n}"
puts "mean: #{mean}"
puts "stddev: #{stddev}"
hist.each_with_index {|x,i| puts "%.1f:%s" % [0.1*i, "=" * (70*x/hist.max)]}
puts
end
[100, 1000, 10000].each {|n| generate_statistics n}</lang>
- Output:
size: 100 mean: 0.5565132836634081 stddev: 0.30678831716883026 0.0:================================ 0.1:============================================================ 0.2:================================ 0.3:============================ 0.4:============================================== 0.5:======================= 0.6:======================================================== 0.7:======================================================== 0.8:============================================================ 0.9:====================================================================== size: 1000 mean: 0.4910962662424557 stddev: 0.28325915710008404 0.0:====================================================== 0.1:================================================== 0.2:======================================================= 0.3:====================================================================== 0.4:===================================================== 0.5:================================================= 0.6:================================================= 0.7:============================================================= 0.8:================================================ 0.9:================================================= size: 10000 mean: 0.5036461506004852 stddev: 0.28754747617166443 0.0:============================================================== 0.1:================================================================= 0.2:==================================================================== 0.3:================================================================ 0.4:================================================================ 0.5:================================================================= 0.6:====================================================================== 0.7:=================================================================== 0.8:=================================================================== 0.9:=================================================================
Run BASIC
<lang runbasic>call sample 100 call sample 1000 call sample 10000
end
sub sample n
dim samp(n)
for i =1 to n
samp(i) =rnd(1)
next i
' calculate mean, standard deviation
sum = 0
sumSq = 0
for i = 1 to n
sum = sum + samp(i)
sumSq = sumSq + samp(i)^2
next i
print n; " Samples used."
mean = sum / n
print "Mean = "; mean
print "Std Dev = "; (sumSq /n -mean^2)^0.5
'------- Show histogram
bins = 10
dim bins(bins)
for i = 1 to n
z = int(bins * samp(i))
bins(z) = bins(z) +1
next i
for b = 0 to bins -1
print b;" ";
for j = 1 to int(bins *bins(b)) /n *70
print "*";
next j
print
next b
print
end sub</lang>
100 Samples used. Mean = 0.514312738 Std Dev = 0.291627558 0 ************************************************************************************************** 1 ********************************************************************** 2 ********************* 3 *********************************** 4 *************************************************************** 5 ******************************************************************************************* 6 *********************************************************************************************************************** 7 ********************************************************************** 8 *************************************************************** 9 ********************************************************************** 1000 Samples used. Mean = 0.495704208 Std Dev = 0.281389168 0 *************************************************************** 1 ******************************************************************** 2 ************************************************************************** 3 ******************************************************************************* 4 ************************************************************************** 5 ********************************************************************** 6 ************************************************************************ 7 ********************************************************************** 8 ******************************************************** 9 ********************************************************************** 10000 Samples used. Mean = 0.493594211 Std Dev = 0.288635912 0 ************************************************************************ 1 ************************************************************************ 2 ********************************************************************** 3 ******************************************************************* 4 ********************************************************************** 5 ************************************************************************ 6 ************************************************************************ 7 ***************************************************************** 8 ********************************************************************** 9 ******************************************************************
