Benford's law

From Rosetta Code
Task
Benford's law
You are encouraged to solve this task according to the task description, using any language you may know.
This page uses content from Wikipedia. The original article was at Benford's_law. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)


Benford's law, also called the first-digit law, refers to the frequency distribution of digits in many (but not all) real-life sources of data.

In this distribution, the number 1 occurs as the first digit about 30% of the time, while larger numbers occur in that position less frequently: 9 as the first digit less than 5% of the time. This distribution of first digits is the same as the widths of gridlines on a logarithmic scale.

Benford's law also concerns the expected distribution for digits beyond the first, which approach a uniform distribution.

This result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude.

A set of numbers is said to satisfy Benford's law if the leading digit   () occurs with probability

For this task, write (a) routine(s) to calculate the distribution of first significant (non-zero) digits in a collection of numbers, then display the actual vs. expected distribution in the way most convenient for your language (table / graph / histogram / whatever).

Use the first 1000 numbers from the Fibonacci sequence as your data set. No need to show how the Fibonacci numbers are obtained.

You can generate them or load them from a file; whichever is easiest.

Display your actual vs expected distribution.


For extra credit: Show the distribution for one other set of numbers from a page on Wikipedia. State which Wikipedia page it can be obtained from and what the set enumerates. Again, no need to display the actual list of numbers or the code to load them.


See also:



8th[edit]

 
: n:log10e ` 1 10 ln / ` ;
 
with: n
 
: n:log10 \ n -- n
ln log10e * ;
 
: benford \ x -- x
1 swap / 1+ log10 ;
 
: fibs \ xt n
swap >r
0.0 1.0 rot
( dup [email protected] w:exec tuck + ) swap times
2drop rdrop ;
 
var counts
 
: init
a:new ( 0 a:push ) 9 times counts ! ;
 
: leading \ n -- n
"%g" s:strfmt
0 s:@ '0 - nip ;
 
: bump-digit \ n --
counts @ swap 1- dup >r a:@
1+ r> swap a:! drop ;
 
: count-fibs \ --
( leading bump-digit ) 1000 fibs ;
 
: adjust \ --
counts @ ( 0.001 * ) a:map counts ! ;
 
: spaces \ n --
' space swap times ;
 
: .fw \ s n --
>r s:len r> rot . swap - spaces ;
 
: .header \ --
"Digit" 8 .fw "Expected" 10 .fw "Actual" 10 .fw cr ;
 
: .digit \ n --
>s 8 .fw ;
 
: .actual \ n --
"%.3f" s:strfmt 10 .fw ;
 
: .expected \ n --
"%.4f" s:strfmt 10 .fw ;
 
: report \ --
.header
counts @
( swap 1+ dup benford swap
.digit .expected .actual cr )
a:each drop ;
 
: benford-test
init count-fibs adjust report ;
 
;with
 
benford-test
bye
 
Output:
Digit   Expected  Actual
1       0.3010    0.301
2       0.1761    0.177
3       0.1249    0.125
4       0.0969    0.096
5       0.0792    0.080
6       0.0669    0.067
7       0.0580    0.056
8       0.0512    0.053
9       0.0458    0.045

Ada[edit]

The program reads the Fibonacci-Numbers from the standard input. Each input line is supposed to hold N, followed by Fib(N).

with Ada.Text_IO, Ada.Numerics.Generic_Elementary_Functions; 
 
procedure Benford is
 
subtype Nonzero_Digit is Natural range 1 .. 9;
function First_Digit(S: String) return Nonzero_Digit is
(if S(S'First) in '1' .. '9'
then Nonzero_Digit'Value(S(S'First .. S'First))
else First_Digit(S(S'First+1 .. S'Last)));
 
package N_IO is new Ada.Text_IO.Integer_IO(Natural);
 
procedure Print(D: Nonzero_Digit; Counted, Sum: Natural) is
package Math is new Ada.Numerics.Generic_Elementary_Functions(Float);
package F_IO is new Ada.Text_IO.Float_IO(Float);
Actual: constant Float := Float(Counted) / Float(Sum);
Expected: constant Float := Math.Log(1.0 + 1.0 / Float(D), Base => 10.0);
Deviation: constant Float := abs(Expected-Actual);
begin
N_IO.Put(D, 5);
N_IO.Put(Counted, 14);
F_IO.Put(Float(Sum)*Expected, Fore => 16, Aft => 1, Exp => 0);
F_IO.Put(100.0*Actual, Fore => 9, Aft => 2, Exp => 0);
F_IO.Put(100.0*Expected, Fore => 11, Aft => 2, Exp => 0);
F_IO.Put(100.0*Deviation, Fore => 13, Aft => 2, Exp => 0);
end Print;
 
Cnt: array(Nonzero_Digit) of Natural := (1 .. 9 => 0);
D: Nonzero_Digit;
Sum: Natural := 0;
Counter: Positive;
 
begin
while not Ada.Text_IO.End_Of_File loop
-- each line in the input file holds Counter, followed by Fib(Counter)
N_IO.Get(Counter);
-- Counter and skip it, we just don't need it
D := First_Digit(Ada.Text_IO.Get_Line);
-- read the rest of the line and extract the first digit
Cnt(D) := Cnt(D)+1;
Sum := Sum + 1;
end loop;
Ada.Text_IO.Put_Line(" Digit Found[total] Expected[total] Found[%]"
& " Expected[%] Difference[%]");
for I in Nonzero_Digit loop
Print(I, Cnt(I), Sum);
Ada.Text_IO.New_Line;
end loop;
end Benford;
Output:
>./benford < fibo.txt
 Digit  Found[total]   Expected[total]    Found[%]   Expected[%]   Difference[%]
    1           301             301.0       30.10         30.10            0.00
    2           177             176.1       17.70         17.61            0.09
    3           125             124.9       12.50         12.49            0.01
    4            96              96.9        9.60          9.69            0.09
    5            80              79.2        8.00          7.92            0.08
    6            67              66.9        6.70          6.69            0.01
    7            56              58.0        5.60          5.80            0.20
    8            53              51.2        5.30          5.12            0.18
    9            45              45.8        4.50          4.58            0.08

Extra Credit[edit]

Input is the list of primes below 100,000 from [1]. Since each line in that file holds prime and only a prime, but no ongoing counter, we must slightly modify the program by commenting out a single line:

      -- N_IO.Get(Counter);

We can also edit out the declaration of the variable "Counter" ...or live with a compiler warning about never reading or assigning that variable.

Output:

As it turns out, the distribution of the first digits of primes is almost flat and does not seem follow Benford's law:

>./benford < primes-to-100k.txt 
 Digit  Found[total]   Expected[total]    Found[%]   Expected[%]   Difference[%]
    1          1193            2887.5       12.44         30.10           17.67
    2          1129            1689.1       11.77         17.61            5.84
    3          1097            1198.4       11.44         12.49            1.06
    4          1069             929.6       11.14          9.69            1.45
    5          1055             759.5       11.00          7.92            3.08
    6          1013             642.2       10.56          6.69            3.87
    7          1027             556.3       10.71          5.80            4.91
    8          1003             490.7       10.46          5.12            5.34
    9          1006             438.9       10.49          4.58            5.91

Aime[edit]

text
sum(text a, text b)
{
data d;
integer e, f, n, r;
 
e = length(a);
f = length(b);
 
r = 0;
 
n = min(e, f);
while (n) {
n -= 1;
e -= 1;
f -= 1;
r += a[e] - '0';
r += b[f] - '0';
b_insert(d, 0, r % 10 + '0');
r /= 10;
}
 
if (f) {
e = f;
a = b;
}
 
while (e) {
e -= 1;
r += a[e] - '0';
b_insert(d, 0, r % 10 + '0');
r /= 10;
}
 
if (r) {
b_insert(d, 0, r + '0');
}
 
return b_string(d);
}
 
text
fibs(list l, integer n)
{
integer c, i;
text a, b, w;
 
l_r_integer(l, 1, 1);
 
a = "0";
b = "1";
i = 1;
while (i < n) {
w = sum(a, b);
a = b;
b = w;
c = w[0] - '0';
l_r_integer(l, c, 1 + l_q_integer(l, c));
i += 1;
}
 
return w;
}
 
integer
main(void)
{
integer i, n;
list f;
real m;
 
n = 1000;
 
i = 10;
while (i) {
i -= 1;
lb_p_integer(f, 0);
}
 
fibs(f, n);
 
m = 100r / n;
 
o_text("\t\texpected\t found\n");
i = 0;
while (i < 9) {
i += 1;
o_winteger(8, i);
o_wpreal(16, 3, 3, 100 * log10(1 + 1r / i));
o_wpreal(16, 3, 3, l_q_integer(f, i) * m);
o_text("\n");
}
 
return 0;
}
Output:
		expected	   found
       1          30.102          30.1  
       2          17.609          17.7  
       3          12.493          12.5  
       4           9.691           9.600
       5           7.918           8    
       6           6.694           6.7  
       7           5.799           5.6  
       8           5.115           5.300
       9           4.575           4.5  

ALGOL 68[edit]

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Uses Algol 68G's LONG LONG INT which has programmer specifiable precision.

BEGIN
# set the number of digits for LONG LONG INT values #
PR precision 256 PR
# returns the probability of the first digit of each non-zero number in s #
PROC digit probability = ( []LONG LONG INT s )[]REAL:
BEGIN
[ 1 : 9 ]REAL result;
# count the number of times each digit is the first #
[ 1 : 9 ]INT count := ( 0, 0, 0, 0, 0, 0, 0, 0, 0 );
FOR i FROM LWB s TO UPB s DO
LONG LONG INT v := ABS s[ i ];
IF v /= 0 THEN
WHILE v > 9 DO v OVERAB 10 OD;
count[ SHORTEN SHORTEN v ] +:= 1
FI
OD;
# calculate the probability of each digit #
INT number of elements = ( UPB s + 1 ) - LWB s;
FOR i TO 9 DO
result[ i ] := IF number of elements = 0 THEN 0 ELSE count[ i ] / number of elements FI
OD;
result
END # digit probability # ;
# outputs the digit probabilities of some numbers and those expected by Benford's law #
PROC compare to benford = ( []REAL actual )VOID:
FOR i TO 9 DO
print( ( "Benford: ", fixed( log( 1 + ( 1 / i ) ), -7, 3 ), " actual: ", fixed( actual[ i ], -7, 3 ), newline ) )
OD # compare to benford # ;
# generate 1000 fibonacci numbers #
[ 0 : 1000 ]LONG LONG INT fn;
fn[ 0 ] := 0;
fn[ 1 ] := 1;
FOR i FROM 2 TO UPB fn DO fn[ i ] := fn[ i - 1 ] + fn[ i - 2 ] OD;
# get the probabilities of each first digit of the fibonacci numbers and #
# compare to the probabilities expected by Benford's law #
compare to benford( digit probability( fn ) )
END
Output:
Benford:   0.301 actual:   0.301
Benford:   0.176 actual:   0.177
Benford:   0.125 actual:   0.125
Benford:   0.097 actual:   0.096
Benford:   0.079 actual:   0.080
Benford:   0.067 actual:   0.067
Benford:   0.058 actual:   0.056
Benford:   0.051 actual:   0.053
Benford:   0.046 actual:   0.045

AutoHotkey[edit]

Works with: AutoHotkey_L
(AutoHotkey1.1+)
SetBatchLines, -1
fib := NStepSequence(1, 1, 2, 1000)
Out := "Digit`tExpected`tObserved`tDeviation`n"
n := []
for k, v in fib
d := SubStr(v, 1, 1)
, n[d] := n[d] ? n[d] + 1 : 1
for k, v in n
Exppected := 100 * Log(1+ (1 / k))
, Observed := (v / fib.MaxIndex()) * 100
, Out .= k "`t" Exppected "`t" Observed "`t" Abs(Exppected - Observed) "`n"
MsgBox, % Out
 
NStepSequence(v1, v2, n, k) {
a := [v1, v2]
Loop, % k - 2 {
a[j := A_Index + 2] := 0
Loop, % j < n + 2 ? j - 1 : n
a[j] := BigAdd(a[j - A_Index], a[j])
}
return, a
}
 
BigAdd(a, b) {
if (StrLen(b) > StrLen(a))
t := a, a := b, b := t
LenA := StrLen(a) + 1, LenB := StrLen(B) + 1, Carry := 0
Loop, % LenB - 1
Sum := SubStr(a, LenA - A_Index, 1) + SubStr(B, LenB - A_Index, 1) + Carry
, Carry := Sum // 10
, Result := Mod(Sum, 10) . Result
Loop, % I := LenA - LenB {
if (!Carry) {
Result := SubStr(a, 1, I) . Result
break
}
Sum := SubStr(a, I, 1) + Carry
, Carry := Sum // 10
, Result := Mod(Sum, 10) . Result
, I--
}
return, (Carry ? Carry : "") . Result
}
NStepSequence() is available here.