Rust
<lang rust>#![feature(iter_arith)] extern crate rand;
use rand::distributions::{IndependentSample, Range};
pub fn mean(data: &[f32]) -> Option<f32> {
if data.is_empty() {
None
} else {
let sum: f32 = data.iter().sum();
Some(sum / data.len() as f32)
}
}
pub fn variance(data: &[f32]) -> Option<f32> {
if data.is_empty() {
None
} else {
let mean = mean(data).unwrap();
let mut sum = 0f32;
for &x in data {
sum += (x - mean).powi(2);
}
Some(sum / data.len() as f32)
}
}
pub fn standard_deviation(data: &[f32]) -> Option<f32> {
if data.is_empty() {
None
} else {
let variance = variance(data).unwrap();
Some(variance.sqrt())
}
}
fn print_histogram(width: u32, data: &[f32]) {
let mut histogram = [0; 10];
let len = histogram.len() as f32;
for &x in data {
histogram[(x * len) as usize] += 1;
}
let max_frequency = *histogram.iter().max().unwrap() as f32;
for (i, &frequency) in histogram.iter().enumerate() {
let bar_width = frequency as f32 * width as f32 / max_frequency;
print!("{:3.1}: ", i as f32 / len);
for _ in 0..bar_width as usize {
print!("*");
}
println!("");
}
}
fn main() {
let range = Range::new(0f32, 1f32); let mut rng = rand::thread_rng();
for &number_of_samples in [1000, 10_000, 1_000_000].iter() {
let mut data = vec![];
for _ in 0..number_of_samples {
let x = range.ind_sample(&mut rng);
data.push(x);
}
println!(" Statistics for sample size {}", number_of_samples);
println!("Mean: {:?}", mean(&data));
println!("Variance: {:?}", variance(&data));
println!("Standard deviation: {:?}", standard_deviation(&data));
print_histogram(40, &data);
}
}</lang>
- Output:
Statistics for sample size 1000 Mean: Some(0.50145197) Variance: Some(0.08201705) Standard deviation: Some(0.2863862) 0.0: ********************************* 0.1: **************************** 0.2: ********************************** 0.3: ************************************ 0.4: ************************************** 0.5: ********************************* 0.6: ****************************** 0.7: ****************************** 0.8: **************************************** 0.9: ****************************** Statistics for sample size 10000 Mean: Some(0.49700406) Variance: Some(0.08357173) Standard deviation: Some(0.28908777) 0.0: ************************************** 0.1: *************************************** 0.2: *************************************** 0.3: *************************************** 0.4: *********************************** 0.5: *************************************** 0.6: ************************************* 0.7: **************************************** 0.8: ************************************** 0.9: ************************************* Statistics for sample size 1000000 Mean: Some(0.50038373) Variance: Some(0.08325759) Standard deviation: Some(0.2885439) 0.0: *************************************** 0.1: *************************************** 0.2: *************************************** 0.3: **************************************** 0.4: *************************************** 0.5: *************************************** 0.6: *************************************** 0.7: *************************************** 0.8: *************************************** 0.9: ***************************************
Scala
<lang scala>def mean(a:Array[Double])=a.sum / a.size def stddev(a:Array[Double])={
val sum = a.fold(0.0)((a, b) => a + math.pow(b,2)) math.sqrt((sum/a.size) - math.pow(mean(a),2))
} def hist(a:Array[Double]) = {
val grouped=(SortedMap[Double, Array[Double]]() ++ (a groupBy (x => math.rint(x*10)/10))) grouped.map(v => (v._1, v._2.size))
} def printHist(a:Array[Double])=for((g,v) <- hist(a)){
println(s"$g: ${"*"*(205*v/a.size)} $v")
}
for(n <- Seq(100,1000,10000)){
val a = Array.fill(n)(Random.nextDouble)
println(s"$n numbers")
println(s"Mean: ${mean(a)}")
println(s"StdDev: ${stddev(a)}")
printHist(a)
println
}</lang>
- Output:
100 numbers Mean: 0.5151424022100874 StdDev: 0.25045766440922146 0.0: **** 2 0.1: **************** 8 0.2: **************** 8 0.3: ******************** 10 0.4: ************************ 12 0.5: ****************************** 15 0.6: ****************************** 15 0.7: **************** 8 0.8: ******************** 10 0.9: ********************** 11 1.0: ** 1 1000 numbers Mean: 0.4954605718792786 StdDev: 0.28350795290401604 0.0: ********* 48 0.1: ******************* 93 0.2: *********************** 117 0.3: ******************** 99 0.4: ***************** 87 0.5: ********************** 108 0.6: ************************* 122 0.7: ****************** 88 0.8: ******************** 100 0.9: ****************** 88 1.0: ********** 50 10000 numbers Mean: 0.502395544726441 StdDev: 0.2874443665645294 0.0: ********** 496 0.1: ******************** 979 0.2: ******************* 962 0.3: ******************** 1010 0.4: ******************** 998 0.5: ********************* 1035 0.6: ******************** 984 0.7: ********************* 1031 0.8: ********************* 1027 0.9: ******************** 991 1.0: ********* 487