Output:

Digit	Expected	Observed	Deviation
1	30.103000	30.100000	0.003000
2	17.609126	17.700000	0.090874
3	12.493874	12.500000	0.006126
4	9.691001	9.600000	0.091001
5	7.918125	8.000000	0.081875
6	6.694679	6.700000	0.005321
7	5.799195	5.600000	0.199195
8	5.115252	5.300000	0.184748
9	4.575749	4.500000	0.075749

AWK[edit]

 
# syntax: GAWK -f BENFORDS_LAW.AWK
BEGIN {
n = 1000
for (i=1; i<=n; i++) {
arr[substr(fibonacci(i),1,1)]++
}
print("digit expected observed deviation")
for (i=1; i<=9; i++) {
expected = log10(i+1) - log10(i)
actual = arr[i] / n
deviation = expected - actual
printf("%5d %8.4f %8.4f %9.4f\n",i,expected*100,actual*100,abs(deviation*100))
}
exit(0)
}
function fibonacci(n, a,b,c,i) {
a = 0
b = 1
for (i=1; i<=n; i++) {
c = a + b
a = b
b = c
}
return(c)
}
function abs(x) { if (x >= 0) { return x } else { return -x } }
function log10(x) { return log(x)/log(10) }
 
Output:
digit expected observed deviation
    1  30.1030  30.0000    0.1030
    2  17.6091  17.7000    0.0909
    3  12.4939  12.5000    0.0061
    4   9.6910   9.6000    0.0910
    5   7.9181   8.0000    0.0819
    6   6.6947   6.7000    0.0053
    7   5.7992   5.7000    0.0992
    8   5.1153   5.3000    0.1847
    9   4.5757   4.5000    0.0757

C[edit]

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
 
float *benford_distribution(void)
{
static float prob[9];
for (int i = 1; i < 10; i++)
prob[i - 1] = log10f(1 + 1.0 / i);
 
return prob;
}
 
float *get_actual_distribution(char *fn)
{
FILE *input = fopen(fn, "r");
if (!input)
{
perror("Can't open file");
exit(EXIT_FAILURE);
}
 
int tally[9] = { 0 };
char c;
int total = 0;
while ((c = getc(input)) != EOF)
{
/* get the first nonzero digit on the current line */
while (c < '1' || c > '9')
c = getc(input);
 
tally[c - '1']++;
total++;
 
/* discard rest of line */
while ((c = getc(input)) != '\n' && c != EOF)
;
}
fclose(input);
 
static float freq[9];
for (int i = 0; i < 9; i++)
freq[i] = tally[i] / (float) total;
 
return freq;
}
 
int main(int argc, char **argv)
{
if (argc != 2)
{
printf("Usage: benford <file>\n");
return EXIT_FAILURE;
}
 
float *actual = get_actual_distribution(argv[1]);
float *expected = benford_distribution();
 
puts("digit\tactual\texpected");
for (int i = 0; i < 9; i++)
printf("%d\t%.3f\t%.3f\n", i + 1, actual[i], expected[i]);
 
return EXIT_SUCCESS;
}
Output:

Use with a file which should contain a number on each line.

$ ./benford fib1000.txt
digit   actual  expected
1       0.301   0.301
2       0.177   0.176
3       0.125   0.125
4       0.096   0.097
5       0.080   0.079
6       0.067   0.067
7       0.056   0.058
8       0.053   0.051
9       0.045   0.046

C++[edit]

//to cope with the big numbers , I used the Class Library for Numbers( CLN ) 
//if used prepackaged you can compile writing "g++ -std=c++11 -lcln yourprogram.cpp -o yourprogram"
#include <cln/integer.h>
#include <cln/integer_io.h>
#include <iostream>
#include <algorithm>
#include <vector>
#include <iomanip>
#include <sstream>
#include <string>
#include <cstdlib>
#include <cmath>
#include <map>
using namespace cln ;
 
class NextNum {
public :
NextNum ( cl_I & a , cl_I & b ) : first( a ) , second ( b ) { }
cl_I operator( )( ) {
cl_I result = first + second ;
first = second ;
second = result ;
return result ;
}
private :
cl_I first ;
cl_I second ;
} ;
 
void findFrequencies( const std::vector<cl_I> & fibos , std::map<int , int> &numberfrequencies ) {
for ( cl_I bignumber : fibos ) {
std::ostringstream os ;
fprintdecimal ( os , bignumber ) ;//from header file cln/integer_io.h
int firstdigit = std::atoi( os.str( ).substr( 0 , 1 ).c_str( )) ;
auto result = numberfrequencies.insert( std::make_pair( firstdigit , 1 ) ) ;
if ( ! result.second )
numberfrequencies[ firstdigit ]++ ;
}
}
 
int main( ) {
std::vector<cl_I> fibonaccis( 1000 ) ;
fibonaccis[ 0 ] = 0 ;
fibonaccis[ 1 ] = 1 ;
cl_I a = 0 ;
cl_I b = 1 ;
//since a and b are passed as references to the generator's constructor
//they are constantly changed !
std::generate_n( fibonaccis.begin( ) + 2 , 998 , NextNum( a , b ) ) ;
std::cout << std::endl ;
std::map<int , int> frequencies ;
findFrequencies( fibonaccis , frequencies ) ;
std::cout << " found expected\n" ;
for ( int i = 1 ; i < 10 ; i++ ) {
double found = static_cast<double>( frequencies[ i ] ) / 1000 ;
double expected = std::log10( 1 + 1 / static_cast<double>( i )) ;
std::cout << i << " :" << std::setw( 16 ) << std::right << found * 100 << " %" ;
std::cout.precision( 3 ) ;
std::cout << std::setw( 26 ) << std::right << expected * 100 << " %\n" ;
}
return 0 ;
}
 
Output:
                found                    expected
1 :            30.1 %                      30.1 %
2 :            17.7 %                      17.6 %
3 :            12.5 %                      12.5 %
4 :             9.5 %                      9.69 %
5 :               8 %                      7.92 %
6 :             6.7 %                      6.69 %
7 :             5.6 %                       5.8 %
8 :             5.3 %                      5.12 %
9 :             4.5 %                      4.58 %

Clojure[edit]

(ns example
(:gen-class))
 
(defn abs [x]
(if (> x 0)
x
(- x)))
 
(defn calc-benford-stats [digits]
" Frequencies of digits in data "
(let [y (frequencies digits)
tot (reduce + (vals y))]
[y tot]))
 
(defn show-benford-stats [v]
" Prints in percent the actual, Benford expected, and difference"
(let [fd (map (comp first str) v)] ; first digit of each record
(doseq [q (range 1 10)
:let [[y tot] (calc-benford-stats fd)
d (first (str q)) ; reference digit
f (/ (get y d 0) tot 0.01) ; percent of occurence of digit
p (* (Math/log10 (/ (inc q) q)) 100) ; Benford expected percent
e (abs (- f p))]] ; error (difference)
(println (format "%3d %10.2f %10.2f %10.2f"
q
f
p
e)))))
 
; Generate fibonacci results
(def fib (lazy-cat [0N 1N] (map + fib (rest fib))))
 
;(def fib-digits (map (comp first str) (take 10000 fib)))
(def fib-digits (take 10000 fib))
(def header " found-% expected-% diff")
 
(println "Fibonacci Results")
(println header)
(show-benford-stats fib-digits)
;
; Universal Constants from Physics (using first column of data)
(println "Universal Constants from Physics")
(println header)
(let [
data-parser (fn [s]
(let [x (re-find #"\s{10}-?[0|/\.]*([1-9])" s)]
(if (not (nil? x)) ; Skips records without number
(second x)
x)))
 
input (slurp "http://physics.nist.gov/cuu/Constants/Table/allascii.txt")
 
y (for [line (line-seq (java.io.BufferedReader.
(java.io.StringReader. input)))]
(data-parser line))
z (filter identity y)]
(show-benford-stats z))
 
; Sunspots
(println "Sunspots average count per month since 1749")
(println header)
(let [
data-parser (fn [s]
(nth (re-find #"(.+?\s){3}([1-9])" s) 2))
 
; Sunspot data loaded from file (saved from ;https://solarscience.msfc.nasa.gov/greenwch/SN_m_tot_V2.0.txt")
; (note: attempting to load directly from url causes https Trust issues, so saved to file after loading to Browser)
input (slurp "SN_m_tot_V2.0.txt")
y (for [line (line-seq (java.io.BufferedReader.
(java.io.StringReader. input)))]
(data-parser line))]
 
(show-benford-stats y))
 
 
Output:
Fibonacci Results
         found-%    expected-%  diff
  1      30.11      30.10       0.01
  2      17.62      17.61       0.01
  3      12.49      12.49       0.00
  4       9.68       9.69       0.01
  5       7.92       7.92       0.00
  6       6.68       6.69       0.01
  7       5.80       5.80       0.00
  8       5.13       5.12       0.01
  9       4.56       4.58       0.02
Universal Constants from Physics
         found-%    expected-%  diff
  1      34.34      30.10       4.23
  2      18.67      17.61       1.07
  3       9.04      12.49       3.46
  4       8.43       9.69       1.26
  5       8.43       7.92       0.52
  6       7.23       6.69       0.53
  7       3.31       5.80       2.49
  8       5.12       5.12       0.01
  9       5.42       4.58       0.85
Sunspots average count per month since 1749
         found-%    expected-%  diff
  1      37.44      30.10       7.34
  2      16.28      17.61       1.33
  3       7.16      12.49       5.34
  4       6.88       9.69       2.81
  5       6.35       7.92       1.57
  6       6.04       6.69       0.66
  7       7.25       5.80       1.45
  8       5.57       5.12       0.46
  9       5.76       4.58       1.18

CoffeeScript[edit]

fibgen = () ->
a = 1; b = 0
return () ->
([a, b] = [b, a+b])[1]
 
leading = (x) -> x.toString().charCodeAt(0) - 0x30
 
f = fibgen()
 
benford = (0 for i in [1..9])
benford[leading(f()) - 1] += 1 for i in [1..1000]
 
log10 = (x) -> Math.log(x) * Math.LOG10E
 
actual = benford.map (x) -> x * 0.001
expected = (log10(1 + 1/x) for x in [1..9])
 
console.log "Leading digital distribution of the first 1,000 Fibonacci numbers"
console.log "Digit\tActual\tExpected"
for i in [1..9]
console.log i + "\t" + actual[i - 1].toFixed(3) + '\t' + expected[i - 1].toFixed(3)
Output:
Leading digital distribution of the first 1,000 Fibonacci numbers
Digit   Actual  Expected
1       0.301   0.301
2       0.177   0.176
3       0.125   0.125
4       0.096   0.097
5       0.080   0.079
6       0.067   0.067
7       0.056   0.058
8       0.053   0.051
9       0.045   0.046

Common Lisp[edit]

(defun calculate-distribution (numbers)
"Return the frequency distribution of the most significant nonzero
digits in the given list of numbers. The first element of the list
is the frequency for digit 1, the second for digit 2, and so on."

 
(defun nonzero-digit-p (c)
"Check whether the character is a nonzero digit"
(and (digit-char-p c) (char/= c #\0)))
 
(defun first-digit (n)
"Return the most significant nonzero digit of the number or NIL if
there is none."

(let* ((s (write-to-string n))
(c (find-if #'nonzero-digit-p s)))
(when c
(digit-char-p c))))
 
(let ((tally (make-array 9 :element-type 'integer :initial-element 0)))
(loop for n in numbers
for digit = (first-digit n)
when digit
do (incf (aref tally (1- digit))))
(loop with total = (length numbers)
for digit-count across tally
collect (/ digit-count total))))
 
(defun calculate-benford-distribution ()
"Return the frequency distribution according to Benford's law.
The first element of the list is the probability for digit 1, the second
element the probability for digit 2, and so on."

(loop for i from 1 to 9
collect (log (1+ (/ i)) 10)))
 
(defun benford (numbers)
"Print a table of the actual and expected distributions for the given
list of numbers."

(let ((actual-distribution (calculate-distribution numbers))
(expected-distribution (calculate-benford-distribution)))
(write-line "digit actual expected")
(format T "~:{~3D~9,3F~8,3F~%~}"
(map 'list #'list '(1 2 3 4 5 6 7 8 9)
actual-distribution
expected-distribution))))
; *fib1000* is a list containing the first 1000 numbers in the Fibonnaci sequence
> (benford *fib1000*)
digit actual expected
  1    0.301   0.301
  2    0.177   0.176
  3    0.125   0.125
  4    0.096   0.097
  5    0.080   0.079
  6    0.067   0.067
  7    0.056   0.058
  8    0.053   0.051
  9    0.045   0.046

D[edit]

Translation of: Scala
import std.stdio, std.range, std.math, std.conv, std.bigint;
 
double[2][9] benford(R)(R seq) if (isForwardRange!R && !isInfinite!R) {
typeof(return) freqs = 0;
uint seqLen = 0;
foreach (d; seq)
if (d != 0) {
freqs[d.text[0] - '1'][1]++;
seqLen++;
}
 
foreach (immutable i, ref p; freqs)
p = [log10(1.0 + 1.0 / (i + 1)), p[1] / seqLen];
return freqs;
}
 
void main() {
auto fibs = recurrence!q{a[n - 1] + a[n - 2]}(1.BigInt, 1.BigInt);
 
writefln("%9s %9s %9s", "Actual", "Expected", "Deviation");
foreach (immutable i, immutable p; fibs.take(1000).benford)
writefln("%d: %5.2f%% | %5.2f%% | %5.4f%%",
i+1, p[1] * 100, p[0] * 100, abs(p[1] - p[0]) * 100);
}
Output:
   Actual  Expected Deviation
1: 30.10% | 30.10% | 0.0030%
2: 17.70% | 17.61% | 0.0908%
3: 12.50% | 12.49% | 0.0061%
4:  9.60% |  9.69% | 0.0910%
5:  8.00% |  7.92% | 0.0818%
6:  6.70% |  6.69% | 0.0053%
7:  5.60% |  5.80% | 0.1992%
8:  5.30% |  5.12% | 0.1847%
9:  4.50% |  4.58% | 0.0757%

Alternative Version[edit]

The output is the same.

import std.stdio, std.range, std.math, std.conv, std.bigint,
std.algorithm, std.array;
 
auto benford(R)(R seq) if (isForwardRange!R && !isInfinite!R) {
return seq.filter!q{a != 0}.map!q{a.text[0]-'1'}.array.sort().group;
}
 
void main() {
auto fibs = recurrence!q{a[n - 1] + a[n - 2]}(1.BigInt, 1.BigInt);
auto expected = iota(1, 10).map!(d => log10(1.0 + 1.0 / d));
 
enum N = 1_000;
writefln("%9s %9s %9s", "Actual", "Expected", "Deviation");
foreach (immutable i, immutable f; fibs.take(N).benford)
writefln("%d: %5.2f%% | %5.2f%% | %5.4f%%", i + 1,
f * 100.0 / N, expected[i] * 100,
abs((f / double(N)) - expected[i]) * 100);
}

Elixir[edit]

defmodule Benfords_law do
def distribution(n), do: :math.log10( 1 + (1 / n) )
 
def task(total \\ 1000) do
IO.puts "Digit Actual Benfords expected"
fib(total)
|> Enum.group_by(fn i -> hd(to_char_list(i)) end)
|> Enum.map(fn {key,list} -> {key - ?0, length(list)} end)
|> Enum.sort
|> Enum.each(fn {x,len} -> IO.puts "#{x} #{len / total} #{distribution(x)}" end)
end
 
defp fib(n) do # suppresses zero
Stream.unfold({1,1}, fn {a,b} -> {a,{b,a+b}} end) |> Enum.take(n)
end
end
 
Benfords_law.task
Output:
Digit   Actual  Benfords expected
1       0.301   0.3010299956639812
2       0.177   0.17609125905568124
3       0.125   0.12493873660829993
4       0.096   0.09691001300805642
5       0.08    0.07918124604762482
6       0.067   0.06694678963061322
7       0.056   0.05799194697768673
8       0.053   0.05115252244738129
9       0.045   0.04575749056067514

Erlang[edit]

 
-module( benfords_law ).
-export( [actual_distribution/1, distribution/1, task/0] ).
 
actual_distribution( Ns ) -> lists:foldl( fun first_digit_count/2, dict:new(), Ns ).
 
distribution( N ) -> math:log10( 1 + (1 / N) ).
 
task() ->
Total = 1000,
Fibonaccis = fib( Total ),
Actual_dict = actual_distribution( Fibonaccis ),
Keys = lists:sort( dict:fetch_keys( Actual_dict) ),
io:fwrite( "Digit Actual Benfords expected~n" ),
[io:fwrite("~p ~p ~p~n", [X, dict:fetch(X, Actual_dict) / Total, distribution(X)]) || X <- Keys].
 
 
 
fib( N ) -> fib( N, 0, 1, [] ).
fib( 0, Current, _, Acc ) -> lists:reverse( [Current | Acc] );
fib( N, Current, Next, Acc ) -> fib( N-1, Next, Current+Next, [Current | Acc] ).
 
first_digit_count( 0, Dict ) -> Dict;
first_digit_count( N, Dict ) ->
[Key | _] = erlang:integer_to_list( N ),
dict:update_counter( Key - 48, 1, Dict ).
 
Output:
7> benfords_law:task().
Digit   Actual  Benfords expected
1       0.301   0.3010299956639812
2       0.177   0.17609125905568124
3       0.125   0.12493873660829993
4       0.096   0.09691001300805642
5       0.08    0.07918124604762482
6       0.067   0.06694678963061322
7       0.056   0.05799194697768673
8       0.053   0.05115252244738129
9       0.045   0.04575749056067514

Forth[edit]

: 3drop   drop 2drop ;
: f2drop fdrop fdrop ;
 
: int-array create cells allot does> swap cells + ;
 
: 1st-fib 0e 1e ;
: next-fib ftuck f+ ;
 
: 1st-digit ( fp -- n )
pad 6 represent 3drop pad [email protected] [char] 0 - ;
 
10 int-array counts
 
: tally
0 counts 10 cells erase
1st-fib
1000 0 DO
1 fdup 1st-digit counts +!
next-fib
LOOP f2drop ;
 
: benford ( d -- fp )
s>f 1/f 1e f+ flog ;
 
: tab 9 emit ;
 
: heading ( -- )
cr ." Leading digital distribution of the first 1,000 Fibonacci numbers:"
cr ." Digit" tab ." Actual" tab ." Expected" ;
 
: .fixed ( n -- ) \ print count as decimal fraction
s>d <# # # # [char] . hold #s #> type space ;
 
: report ( -- )
precision 3 set-precision
heading
10 1 DO
cr i 3 .r
tab i counts @ .fixed
tab i benford f.
LOOP
set-precision ;
 
: compute-benford tally report ;
Output:
Gforth 0.7.2, Copyright (C) 1995-2008 Free Software Foundation, Inc.
Gforth comes with ABSOLUTELY NO WARRANTY; for details type `license'
Type `bye' to exit
compute-benford 
Leading digital distribution of the first 1,000 Fibonacci numbers:
Digit	Actual	Expected
  1	0.301 	0.301 
  2	0.177 	0.176 
  3	0.125 	0.125 
  4	0.096 	0.0969 
  5	0.080 	0.0792 
  6	0.067 	0.0669 
  7	0.056 	0.058 
  8	0.053 	0.0512 
  9	0.045 	0.0458  ok

Fortran[edit]

FORTRAN 90. Compilation and output of this program using emacs compile command and a fairly obvious Makefile entry:

-*- mode: compilation; default-directory: "/tmp/" -*-
Compilation started at Sat May 18 01:13:00
 
a=./f && make $a && $a
f95 -Wall -ffree-form f.F -o f
0.301030010 0.176091254 0.124938756 9.69100147E-02 7.91812614E-02 6.69467747E-02 5.79919666E-02 5.11525236E-02 4.57575098E-02 THE LAW
0.300999999 0.177000001 0.125000000 9.60000008E-02 7.99999982E-02 6.70000017E-02 5.70000000E-02 5.29999994E-02 4.50000018E-02 LEADING FIBONACCI DIGIT
 
Compilation finished at Sat May 18 01:13:00
subroutine fibber(a,b,c,d)
! compute most significant digits, Fibonacci like.
implicit none
integer (kind=8), intent(in) :: a,b
integer (kind=8), intent(out) :: c,d
d = a + b
if (15 .lt. log10(float(d))) then
c = b/10
d = d/10
else
c = b
endif
end subroutine fibber
 
integer function leadingDigit(a)
implicit none
integer (kind=8), intent(in) :: a
integer (kind=8) :: b
b = a
do while (9 .lt. b)
b = b/10
end do
leadingDigit = transfer(b,leadingDigit)
end function leadingDigit
 
real function benfordsLaw(a)
implicit none
integer, intent(in) :: a
benfordsLaw = log10(1.0 + 1.0 / a)
end function benfordsLaw
 
program benford
 
implicit none
 
interface
 
subroutine fibber(a,b,c,d)
implicit none
integer (kind=8), intent(in) :: a,b
integer (kind=8), intent(out) :: c,d
end subroutine fibber
 
integer function leadingDigit(a)
implicit none
integer (kind=8), intent(in) :: a
end function leadingDigit
 
real function benfordsLaw(a)
implicit none
integer, intent(in) :: a
end function benfordsLaw
 
end interface
 
integer (kind=8) :: a, b, c, d
integer :: i, count(10)
data count/10*0/
a = 1
b = 1
do i = 1, 1001
count(leadingDigit(a)) = count(leadingDigit(a)) + 1
call fibber(a,b,c,d)
a = c
b = d
end do
write(6,*) (benfordsLaw(i),i=1,9),'THE LAW'
write(6,*) (count(i)/1000.0 ,i=1,9),'LEADING FIBONACCI DIGIT'
end program benford

FreeBASIC[edit]

Library: GMP
' version 27-10-2016
' compile with: fbc -s console
 
#Define max 1000 ' total number of Fibonacci numbers
#Define max_sieve 15485863 ' should give 1,000,000
 
#Include Once "gmp.bi" ' uses the GMP libary
 
Dim As ZString Ptr z_str
Dim As ULong n, d
ReDim As ULong digit(1 To 9)
Dim As Double expect, found
 
Dim As mpz_ptr fib1, fib2
fib1 = Allocate(Len(__mpz_struct)) : Mpz_init_set_ui(fib1, 0)
fib2 = Allocate(Len(__mpz_struct)) : Mpz_init_set_ui(fib2, 1)
 
digit(1) = 1 ' fib2
For n = 2 To max
Swap fib1, fib2 ' fib1 = 1, fib2 = 0
mpz_add(fib2, fib1, fib2) ' fib1 = 1, fib2 = 1 (fib1 + fib2)
z_str = mpz_get_str(0, 10, fib2)
d = Val(Left(*z_str, 1)) ' strip the 1 digit on the left off
digit(d) = digit(d) +1
Next
 
mpz_clear(fib1) : DeAllocate(fib1)
mpz_clear(fib2) : DeAllocate(fib2)
 
Print
Print "First 1000 Fibonacci numbers"
Print "nr: total found expected difference"
 