Sidef
<lang ruby>func generate_statistics(n) {
var(sum=0, sum2=0); var hist = 10.of(0);
n.times {
var r = 1.rand;
sum += r;
sum2 += r**2;
hist[10*r] += 1;
}
var mean = sum/n; var stddev = Math.sqrt(sum2/n - mean**2);
say "size: #{n}";
say "mean: #{mean}";
say "stddev: #{stddev}";
var max = hist.max;
hist.range.each {|i|
printf("%.1f:%s\n", 0.1*i, "=" * 70*hist[i]/max);
}
print "\n";
}
[100, 1000, 10000].each {|n| generate_statistics(n) }</lang>
- Output:
size: 100 mean: 0.4585051431752446588 stddev: 0.2870559459562831101619581273667538623484 0.0:================================================================= 0.1:================================================== 0.2:====================================================================== 0.3:============================================= 0.4:======================================================= 0.5:============================== 0.6:================================================== 0.7:================================================== 0.8:================================================== 0.9:=================================== size: 1000 mean: 0.51292239343467439552 stddev: 0.2832968595790956540009121237087699143503 0.0:=================================================== 0.1:======================================================== 0.2:======================================================== 0.3:======================================================== 0.4:====================================================================== 0.5:================================================================== 0.6:=============================================================== 0.7:========================================================= 0.8:======================================================== 0.9:==================================================================== size: 10000 mean: 0.49883638025449614521145 stddev: 0.2898083000452161646017460189689302069547 0.0:==================================================================== 0.1:============================================================ 0.2:====================================================================== 0.3:============================================================== 0.4:=============================================================== 0.5:================================================================= 0.6:=============================================================== 0.7:================================================================= 0.8:================================================================== 0.9:===============================================================
Stata
For a uniform distribution on [0,1], the mean is 1/2 and the variance is 1/12 (hence the standard deviation is 0.28867513). With a large sample, one can check the convergence to these values.
<lang stata>. clear all . set obs 100000 number of observations (_N) was 0, now 100,000 . gen x=runiform() . summarize x
Variable | Obs Mean Std. Dev. Min Max
+---------------------------------------------------------
x | 100,000 .4991874 .2885253 1.18e-06 .9999939
. hist x</lang>
Tcl
<lang tcl>package require Tcl 8.5 proc stats {size} {
set sum 0.0
set sum2 0.0
for {set i 0} {$i < $size} {incr i} {
set r [expr {rand()}]
incr histo([expr {int(floor($r*10))}]) 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: $stddev"
foreach i {0 1 2 3 4 5 6 7 8 9} {
# The 205 is a magic factor stolen from the Go solution puts [string repeat "*" [expr {$histo($i)*205/int($size)}]]
}
}
stats 100 puts "" stats 1000 puts "" stats 10000</lang>
- Output:
100 numbers Mean: 0.4801193240797704 StdDev: 0.28697057708153784 ************** ********************************** ******************** ************** **************************** **************** ************** **************************** **************** **************** 1000 numbers Mean: 0.49478823525495275 StdDev: 0.2821543810265757 ******************* ****************** ************************ ******************** ******************* ********************** ********************* ******************** ****************** ****************** 10000 numbers Mean: 0.49928563715870816 StdDev: 0.2888258479070212 ******************** ********************* ******************** ******************** ******************* ********************* ******************* ******************** ********************* ********************
As can be seen, increasing the sample size reduces the variation between the buckets, showing that the rand() function at least approximates a uniform distribution. (Because Tcl 8.5 supports arbitrary precision integer arithmetic there is no reason in principle why the details for a trillion numbers couldn't be calculated, but it would take quite a while.)