For d = 1 To 9
n = digit(d)
found = n / 10
expect = (Log(1 + 1 / d) / Log(10)) * 100
Print Using " ## ##### ###.## % ###.## % ##.### %"; _
d; n ; found; expect; expect - found
Next
 
 
ReDim digit(1 To 9)
ReDim As UByte sieve(max_sieve)
 
'For d = 4 To max_sieve Step 2
' sieve(d) = 1
'Next
Print : Print "start sieve"
For d = 3 To sqr(max_sieve)
If sieve(d) = 0 Then
For n = d * d To max_sieve Step d * 2
sieve(n) = 1
Next
End If
Next
 
digit(2) = 1 ' 2
 
Print "start collecting first digits"
For n = 3 To max_sieve Step 2
If sieve(n) = 0 Then
d = Val(Left(Trim(Str(n)), 1))
digit(d) = digit(d) +1
End If
Next
 
Dim As ulong total
For n = 1 To 9
total = total + digit(n)
Next
 
Print
Print "First";total; " primes"
Print "nr: total found expected difference"
 
For d = 1 To 9
n = digit(d)
found = n / total * 100
expect = (Log(1 + 1 / d) / Log(10)) * 100
Print Using " ## ######## ###.## % ###.## % ###.### %"; _
d; n ; found; expect; expect - found
Next
 
 
' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
Output:
First 1000 Fibonacci numbers
nr:  total     found   expected  difference
  1    301   30.10 %    30.10 %     0.003 %
  2    177   17.70 %    17.61 %    -0.091 %
  3    125   12.50 %    12.49 %    -0.006 %
  4     96    9.60 %     9.69 %     0.091 %
  5     80    8.00 %     7.92 %    -0.082 %
  6     67    6.70 %     6.69 %    -0.005 %
  7     56    5.60 %     5.80 %     0.199 %
  8     53    5.30 %     5.12 %    -0.185 %
  9     45    4.50 %     4.58 %     0.076 %

start sieve
start collecting first digits

First1000000 primes
nr:     total     found   expected   difference
  1    415441   41.54 %    30.10 %    -11.441 %
  2     77025    7.70 %    17.61 %      9.907 %
  3     75290    7.53 %    12.49 %      4.965 %
  4     74114    7.41 %     9.69 %      2.280 %
  5     72951    7.30 %     7.92 %      0.623 %
  6     72257    7.23 %     6.69 %     -0.531 %
  7     71564    7.16 %     5.80 %     -1.357 %
  8     71038    7.10 %     5.12 %     -1.989 %
  9     70320    7.03 %     4.58 %     -2.456 %

Go[edit]

package main
 
import (
"fmt"
"math"
)
 
func Fib1000() []float64 {
a, b, r := 0., 1., [1000]float64{}
for i := range r {
r[i], a, b = b, b, b+a
}
return r[:]
}
 
func main() {
show(Fib1000(), "First 1000 Fibonacci numbers")
}
 
func show(c []float64, title string) {
var f [9]int
for _, v := range c {
f[fmt.Sprintf("%g", v)[0]-'1']++
}
fmt.Println(title)
fmt.Println("Digit Observed Predicted")
for i, n := range f {
fmt.Printf("  %d  %9.3f  %8.3f\n", i+1, float64(n)/float64(len(c)),
math.Log10(1+1/float64(i+1)))
}
}
Output:
First 1000 Fibonacci numbers
Digit  Observed  Predicted
  1      0.301     0.301
  2      0.177     0.176
  3      0.125     0.125
  4      0.096     0.097
  5      0.080     0.079
  6      0.067     0.067
  7      0.056     0.058
  8      0.053     0.051
  9      0.045     0.046

Haskell[edit]

import qualified Data.Map as M
import Data.Char (digitToInt)
 
fstdigit :: Integer -> Int
fstdigit = digitToInt . head . show
 
n = 1000::Int
fibs = 1:1:zipWith (+) fibs (tail fibs)
fibdata = map fstdigit $ take n fibs
freqs = M.fromListWith (+) $ zip fibdata (repeat 1)
 
tab :: [(Int, Double, Double)]
tab = [(d,
(fromIntegral (M.findWithDefault 0 d freqs) /(fromIntegral n) ),
logBase 10.0 $ 1 + 1/(fromIntegral d) ) | d<-[1..9]]
 
main = print tab
Output:
[(1,0.301,0.301029995663981),
(2,0.177,0.176091259055681),
(3,0.125,0.1249387366083),
(4,0.096,0.0969100130080564),
(5,0.08,0.0791812460476248),
(6,0.067,0.0669467896306132),
(7,0.056,0.0579919469776867),
(8,0.053,0.0511525224473813),
(9,0.045,0.0457574905606751)]

Icon and Unicon[edit]

The following solution works in both languages.

global counts, total
 
procedure main()
 
counts := table(0)
total := 0.0
every benlaw(fib(1 to 1000))
 
every i := 1 to 9 do
write(i,": ",right(100*counts[string(i)]/total,9)," ",100*P(i))
 
end
 
procedure benlaw(n)
if counts[n ? (tab(upto('123456789')),move(1))] +:= 1 then total +:= 1
end
 
procedure P(d)
return log(1+1.0/d, 10)
end
 
procedure fib(n) # From Fibonacci Sequence task
return fibMat(n)[1]
end
 
procedure fibMat(n)
if n <= 0 then return [0,0]
if n = 1 then return [1,0]
fp := fibMat(n/2)
c := fp[1]*fp[1] + fp[2]*fp[2]
d := fp[1]*(fp[1]+2*fp[2])
if n%2 = 1 then return [c+d, d]
else return [d, c]
end

Sample run:

->benlaw
1:      30.1 30.10299956639811
2:      17.7 17.60912590556812
3:      12.5 12.49387366082999
4:       9.6 9.69100130080564
5:       8.0 7.918124604762481
6:       6.7 6.694678963061322
7:       5.6 5.799194697768673
8:       5.3 5.115252244738128
9:       4.5 4.575749056067514
->

J[edit]

We show the correlation coefficient of Benford's law with the leading digits of the first 1000 Fibonacci numbers is almost unity.

log10 =: 10&^.
benford =: [email protected]:(1+%)
assert '0.30 0.18 0.12 0.10 0.08 0.07 0.06 0.05 0.05' -: 5j2 ": benford >: i. 9
 
 
append_next_fib =: , +/@:(_2&{.)
assert 5 8 13 -: append_next_fib 5 8
 
leading_digits =: {[email protected]":&>
assert '581' -: leading_digits 5 8 13x
 
count =: #/.~ /: ~.
assert 2 1 3 4 -: count 'XCXBAXACXC' NB. 2 A's, 1 B, 3 C's, and some X's.
 
normalize =: % +/
assert 1r3 2r3 -: normalize 1 2x
 
FIB =: append_next_fib ^: (1000-#) 1 1
LDF =: leading_digits FIB
 
 
TALLY_BY_KEY =: count LDF
assert 9 -: # TALLY_BY_KEY NB. If all of [1-9] are present then we know what the digits are.
 
mean =: +/ % #
center=: - mean
mp =: $:~ :(+/ .*)
num =: mp&:center
den =: %:@:(*&:(+/@:(*:@:center)))
r =: num % den NB. r is the LibreOffice correl function
assert '_0.982' -: 6j3 ": 1 2 3 r 6 5 3 NB. confirmed using LibreOffice correl function
 
 
assert '0.9999' -: 6j4 ": (normalize TALLY_BY_KEY) r benford >: i.9
 
assert '0.9999' -: 6j4 ": TALLY_BY_KEY r benford >: i.9 NB. Of course we don't need normalization

Java[edit]

import java.math.BigInteger;
import java.util.Locale;
 
public class BenfordsLaw {
 
private static BigInteger[] generateFibonacci(int n) {
BigInteger[] fib = new BigInteger[n];
fib[0] = BigInteger.ONE;
fib[1] = BigInteger.ONE;
for (int i = 2; i < fib.length; i++) {
fib[i] = fib[i - 2].add(fib[i - 1]);
}
return fib;
}
 
public static void main(String[] args) {
BigInteger[] numbers = generateFibonacci(1000);
 
int[] firstDigits = new int[10];
for (BigInteger number : numbers) {
firstDigits[Integer.valueOf(number.toString().substring(0, 1))]++;
}
 
for (int i = 1; i < firstDigits.length; i++) {
System.out.printf(Locale.ROOT, "%d %10.6f %10.6f%n",
i, (double) firstDigits[i] / numbers.length, Math.log10(1.0 + 1.0 / i));
}
}
}

The output is:

1   0.301000   0.301030
2   0.177000   0.176091
3   0.125000   0.124939
4   0.096000   0.096910
5   0.080000   0.079181
6   0.067000   0.066947
7   0.056000   0.057992
8   0.053000   0.051153
9   0.045000   0.045757

To use other number sequences, implement a suitable NumberGenerator, construct a Benford instance with it and print it.

jq[edit]

Works with: jq version 1.4

This implementation shows the observed and expected number of occurrences together with the χ² statistic.

For the sake of being self-contained, the following includes a generator for Fibonacci numbers, and a prime number generator that is inefficient but brief and can generate numbers within an arbitrary range.
# Generate the first n Fibonacci numbers: 1, 1, ...
# Numerical accuracy is insufficient beyond about 1450.
def fibonacci(n):
# input: [f(i-2), f(i-1), countdown]
def fib: (.[0] + .[1]) as $sum
| if .[2] <= 0 then empty
elif .[2] == 1 then $sum
else $sum, ([ .[1], $sum, .[2] - 1 ] | fib)
end;
[1, 0, n] | fib ;
 
# is_prime is tailored to work with jq 1.4
def is_prime:
if . == 2 then true
else 2 < . and . % 2 == 1 and
. as $in
| (($in + 1) | sqrt) as $m
| (((($m - 1) / 2) | floor) + 1) as $max
| reduce range(1; $max) as $i
(true; if . then ($in % ((2 * $i) + 1)) > 0 else false end)
end ;
 
# primes in [m,n)
def primes(m;n):
range(m;n) | select(is_prime);
 
def runs:
reduce .[] as $item
( [];
if . == [] then [ [ $item, 1] ]
else .[length-1] as $last
| if $last[0] == $item
then (.[0:length-1] + [ [$item, $last[1] + 1] ] )
else . + [[$item, 1]]
end
end ) ;
 
# Inefficient but brief:
def histogram: sort | runs;
 
def benford_probability:
tonumber
| if . > 0 then ((1 + (1 /.)) | log) / (10|log)
else 0
end ;
 
# benford takes a stream and produces an array of [ "d", observed, expected ]
def benford(stream):
[stream | tostring | .[0:1] ] | histogram as $histogram
| reduce ($histogram | .[] | .[0]) as $digit
([]; . + [$digit, ($digit|benford_probability)] )
| map(select(type == "number")) as $probabilities
| ([ $histogram | .[] | .[1] ] | add) as $total
| reduce range(0; $histogram|length) as $i
([]; . + ([$histogram[$i] + [$total * $probabilities[$i]] ] ) ) ;
 
# given an array of [value, observed, expected] values,
# produce the χ² statistic
def chiSquared:
reduce .[] as $triple
(0;
if $triple[2] == 0 then .
else . + ($triple[1] as $o | $triple[2] as $e | ($o - $e) | (.*.)/$e)
end) ;
 
# truncate n places after the decimal point;
# return a string since it can readily be converted back to a number
def precision(n):
tostring as $s | $s | index(".")
| if . then $s[0:.+n+1] else $s end ;
 
# Right-justify but do not truncate
def rjustify(n):
length as $length | if n <= $length then . else " " * (n-$length) + . end;
 
# Attempt to align decimals so integer part is in a field of width n
def align(n):
index(".") as $ix
| if n < $ix then .
elif $ix then (.[0:$ix]|rjustify(n)) +.[$ix:]
else rjustify(n)
end ;
 
# given an array of [value, observed, expected] values,
# produce rows of the form: value observed expected
def print_rows(prec):
.[] | map( precision(prec)|align(5) + " ") | add ;
 
def report(heading; stream):
benford(stream) as $array
| heading,
" Digit Observed Expected",
( $array | print_rows(2) ),
"",
" χ² = \( $array | chiSquared | precision(4))",
""
;
 
def task:
report("First 100 fibonacci numbers:"; fibonacci( 100) ),
report("First 1000 fibonacci numbers:"; fibonacci(1000) ),
report("Primes less than 1000:"; primes(2;1000)),
report("Primes between 1000 and 10000:"; primes(1000;10000)),
report("Primes less than 100000:"; primes(2;100000))
;
 
task
Output:
First 100 fibonacci numbers:
 Digit Observed Expected
    1     30     30.10  
    2     18     17.60  
    3     13     12.49  
    4      9      9.69  
    5      8      7.91  
    6      6      6.69  
    7      5      5.79  
    8      7      5.11  
    9      4      4.57  

 χ² = 1.0287

First 1000 fibonacci numbers:
 Digit Observed Expected
    1    301    301.02  
    2    177    176.09  
    3    125    124.93  
    4     96     96.91  
    5     80     79.18  
    6     67     66.94  
    7     56     57.99  
    8     53     51.15  
    9     45     45.75  

 χ² = 0.1694

Primes less than 1000:
 Digit Observed Expected
    1     25     50.57  
    2     19     29.58  
    3     19     20.98  
    4     20     16.28  
    5     17     13.30  
    6     18     11.24  
    7     18      9.74  
    8     17      8.59  
    9     15      7.68  

 χ² = 45.0162

Primes between 1000 and 10000:
 Digit Observed Expected
    1    135    319.39  
    2    127    186.83  
    3    120    132.55  
    4    119    102.82  
    5    114     84.01  
    6    117     71.03  
    7    107     61.52  
    8    110     54.27  
    9    112     48.54  

 χ² = 343.5583

Primes less than 100000:
 Digit Observed Expected
    1   1193   2887.47  
    2   1129   1689.06  
    3   1097   1198.41  
    4   1069    929.56  
    5   1055    759.50  
    6   1013    642.15  
    7   1027    556.25  
    8   1003    490.65  
    9   1006    438.90  

 χ² = 3204.8072

Julia[edit]

fib(n) = ([one(n) one(n) ; one(n) zero(n)]^n)[1,2]
 
ben(l) = [count(x->x==i, map(n->string(n)[1],l)) for i='1':'9']./length(l)
 
benford(l) = [Number[1:9;] ben(l) log10(1.+1./[1:9;])]
Output:
julia> benford([fib(big(n)) for n = 1:1000])
9x3 Array{Number,2}:
 1  0.301  0.30103  
 2  0.177  0.176091 
 3  0.125  0.124939 
 4  0.096  0.09691  
 5  0.08   0.0791812
 6  0.067  0.0669468
 7  0.056  0.0579919
 8  0.053  0.0511525
 9  0.045  0.0457575

Kotlin[edit]

import java.math.BigInteger
 
interface NumberGenerator {
val numbers: Array<BigInteger>
}
 
class Benford(ng: NumberGenerator) {
override fun toString() = str
 
private val firstDigits = IntArray(9)
private val count = ng.numbers.size.toDouble()
private val str: String
 
init {
for (n in ng.numbers) {
firstDigits[n.toString().substring(0, 1).toInt() - 1]++
}
 
str = with(StringBuilder()) {
for (i in firstDigits.indices) {
append(i + 1).append('\t').append(firstDigits[i] / count)
append('\t').append(Math.log10(1 + 1.0 / (i + 1))).append('\n')
}
 
toString()
}
}
}
 
object FibonacciGenerator : NumberGenerator {
override val numbers: Array<BigInteger> by lazy {
val fib = Array<BigInteger>(1000, { BigInteger.ONE })
for (i in 2 until fib.size)
fib[i] = fib[i - 2].add(fib[i - 1])
fib
}
}
 
fun main(a: Array<String>) = println(Benford(FibonacciGenerator))

Liberty BASIC[edit]

Using function from http://rosettacode.org/wiki/Fibonacci_sequence#Liberty_BASIC

 
dim bin(9)
 
N=1000
for i = 0 to N-1
num$ = str$(fiboI(i))
d=val(left$(num$,1))
'print num$, d
bin(d)=bin(d)+1
next
print
 
print "Digit", "Actual freq", "Expected freq"
for i = 1 to 9
print i, bin(i)/N, using("#.###", P(i))
next
 
 
function P(d)
P = log10(d+1)-log10(d)
end function
 
function log10(x)
log10 = log(x)/log(10)
end function
 
function fiboI(n)
a = 0
b = 1
for i = 1 to n
temp = a + b
a = b
b = temp
next i
fiboI = a
end function
 
Output:
Digit         Actual freq   Expected freq
1             0.301         0.301
2             0.177         0.176
3             0.125         0.125
4             0.095         0.097
5             0.08          0.079
6             0.067         0.067
7             0.056         0.058
8             0.053         0.051
9             0.045         0.046

Lua[edit]

actual = {}
expected = {}
for i = 1, 9 do
actual[i] = 0
expected[i] = math.log10(1 + 1 / i)
end
 
n = 0
file = io.open("fibs1000.txt", "r")
for line in file:lines() do
digit = string.byte(line, 1) - 48
actual[digit] = actual[digit] + 1
n = n + 1
end
file:close()
 
print("digit actual expected")
for i = 1, 9 do
print(i, actual[i] / n, expected[i])
end
Output:
digit   actual  expected
1       0.301   0.30102999566398
2       0.177   0.17609125905568
3       0.125   0.1249387366083
4       0.096   0.096910013008056
5       0.08    0.079181246047625
6       0.067   0.066946789630613
7       0.056   0.057991946977687
8       0.053   0.051152522447381
9       0.045   0.045757490560675

Mathematica / Wolfram Language[edit]

fibdata = Array[[email protected]@[email protected]# &, 1000]; 
Table[{d, [email protected][fibdata, d]/[email protected], Log10[1. + 1/d]}, {d, 1,
9}] // Grid
Output:
1	0.301	0.30103
2	0.177	0.176091
3	0.125	0.124939
4	0.096	0.09691
5	0.08	0.0791812
6	0.067	0.0669468
7	0.056	0.0579919
8	0.053	0.0511525
9	0.045	0.0457575

NetRexx[edit]

/* NetRexx */
options replace format comments java crossref symbols nobinary
 
runSample(arg)
return
 
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
method brenfordDeveation(nlist = Rexx[]) public static
observed = 0
loop n_ over nlist
d1 = n_.left(1)
if d1 = 0 then iterate n_
observed[d1] = observed[d1] + 1
end n_
say ' '.right(4) 'Observed'.right(11) 'Expected'.right(11) 'Deviation'.right(11)
loop n_ = 1 to 9
actual = (observed[n_] / (nlist.length - 1))
expect = Rexx(Math.log10(1 + 1 / n_))
deviat = expect - actual
say n_.right(3)':' (actual * 100).format(3, 6)'%' (expect * 100).format(3, 6)'%' (deviat * 100).abs().format(3, 6)'%'
end n_
return
 
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
method fibonacciList(size = 1000) public static returns Rexx[]
fibs = Rexx[size + 1]
fibs[0] = 0
fibs[1] = 1
loop n_ = 2 to size
fibs[n_] = fibs[n_ - 1] + fibs[n_ - 2]
end n_
return fibs
 
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
method runSample(arg) private static
parse arg n_ .
if n_ = '' then n_ = 1000
fibList = fibonacciList(n_)
say 'Fibonacci sequence to' n_
brenfordDeveation(fibList)
return
 
Output:
Fibonacci sequence to 1000
        Observed    Expected   Deviation
  1:  30.100000%  30.103000%   0.003000%
  2:  17.700000%  17.609126%   0.090874%
  3:  12.500000%  12.493874%   0.006127%
  4:   9.600000%   9.691001%   0.091001%
  5:   8.000000%   7.918125%   0.081875%
  6:   6.700000%   6.694679%   0.005321%
  7:   5.600000%   5.799195%   0.199195%
  8:   5.300000%   5.115252%   0.184748%
  9:   4.500000%   4.575749%   0.075749%

Oberon-2[edit]

Works with: oo2c version 2
 
MODULE BenfordLaw;
IMPORT
LRealStr,
LRealMath,
Out := NPCT:Console;
 
VAR
r: ARRAY 1000 OF LONGREAL;
d: ARRAY 10 OF LONGINT;
a: LONGREAL;
i: LONGINT;
 
PROCEDURE Fibb(VAR r: ARRAY OF LONGREAL);
VAR
i: LONGINT;
BEGIN
r[0] := 1.0;r[1] := 1.0;
FOR i := 2 TO LEN(r) - 1 DO
r[i] := r[i - 2] + r[i - 1]
END
END Fibb;
 
PROCEDURE Dist(r [NO_COPY]: ARRAY OF LONGREAL; VAR d: ARRAY OF LONGINT);
VAR
i: LONGINT;
str: ARRAY 256 OF CHAR;
BEGIN
FOR i := 0 TO LEN(r) - 1 DO
LRealStr.RealToStr(r[i],str);
INC(d[ORD(str[0]) - ORD('0')])
END
END Dist;
 
BEGIN
Fibb(r);
Dist(r,d);
Out.String("First 1000 fibonacci numbers: ");Out.Ln;
Out.String(" digit ");Out.String(" observed ");Out.String(" predicted ");Out.Ln;
FOR i := 1 TO LEN(d) - 1 DO
a := LRealMath.ln(1.0 + 1.0 / i ) / LRealMath.ln(10);
Out.Int(i,5);Out.LongRealFix(d[i] / 1000.0,9,3);Out.LongRealFix(a,10,3);Out.Ln
END
END BenfordLaw.
 
Output:
First 1000 fibonacci numbers: 
 digit  observed  predicted 
    1    0.301     0.301
    2    0.177     0.176
    3    0.125     0.125
    4    0.096     0.097
    5    0.080     0.079
    6    0.067     0.067
    7    0.056     0.058
    8    0.053     0.051
    9    0.045     0.046

OCaml[edit]

For the Fibonacci sequence, we use the function from https://rosettacode.org/wiki/Fibonacci_sequence#Arbitrary_Precision
Note the remark about the compilation of the program there.

 
open Num
 
let fib =
let rec fib_aux f0 f1 = function
| 0 -> f0
| 1 -> f1
| n -> fib_aux f1 (f1 +/ f0) (n - 1)
in
fib_aux (num_of_int 0) (num_of_int 1) ;;
 
let create_fibo_string = function n -> string_of_num (fib n) ;;
let rec range i j = if i > j then [] else i :: (range (i + 1) j)
 
let n_max = 1000 ;;
 
let numbers = range 1 n_max in
let get_first_digit = function s -> Char.escaped (String.get s 0) in
let first_digits = List.map get_first_digit (List.map create_fibo_string numbers) in
let data = Array.create 9 0 in
let fill_data vec = function n -> vec.(n - 1) <- vec.(n - 1) + 1 in
List.iter (fill_data data) (List.map int_of_string first_digits) ;
Printf.printf "\nFrequency of the first digits in the Fibonacci sequence:\n" ;
Array.iter (Printf.printf "%f ")
(Array.map (fun x -> (float x) /. float (n_max)) data) ;
 
let xvalues = range 1 9 in
let benfords_law = function x -> log10 (1.0 +. 1.0 /. float (x)) in
Printf.printf "\nPrediction of Benford's law:\n " ;
List.iter (Printf.printf "%f ") (List.map benfords_law xvalues) ;
Printf.printf "\n" ;;
 
Output:
Frequency of the first digits in the Fibonacci sequence:
0.301000 0.177000 0.125000 0.096000 0.080000 0.067000 0.056000 0.053000 0.045000 
Prediction of Benford's law:
 0.301030 0.176091 0.124939 0.096910 0.079181 0.066947 0.057992 0.051153 0.045757