VBA
<lang vb>Option Base 1 Private Function mean(s() As Variant) As Double
mean = WorksheetFunction.Average(s)
End Function Private Function standard_deviation(s() As Variant) As Double
standard_deviation = WorksheetFunction.StDev(s)
End Function Public Sub basic_statistics()
Dim s() As Variant
For e = 2 To 4
ReDim s(10 ^ e)
For i = 1 To 10 ^ e
s(i) = Rnd()
Next i
Debug.Print "sample size"; UBound(s), "mean"; mean(s), "standard deviation"; standard_deviation(s)
t = WorksheetFunction.Frequency(s, [{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}])
For i = 1 To 10
Debug.Print Format((i - 1) / 10, "0.00");
Debug.Print "-"; Format(i / 10, "0.00"),
Debug.Print String$(t(i, 1) / (10 ^ (e - 2)), "X");
Debug.Print
Next i
Debug.Print
Next e
End Sub</lang>
- Output:
sample size 100 mean 0,472405961751938 standard deviation 0,260463885857138 0,00-0,10 XXXXXX 0,10-0,20 XXXXXXXXX 0,20-0,30 XXXXXXXXXXXXXXX 0,30-0,40 XXXXXXXXXXXXXXX 0,40-0,50 XXXXXXXXXXXXXX 0,50-0,60 XXXXXXX 0,60-0,70 XXXXXXXXXXX 0,70-0,80 XXXXXXXX 0,80-0,90 XXXXXXXXXX 0,90-1,00 XXXXX sample size 1000 mean 0,500459910154343 standard deviation 0,278991757028358 0,00-0,10 XXXXXXXX 0,10-0,20 XXXXXXXXXX 0,20-0,30 XXXXXXXXXX 0,30-0,40 XXXXXXXXXX 0,40-0,50 XXXXXXXXXX 0,50-0,60 XXXXXXXXXXXX 0,60-0,70 XXXXXXXXXXX 0,70-0,80 XXXXXXXXX 0,80-0,90 XXXXXXXXX 0,90-1,00 XXXXXXXXXX sample size 10000 mean 0,496753623914719 standard deviation 0,28740805585887 0,00-0,10 XXXXXXXXXX 0,10-0,20 XXXXXXXXXX 0,20-0,30 XXXXXXXXXX 0,30-0,40 XXXXXXXXXX 0,40-0,50 XXXXXXXXXX 0,50-0,60 XXXXXXXXXX 0,60-0,70 XXXXXXXXXX 0,70-0,80 XXXXXXXXXX 0,80-0,90 XXXXXXXXXX 0,90-1,00 XXXXXXXXXX
zkl
<lang zkl>fcn mean(ns) { ns.sum(0.0)/ns.len() } fcn stdDev(ns){
m:=mean(ns); (ns.reduce('wrap(p,n){ x:=(n-m); p+x*x },0.0)/ns.len()).sqrt()
}</lang> <lang zkl>reg ns; foreach n in (T(100,1000,10000)){
ns=(0).pump(n,List,(0.0).random.fp(1.0));
println("N:%,6d mean:%.5f std dev:%.5f".fmt(n,mean(ns),stdDev(ns)));
} foreach r in ([0.0 .. 0.9, 0.1]){ // using the last data set (10000 randoms)
n:=ns.filter('wrap(x){ r<=x<(r+0.1) }).len();
println("%.2f..%.2f:%4d%s".fmt(r,r+0.1,n,"*"*(n/20)));
}</lang> (0.0).random(1.0) generates a [uniform] random number between 0 (inclusive) and 1 (exclusive).
- Output:
N: 100 mean:0.48521 std dev:0.27073 N: 1,000 mean:0.49362 std dev:0.28921 N:10,000 mean:0.49899 std dev:0.28813 0.00..0.10: 986************************************************* 0.10..0.20:1043**************************************************** 0.20..0.30: 992************************************************* 0.30..0.40: 974************************************************ 0.40..0.50:1001************************************************** 0.50..0.60: 998************************************************* 0.60..0.70: 995************************************************* 0.70..0.80:1043**************************************************** 0.80..0.90:1005************************************************** 0.90..1.00: 963************************************************
For the extra credit, pretend we have a device that spews random numbers in the range [0..1) forever. We connect this device to a measuring device that calculates mean and std deviation, printing results on a regular basis. <lang zkl>var pipe=Thread.Pipe(); // used to connect the two threads fcn{ while(1){ pipe.write((0.0).random(1.0)) } }.launch(); // generator fcn{ // consumer/calculator
N:=0; M:=SD:=sum:=ssum:=0.0;
while(1){
x:=pipe.read(); N+=1; sum+=x; ssum+=x*x;
M=sum/N; SD=(ssum/N - M*M).sqrt();
if(0==N%100000)
println("N:%,10d mean:%.5f std dev:%.5f".fmt(N,M,SD));
}
}.launch();
Atomic.sleep(60*60); // wait because exiting the VM kills the threads</lang>
- Output:
... N:45,800,000 mean:0.49997 std dev:0.28869 N:45,900,000 mean:0.49997 std dev:0.28869 N:46,000,000 mean:0.49997 std dev:0.28869 N:46,100,000 mean:0.49998 std dev:0.28869 N:46,200,000 mean:0.49997 std dev:0.28870 N:46,300,000 mean:0.49997 std dev:0.28870 N:46,400,000 mean:0.49997 std dev:0.28870 ...
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