PARI/GP[edit]

distribution(v)={
my(t=vector(9,n,sum(i=1,#v,v[i]==n)));
print("Digit\tActual\tExpected");
for(i=1,9,print(i, "\t", t[i], "\t", round(#v*(log(i+1)-log(i))/log(10))))
};
dist(f)=distribution(vector(1000,n,digits(f(n))[1]));
lucas(n)=fibonacci(n-1)+fibonacci(n+1);
dist(fibonacci)
dist(lucas)
Output:
Digit   Actual  Expected
1       301     301
2       177     176
3       125     125
4       96      97
5       80      79
6       67      67
7       56      58
8       53      51
9       45      46

Digit   Actual  Expected
1       301     301
2       174     176
3       127     125
4       97      97
5       79      79
6       66      67
7       59      58
8       51      51
9       46      46

Pascal[edit]

program fibFirstdigit;
{$IFDEF FPC}{$MODE Delphi}{$ELSE}{$APPTYPE CONSOLE}{$ENDIF}
uses
sysutils;
type
tDigitCount = array[0..9] of LongInt;
var
s: Ansistring;
dgtCnt,
expectedCnt : tDigitCount;
 
procedure GetFirstDigitFibonacci(var dgtCnt:tDigitCount;n:LongInt=1000);
//summing up only the first 9 digits
//n = 1000 -> difference to first 9 digits complete fib < 100 == 2 digits
var
a,b,c : LongWord;//about 9.6 decimals
Begin
for a in dgtCnt do dgtCnt[a] := 0;
a := 0;b := 1;
while n > 0 do
Begin
c := a+b;
//overflow? round and divide by base 10
IF c < a then
Begin a := (a+5) div 10;b := (b+5) div 10;c := a+b;end;
a := b;b := c;
s := IntToStr(a);inc(dgtCnt[Ord(s[1])-Ord('0')]);
dec(n);
end;
end;
 
procedure InitExpected(var dgtCnt:tDigitCount;n:LongInt=1000);
var
i: integer;
begin
for i := 1 to 9 do
dgtCnt[i] := trunc(n*ln(1 + 1 / i)/ln(10));
end;
 
var
reldiff: double;
i,cnt: integer;
begin
cnt := 1000;
InitExpected(expectedCnt,cnt);
GetFirstDigitFibonacci(dgtCnt,cnt);
writeln('Digit count expected rel diff');
For i := 1 to 9 do
Begin
reldiff := 100*(expectedCnt[i]-dgtCnt[i])/expectedCnt[i];
writeln(i:5,dgtCnt[i]:7,expectedCnt[i]:10,reldiff:10:5,' %');
end;
end.
Digit  Count  Expected  rel Diff
    1    301       301   0.00000 %
    2    177       176  -0.56818 %
    3    125       124  -0.80645 %
    4     96        96   0.00000 %
    5     80        79  -1.26582 %
    6     67        66  -1.51515 %
    7     56        57   1.75439 %
    8     53        51  -3.92157 %
    9     45        45   0.00000 %

Perl[edit]

#!/usr/bin/perl
use strict ;
use warnings ;
use POSIX qw( log10 ) ;
 
my @fibonacci = ( 0 , 1 ) ;
while ( @fibonacci != 1000 ) {
push @fibonacci , $fibonacci[ -1 ] + $fibonacci[ -2 ] ;
}
my @actuals ;
my @expected ;
for my $i( 1..9 ) {
my $sum = 0 ;
map { $sum++ if $_ =~ /\A$i/ } @fibonacci ;
push @actuals , $sum / 1000 ;
push @expected , log10( 1 + 1/$i ) ;
}
print " Observed Expected\n" ;
for my $i( 1..9 ) {
print "$i : " ;
my $result = sprintf ( "%.2f" , 100 * $actuals[ $i - 1 ] ) ;
printf "%11s %%" , $result ;
$result = sprintf ( "%.2f" , 100 * $expected[ $i - 1 ] ) ;
printf "%15s %%\n" , $result ;
}
Output:
 
         Observed         Expected
1 :       30.10 %          30.10 %
2 :       17.70 %          17.61 %
3 :       12.50 %          12.49 %
4 :        9.50 %           9.69 %
5 :        8.00 %           7.92 %
6 :        6.70 %           6.69 %
7 :        5.60 %           5.80 %
8 :        5.30 %           5.12 %
9 :        4.50 %           4.58 %

Perl 6[edit]

Works with: rakudo version 2016-10-24
sub benford(@a) { bag +« @a».substr(0,1) }
 
sub show(%distribution) {
printf "%9s %9s  %s\n", <Actual Expected Deviation>;
for 1 .. 9 -> $digit {
my $actual = %distribution{$digit} * 100 / [+] %distribution.values;
my $expected = (1 + 1 / $digit).log(10) * 100;
printf "%d: %5.2f%% | %5.2f%% | %.2f%%\n",
$digit, $actual, $expected, abs($expected - $actual);
}
}
 
multi MAIN($file) { show benford $file.IO.lines }
multi MAIN() { show benford ( 1, 1, 2, *+* ... * )[^1000] }

Output: First 1000 Fibonaccis

   Actual  Expected  Deviation
1: 30.10% | 30.10% | 0.00%
2: 17.70% | 17.61% | 0.09%
3: 12.50% | 12.49% | 0.01%
4:  9.60% |  9.69% | 0.09%
5:  8.00% |  7.92% | 0.08%
6:  6.70% |  6.69% | 0.01%
7:  5.60% |  5.80% | 0.20%
8:  5.30% |  5.12% | 0.18%
9:  4.50% |  4.58% | 0.08%

Extra credit: Square Kilometers of land under cultivation, by country / territory. First column from Wikipedia: Land use statistics by country.

   Actual  Expected  Deviation
1: 33.33% | 30.10% | 3.23%
2: 18.31% | 17.61% | 0.70%
3: 13.15% | 12.49% | 0.65%
4:  8.45% |  9.69% | 1.24%
5:  9.39% |  7.92% | 1.47%
6:  5.63% |  6.69% | 1.06%
7:  4.69% |  5.80% | 1.10%
8:  5.16% |  5.12% | 0.05%
9:  1.88% |  4.58% | 2.70%

Phix[edit]

Translation of: Go
procedure main(sequence s, string title)
sequence f = repeat(0,9)
for i=1 to length(s) do
f[sprint(s[i])[1]-'0'] += 1
end for
puts(1,title)
puts(1,"Digit Observed% Predicted%\n")
for i=1 to length(f) do
printf(1,"  %d  %9.3f  %8.3f\n", {i, f[i]/length(s)*100, log10(1+1/i)*100})
end for
end procedure
main(fib(1000),"First 1000 Fibonacci numbers\n")
main(primes(10000),"First 10000 Prime numbers\n")
main(threes(500),"First 500 powers of three\n")

Supporting staff:

function fib(integer lim)
atom a=0, b=1
sequence res = repeat(0,lim)
for i=1 to lim do
{res[i], a, b} = {b, b, b+a}
end for
return res
end function
 
function primes(integer lim)
integer n = 1, k, p
sequence res = {2}
while length(res)<lim do
k = 3
p = 1
n += 2
while k*k<=n and p do
p = floor(n/k)*k!=n
k += 2
end while
if p then
res = append(res,n)
end if
end while
return res
end function
 
function threes(integer lim)
sequence res = repeat(0,lim)
for i=1 to lim do
res[i] = power(3,i)
end for
return res
end function
 
constant INVLN10 = 0.43429_44819_03251_82765
function log10(object x1)
return log(x1) * INVLN10
end function
Output:
(put into columns by hand)
First 1000 Fibonacci numbers            First 10000 Prime numbers               First 500 powers of three
Digit  Observed%  Predicted%            Digit  Observed%  Predicted%            Digit  Observed%  Predicted%
  1     30.100    30.103                  1     16.010    30.103                  1     30.000    30.103
  2     17.700    17.609                  2     11.290    17.609                  2     17.600    17.609
  3     12.500    12.494                  3     10.970    12.494                  3     12.400    12.494
  4      9.600     9.691                  4     10.690     9.691                  4      9.800     9.691
  5      8.000     7.918                  5     10.550     7.918                  5      8.000     7.918
  6      6.700     6.695                  6     10.130     6.695                  6      6.600     6.695
  7      5.600     5.799                  7     10.270     5.799                  7      5.800     5.799
  8      5.300     5.115                  8     10.030     5.115                  8      5.200     5.115
  9      4.500     4.576                  9     10.060     4.576                  9      4.600     4.576

PL/I[edit]

 
(fofl, size, subrg):
Benford: procedure options(main); /* 20 October 2013 */
declare sc(1000) char(1), f(1000) float (16);
declare d fixed (1);
 
call Fibonacci(f);
call digits(sc, f);
 
put skip list ('digit expected obtained');
do d= 1 upthru 9;
put skip edit (d, log10(1 + 1/d), tally(sc, trim(d))/1000)
(f(3), 2 f(13,8) );
end;
 
Fibonacci: procedure (f);
declare f(*) float (16);
declare i fixed binary;
 
f(1), f(2) = 1;
do i = 3 to 1000;
f(i) = f(i-1) + f(i-2);
end;
end Fibonacci;
 
digits: procedure (sc, f);
declare sc(*) char(1), f(*) float (16);
sc = substr(trim(f), 1, 1);
end digits;
 
tally: procedure (sc, d) returns (fixed binary);
declare sc(*) char(1), d char(1);
declare (i, t) fixed binary;
t = 0;
do i = 1 to 1000;
if sc(i) = d then t = t + 1;
end;
return (t);
end tally;
end Benford;
 

Results:

digit  expected     obtained 
  1   0.30103000   0.30099487
  2   0.17609126   0.17698669
  3   0.12493874   0.12500000
  4   0.09691001   0.09599304
  5   0.07918125   0.07998657
  6   0.06694679   0.06698608
  7   0.05799195   0.05599976
  8   0.05115252   0.05299377
  9   0.04575749   0.04499817

PL/pgSQL[edit]

 
WITH recursive
constant(val) AS
(
SELECT 1000.
)
,
fib(a,b) AS
(
SELECT CAST(0 AS NUMERIC), CAST(1 AS NUMERIC)
UNION ALL
SELECT b,a+b
FROM fib
)
,
benford(first_digit, probability_real, probability_theoretical) AS
(
SELECT *,
CAST(log(1. + 1./CAST(first_digit AS INT)) AS NUMERIC(5,4)) probability_theoretical
FROM (
SELECT first_digit, CAST(COUNT(1)/(SELECT val FROM constant) AS NUMERIC(5,4)) probability_real FROM
(
SELECT SUBSTRING(CAST(a AS VARCHAR(100)),1,1) first_digit
FROM fib
WHERE SUBSTRING(CAST(a AS VARCHAR(100)),1,1) <> '0'
LIMIT (SELECT val FROM constant)
) t
GROUP BY first_digit
) f
ORDER BY first_digit ASC
)
SELECT *
FROM benford CROSS JOIN
(SELECT CAST(corr(probability_theoretical,probability_real) AS NUMERIC(5,4)) correlation
FROM benford) c
 

PowerShell[edit]

The sample file was not found. I selected another that contained the first two-thousand in the Fibonacci sequence, so there is a small amount of extra filtering.

 
$url = "https://oeis.org/A000045/b000045.txt"
$file = "$env:TEMP\FibonacciNumbers.txt"
(New-Object System.Net.WebClient).DownloadFile($url, $file)
 
$benford = Get-Content -Path $file |
Select-Object -Skip 1 -First 1000 |
ForEach-Object {(($_ -split " ")[1].ToString().ToCharArray())[0]} |
Group-Object |
Select-Object -Property @{Name="Digit"  ; Expression={[int]($_.Name)}},
Count,
@{Name="Actual"  ; Expression={$_.Count/1000}},
@{Name="Expected"; Expression={[double]("{0:f5}" -f [Math]::Log10(1 + 1 / $_.Name))}}
 
$benford | Sort-Object -Property Digit | Format-Table -AutoSize
 
Remove-Item -Path $file -Force -ErrorAction SilentlyContinue
 
Output:
Digit Count Actual Expected
----- ----- ------ --------
    1   301  0.301  0.30103
    2   177  0.177  0.17609
    3   125  0.125  0.12494
    4    96  0.096  0.09691
    5    80   0.08  0.07918
    6    67  0.067  0.06695
    7    56  0.056  0.05799
    8    53  0.053  0.05115
    9    45  0.045  0.04576

Prolog[edit]

Works with: SWI Prolog version 6.2.6 by Jan Wielemaker, University of Amsterdam

Note: SWI Prolog implements arbitrary precision integer arithmetic through use of the GNU MP library

%_________________________________________________________________
% Does the Fibonacci sequence follow Benford's law?
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
% Fibonacci sequence generator
fib(C, [P,S], C, N) :- N is P + S.
fib(C, [P,S], Cv, V) :- succ(C, Cn), N is P + S, !, fib(Cn, [S,N], Cv, V).
 
fib(0, 0).
fib(1, 1).
fib(C, N) :- fib(2, [0,1], C, N). % Generate from 3rd sequence on
 
% The benford law calculated
benford(D, Val) :- Val is log10(1+1/D).
 
% Retrieves the first characters of the first 1000 fibonacci numbers
% (excluding zero)
firstchar(V) :-
fib(C,N), N =\= 0, atom_chars(N, [Ch|_]), number_chars(V, [Ch]),
(C>999-> !; true).
 
% Increment the n'th list item (1 based), result -> third argument.
incNth(1, [Dh|Dt], [Ch|Dt]) :- !, succ(Dh, Ch).
incNth(H, [Dh|Dt], [Dh|Ct]) :- succ(Hn, H), !, incNth(Hn, Dt, Ct).
 
% Calculate the frequency of the all the list items
freq([], D, D).
freq([H|T], D, C) :- incNth(H, D, L), !, freq(T, L, C).
 
freq([H|T], Freq) :-
length([H|T], Len), min_list([H|T], Min), max_list([H|T], Max),
findall(0, between(Min,Max,_), In),
freq([H|T], In, F), % Frequency stored in F
findall(N, (member(V, F), N is V/Len), Freq). % Normalise F->Freq
 
% Output the results
writeHdr :-
format('~t~w~15| - ~t~w\n', ['Benford', 'Measured']).
writeData(Benford, Freq) :-
format('~t~2f%~15| - ~t~2f%\n', [Benford*100, Freq*100]).
 
go :- % main goal
findall(B, (between(1,9,N), benford(N,B)), Benford),
findall(C, firstchar(C), Fc), freq(Fc, Freq),
writeHdr, maplist(writeData, Benford, Freq).
Output:
?- go.
        Benford - Measured
         30.10% - 30.10%
         17.61% - 17.70%
         12.49% - 12.50%
          9.69% - 9.60%
          7.92% - 8.00%
          6.69% - 6.70%
          5.80% - 5.60%
          5.12% - 5.30%
          4.58% - 4.50%

PureBasic[edit]

#MAX_N=1000
NewMap d1.i()
Dim fi.s(#MAX_N)
fi(0)="0" : fi(1)="1"
Declare.s Sigma(sx.s,sy.s)
 
For I=2 To #MAX_N
fi(I)=Sigma(fi(I-2),fi(I-1))
Next
 
For I=1 To #MAX_N
d1(Left(fi(I),1))+1
Next
 
Procedure.s Sigma(sx.s, sy.s)
Define i.i, v1.i, v2.i, r.i
Define s.s, sa.s
sy=ReverseString(sy) : s=ReverseString(sx)
For i=1 To Len(s)*Bool(Len(s)>Len(sy))+Len(sy)*Bool(Len(sy)>=Len(s))
v1=Val(Mid(s,i,1))
v2=Val(Mid(sy,i,1))
r+v1+v2
sa+Str(r%10)
r/10
Next i
If r : sa+Str(r%10) : EndIf
ProcedureReturn ReverseString(sa)
EndProcedure
 
OpenConsole("Benford's law: Fibonacci sequence 1.."+Str(#MAX_N))
 
Print(~"Dig.\t\tCnt."+~"\t\tExp.\t\tDif.\n\n")
ForEach d1()
Print(RSet(MapKey(d1()),4," ")+~"\t:\t"+RSet(Str(d1()),3," ")+~"\t\t")
ex=Int(#MAX_N*Log(1+1/Val(MapKey(d1())))/Log(10))
PrintN(RSet(Str(ex),3," ")+~"\t\t"+RSet(StrF((ex-d1())*100/ex,5),8," ")+" %")
Next
 
PrintN(~"\nPress Enter...")
Input()
Output:
Dig.            Cnt.            Exp.            Dif.

   1    :       301             301              0.00000 %
   2    :       177             176             -0.56818 %
   3    :       125             124             -0.80645 %
   4    :        96              96              0.00000 %
   5    :        80              79             -1.26582 %
   6    :        67              66             -1.51515 %
   7    :        56              57              1.75439 %
   8    :        53              51             -3.92157 %
   9    :        45              45              0.00000 %

Press Enter...

Python[edit]

Works with Python 3.X & 2.7

from __future__ import division
from itertools import islice, count
from collections import Counter
from math import log10
from random import randint
 
expected = [log10(1+1/d) for d in range(1,10)]
 
def fib():
a,b = 1,1
while True:
yield a
a,b = b,a+b
 
# powers of 3 as a test sequence
def power_of_threes():
return (3**k for k in count(0))
 
def heads(s):
for a in s: yield int(str(a)[0])
 
def show_dist(title, s):
c = Counter(s)
size = sum(c.values())
res = [c[d]/size for d in range(1,10)]
 
print("\n%s Benfords deviation" % title)
for r, e in zip(res, expected):
print("%5.1f%% %5.1f%%  %5.1f%%" % (r*100., e*100., abs(r - e)*100.))
 
def rand1000():
while True: yield randint(1,9999)
 
if __name__ == '__main__':
show_dist("fibbed", islice(heads(fib()), 1000))
show_dist("threes", islice(heads(power_of_threes()), 1000))
 
# just to show that not all kind-of-random sets behave like that
show_dist("random", islice(heads(rand1000()), 10000))
Output:
fibbed Benfords deviation
 30.1%  30.1%    0.0%
 17.7%  17.6%    0.1%
 12.5%  12.5%    0.0%
  9.6%   9.7%    0.1%
  8.0%   7.9%    0.1%
  6.7%   6.7%    0.0%
  5.6%   5.8%    0.2%
  5.3%   5.1%    0.2%
  4.5%   4.6%    0.1%

threes Benfords deviation
 30.0%  30.1%    0.1%
 17.7%  17.6%    0.1%
 12.3%  12.5%    0.2%
  9.8%   9.7%    0.1%
  7.9%   7.9%    0.0%
  6.6%   6.7%    0.1%
  5.9%   5.8%    0.1%
  5.2%   5.1%    0.1%
  4.6%   4.6%    0.0%

random Benfords deviation
 11.2%  30.1%   18.9%
 10.9%  17.6%    6.7%
 11.6%  12.5%    0.9%
 11.1%   9.7%    1.4%
 11.6%   7.9%    3.7%
 11.4%   6.7%    4.7%
 10.3%   5.8%    4.5%
 11.0%   5.1%    5.9%
 10.9%   4.6%    6.3%

R[edit]

 
pbenford <- function(d){
return(log10(1+(1/d)))
}
 
get_lead_digit <- function(number){
return(as.numeric(substr(number,1,1)))
}
 
fib_iter <- function(n){
first <- 1
second <- 0
for(i in 1:n){
sum <- first + second
first <- second
second <- sum
}
return(sum)
}
 
fib_sequence <- mapply(fib_iter,c(1:1000))
lead_digits <- mapply(get_lead_digit,fib_sequence)
 
observed_frequencies <- table(lead_digits)/1000
expected_frequencies <- mapply(pbenford,c(1:9))
 
data <- data.frame(observed_frequencies,expected_frequencies)
colnames(data) <- c("digit","obs.frequency","exp.frequency")
dev_percentage <- abs((data$obs.frequency-data$exp.frequency)*100)
data <- data.frame(data,dev_percentage)
 
print(data)
 
Output:
digit obs.frequency exp.frequency dev_percentage
    1         0.301       0.30103       0.003000
    2         0.177       0.17609       0.090874
    3         0.125       0.12494       0.006126
    4         0.096       0.09691       0.091001
    5         0.080       0.07918       0.081875
    6         0.067       0.06695       0.005321
    7         0.056       0.05799       0.199195
    8         0.053       0.05115       0.184748
    9         0.045       0.04576       0.075749

Racket[edit]

#lang racket
 
(define (log10 n) (/ (log n) (log 10)))
 
(define (first-digit n)
(quotient n (expt 10 (inexact->exact (floor (log10 n))))))
 
(define N 10000)
 
(define fibs
(let loop ([n N] [a 0] [b 1])
(if (zero? n) '() (cons b (loop (sub1 n) b (+ a b))))))
 
(define v (make-vector 10 0))
(for ([n fibs])
(define f (first-digit n))
(vector-set! v f (add1 (vector-ref v f))))
 
(printf "N OBS EXP\n")
(define (pct n) (~r (* n 100.0) #:precision 1 #:min-width 4))
(for ([i (in-range 1 10)])
(printf "~a: ~a% ~a%\n" i
(pct (/ (vector-ref v i) N))
(pct (log10 (+ 1 (/ i))))))
 
;; Output:
;; N OBS EXP
;; 1: 30.1% 30.1%
;; 2: 17.6% 17.6%
;; 3: 12.5% 12.5%
;; 4: 9.7% 9.7%
;; 5: 7.9% 7.9%
;; 6: 6.7% 6.7%
;; 7: 5.8% 5.8%
;; 8: 5.1% 5.1%
;; 9: 4.6% 4.6%

REXX[edit]

The REXX language practically hasn't any high math functions, so the   e,   ln,   and log   functions were included herein.

Note that the   e   and   ln10   functions return a limited amount of accuracy, and they could be greater than 50 digits.

Note that prime numbers don't lend themselves to Benford's law.
(Apologies to those people's eyes looking at the prime number generator.   Uf-ta!)

For the extra credit stuff, I choose to generate the primes and factorials rather than find a web-page with them listed, as each list is very easy to generate.

/*REXX program  demonstrates  some common functions  (thirty decimal digits are shown). */
numeric digits 50 /*use only 50 dec digits for LN & LOG.*/
parse arg N .; if N=='' | N=="," then N=1000 /*allow sample size to be specified. */
@benny= "Benford's law applied to" /*a handy-dandy literal for some SAYs. */
w1= max(2+length('observed'), length(N-2) ); pad=" " /*used for aligning output.*/
w2= max(2+length('expected'), length(N ) ) /* " " " " */
 
@.=1; do j=3 to N; jm1=j-1; jm2=j-2; @.[email protected].jm2 + @.jm1; end /*j*/
call show @benny N 'Fibonacci numbers'
p=1
@.1=2; do j=3 by 2 until p==N; if \isPrime(j) then iterate; p=p+1; @.p=j; end /*j*/
call show @benny N 'prime numbers'
 
!=1; do j=1 for N;  !=!*j; end /*j*/
call show @benny N 'factorial products'
exit /*stick a fork in it, we're all done. */
/*───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/
e: return 2.7182818284590452353602874713526624977572470936999595749669676277240766303535
isPrime: procedure; parse arg x; if wordpos(x,'2 3 5 7')\==0 then return 1; if x//2==0 then return 0; if x//3==0 then return 0; do j=5 by 6 until j*j>x; if x//j==0 then return 0; if x//(j+2)==0 then return 0; end; return 1
ln: procedure; parse arg x; _=e(); ig=(x>1.5); is=1 - 2 * (ig \== 1); ii=0; s=x; return .ln()
.ln: do while ig&s>1.5|\ig&s<.5;do k=-1;iz=s*_**-is;if k>=0&(ig&iz<1|\ig&iz>.5) then leave;_=_*_;izz=iz;end;s=izz;ii=ii+is*2**k;end;x=x*e()**-ii-1;z=0;_=-1;p=z;do k=1;_=-_*x;z=z+_/k;if z=p then leave;p=z;end;return z+ii
log: return ln( arg(1) ) / ln(10)
/*──────────────────────────────────────────────────────────────────────────────────────*/
show: say; say pad ' digit ' pad center("observed",w1) pad center('expected',w2)
say pad '───────' pad center("",w1,'─') pad center("",w2,'─') pad arg(1)
 !.=0; do j=1 for N; _=left(@.j,1);  !._=!._+1; end /*get 1st digits.*/
 
do k=1 for 9 /*display results for decimal digits. */
say pad center(k,7) pad center(format(!.k /N , , length(N-2) ), w1),
pad center(format(log(1+1 /k) , , length(N)+2 ), w2)
end /*k*/
return

output when using the default input:

     digit       observed       expected
    ───────     ──────────     ──────────     Benford's law applied to 1000 Fibonacci numbers
       1          0.301         0.301030
       2          0.177         0.176091
       3          0.125         0.124939
       4          0.096         0.096910
       5          0.080         0.079181
       6          0.067         0.066947
       7          0.056         0.057992
       8          0.053         0.051153
       9          0.045         0.045757

     digit       observed       expected
    ───────     ──────────     ──────────     Benford's law applied to 1000 prime numbers
       1          0.160         0.301030
       2          0.146         0.176091
       3          0.139         0.124939
       4          0.139         0.096910
       5          0.131         0.079181
       6          0.135         0.066947
       7          0.118         0.057992
       8          0.017         0.051153
       9          0.015         0.045757

     digit       observed       expected
    ───────     ──────────     ──────────     Benford's law applied to 1000 factorial products
       1          0.293         0.301030
       2          0.176         0.176091
       3          0.124         0.124939
       4          0.102         0.096910
       5          0.069         0.079181
       6          0.087         0.066947
       7          0.051         0.057992
       8          0.051         0.051153
       9          0.047         0.045757

output when using 100,000 primes:

     digit       observed       expected
    ───────     ──────────     ──────────     Benford's law applied to 100000 prime numbers
       1         0.31087       0.30103000
       2         0.09142       0.17609126
       3         0.08960       0.12493874
       4         0.08747       0.09691001
       5         0.08615       0.07918125
       6         0.08458       0.06694679
       7         0.08435       0.05799195
       8         0.08326       0.05115252
       9         0.08230       0.04575749

Ring[edit]

 
# Project : Benford's law
# Date  : 2017/11/29
# Author : Gal Zsolt (~ CalmoSoft ~)
# Email  : <[email protected]>
 
decimals(3)
n= 1000
actual = list(n)
for x = 1 to len(actual)
actual[x] = 0
next
 
for nr = 1 to n
n1 = string(fibonacci(nr))
j = number(left(n1,1))
actual[j] = actual[j] + 1
next
 
see "Digit " + "Actual " + "Expected" + nl
for m = 1 to 9
fr = frequency(m)*100
see "" + m + " " + (actual[m]/10) + " " + fr + nl
next
 
func frequency(n)
freq = log10(n+1) - log10(n)
return freq
 
func log10(n)
log1 = log(n) / log(10)
return log1
 
func fibonacci(y)
if y = 0 return 0 ok
if y = 1 return 1 ok
if y > 1 return fibonacci(y-1) + fibonacci(y-2) ok
 

Output:

Digit	Actual	Expected
1	30.100	30.103
2	17.700	17.609
3	12.500	12.494
4	9.500	9.691
5	8.000	7.918
6	6.700	6.695
7	5.600	5.799
8	5.300	5.115
9	4.500	4.576

Ruby[edit]

Translation of: Python
EXPECTED = (1..9).map{|d| Math.log10(1+1.0/d)}
 
def fib(n)
a,b = 0,1
n.times.map{ret, a, b = a, b, a+b; ret}
end
 
# powers of 3 as a test sequence
def power_of_threes(n)
n.times.map{|k| 3**k}
end
 
def heads(s)
s.map{|a| a.to_s[0].to_i}
end
 
def show_dist(title, s)
s = heads(s)
c = Array.new(10, 0)
s.each{|x| c[x] += 1}
size = s.size.to_f
res = (1..9).map{|d| c[d]/size}
puts "\n  %s Benfords deviation" % title
res.zip(EXPECTED).each.with_index(1) do |(r, e), i|
puts "%2d: %5.1f%%  %5.1f%%  %5.1f%%" % [i, r*100, e*100, (r - e).abs*100]
end
end
 
def random(n)
n.times.map{rand(1..n)}
end
 
show_dist("fibbed", fib(1000))
show_dist("threes", power_of_threes(1000))
 
# just to show that not all kind-of-random sets behave like that
show_dist("random", random(10000))
Output:
    fibbed Benfords deviation
 1:  30.1%   30.1%    0.0%
 2:  17.7%   17.6%    0.1%
 3:  12.5%   12.5%    0.0%
 4:   9.5%    9.7%    0.2%
 5:   8.0%    7.9%    0.1%
 6:   6.7%    6.7%    0.0%
 7:   5.6%    5.8%    0.2%
 8:   5.3%    5.1%    0.2%
 9:   4.5%    4.6%    0.1%

    threes Benfords deviation
 1:  30.0%   30.1%    0.1%
 2:  17.7%   17.6%    0.1%
 3:  12.3%   12.5%    0.2%
 4:   9.8%    9.7%    0.1%
 5:   7.9%    7.9%    0.0%
 6:   6.6%    6.7%    0.1%
 7:   5.9%    5.8%    0.1%
 8:   5.2%    5.1%    0.1%
 9:   4.6%    4.6%    0.0%

    random Benfords deviation
 1:  10.9%   30.1%   19.2%
 2:  10.9%   17.6%    6.7%
 3:  11.7%   12.5%    0.8%
 4:  10.8%    9.7%    1.1%
 5:  11.2%    7.9%    3.3%
 6:  11.9%    6.7%    5.2%
 7:  10.7%    5.8%    4.9%
 8:  11.1%    5.1%    6.0%
 9:  10.8%    4.6%    6.2%

Run BASIC[edit]

 
N = 1000
for i = 0 to N - 1
n$ = str$(fibonacci(i))
j = val(left$(n$,1))
actual(j) = actual(j) +1
next
print
html "<table border=1><TR bgcolor=wheat><TD>Digit<td>Actual<td>Expected</td><tr>"
for i = 1 to 9
html "<tr align=right><td>";i;"</td><td>";using("##.###",actual(i)/10);"</td><td>";using("##.###", frequency(i)*100);"</td></tr>"
next
html "</table>"
end
 
function frequency(n)
frequency = log10(n+1) - log10(n)
end function
 
function log10(n)
log10 = log(n) / log(10)
end function
 
function fibonacci(n)
b = 1
for i = 1 to n
temp = fibonacci + b
fibonacci = b
b = temp
next i
end function
 
DigitActualExpected
130.10030.103
217.70017.609
312.50012.494
4 9.500 9.691
5 8.000 7.918
6 6.700 6.695
7 5.600 5.799
8 5.300 5.115
9 4.500 4.576

Rust[edit]

Works with: rustc version 1.12 stable

This solution uses the num create for arbitrary-precision integers and the num_traits create for the zero and one implementations. It computes the Fibonacci numbers from scratch via the fib function.

 
extern crate num_traits;
extern crate num;
 
use num::bigint::{BigInt, ToBigInt};
use num_traits::{Zero, One};
use std::collections::HashMap;
 
// Return a vector of all fibonacci results from fib(1) to fib(n)
fn fib(n: usize) -> Vec<BigInt> {
let mut result = Vec::with_capacity(n);
let mut a = BigInt::zero();
let mut b = BigInt::one();
 
result.push(b.clone());
 
for i in 1..n {
let t = b.clone();
b = a+b;
a = t;
result.push(b.clone());
}
 
result
}
 
// Return the first digit of a `BigInt`
fn first_digit(x: &BigInt) -> u8 {
let zero = BigInt::zero();
assert!(x > &zero);
 
let s = x.to_str_radix(10);
 
// parse the first digit of the stringified integer
*&s[..1].parse::<u8>().unwrap()
}
 
fn main() {
const N: usize = 1000;
let mut counter: HashMap<u8, u32> = HashMap::new();
for x in fib(N) {
let d = first_digit(&x);
*counter.entry(d).or_insert(0) += 1;
}
 
println!("{:>13} {:>10}", "real", "predicted");
for y in 1..10 {
println!("{}: {:10.3} v. {:10.3}", y, *counter.get(&y).unwrap_or(&0) as f32 / N as f32,
(1.0 + 1.0 / (y as f32)).log10());
}
 
}
 
Output:
         real     predicted
1:      0.301 v.      0.301
2:      0.177 v.      0.176
3:      0.125 v.      0.125
4:      0.096 v.      0.097
5:      0.080 v.      0.079
6:      0.067 v.      0.067
7:      0.056 v.      0.058
8:      0.053 v.      0.051
9:      0.045 v.      0.046

Scala[edit]

// Fibonacci Sequence (begining with 1,1): 1 1 2 3 5 8 13 21 34 55 ...
val fibs : Stream[BigInt] = { def series(i:BigInt,j:BigInt):Stream[BigInt] = i #:: series(j, i+j); series(1,0).tail.tail }
 
 
/**
* Given a numeric sequence, return the distribution of the most-signicant-digit
* as expected by Benford's Law and then by actual distribution.
*/

def benford[N:Numeric]( data:Seq[N] ) : Map[Int,(Double,Double)] = {
 
import scala.math._
 
val maxSize = 10000000 // An arbitrary size to avoid problems with endless streams
 
val size = (data.take(maxSize)).size.toDouble
 
val distribution = data.take(maxSize).groupBy(_.toString.head.toString.toInt).map{ case (d,l) => (d -> l.size) }
 
(for( i <- (1 to 9) ) yield { (i -> (log10(1D + 1D / i), (distribution(i) / size))) }).toMap
}
 
{
println( "Fibonacci Sequence (size=1000): 1 1 2 3 5 8 13 21 34 55 ...\n" )
println( "%9s %9s %9s".format( "Actual", "Expected", "Deviation" ) )
 
benford( fibs.take(1000) ).toList.sorted foreach {
case (k, v) => println( "%d: %5.2f%% | %5.2f%% | %5.4f%%".format(k,v._2*100,v._1*100,math.abs(v._2-v._1)*100) )
}
}
Output:
Fibonacci Sequence (size=1000): 1 1 2 3 5 8 13 21 34 55 ...

   Actual  Expected Deviation
1: 30.10% | 30.10% | 0.0030%
2: 17.70% | 17.61% | 0.0909%
3: 12.50% | 12.49% | 0.0061%
4:  9.60% |  9.69% | 0.0910%
5:  8.00% |  7.92% | 0.0819%
6:  6.70% |  6.69% | 0.0053%
7:  5.60% |  5.80% | 0.1992%
8:  5.30% |  5.12% | 0.1847%
9:  4.50% |  4.58% | 0.0757%

Sidef[edit]

var (actuals, expected) = ([], [])
var fibonacci = 1000.of {|i| fib(i).digit(0) }
 
for i (1..9) {
var num = fibonacci.count_by {|j| j == i }
actuals.append(num / 1000)
expected.append(1 + (1/i) -> log10)
}
 
"%17s%17s\n".printf("Observed","Expected")
for i (1..9) {
"%d : %11s %%%15s %%\n".printf(
i, "%.2f".sprintf(100 * actuals[i - 1]),
"%.2f".sprintf(100 * expected[i - 1]),
)
}
Output:
         Observed         Expected
1 :       30.10 %          30.10 %
2 :       17.70 %          17.61 %
3 :       12.50 %          12.49 %
4 :        9.50 %           9.69 %
5 :        8.00 %           7.92 %
6 :        6.70 %           6.69 %
7 :        5.60 %           5.80 %
8 :        5.30 %           5.12 %
9 :        4.50 %           4.58 %

SQL[edit]

If we load some numbers into a table, we can do the sums without too much difficulty. I tried to make this as database-neutral as possible, but I only had Oracle handy to test it on.

The query is the same for any number sequence you care to put in the benford table.

-- Create table
CREATE TABLE benford (num INTEGER);
 
-- Seed table
INSERT INTO benford (num) VALUES (1);
INSERT INTO benford (num) VALUES (1);
INSERT INTO benford (num) VALUES (2);
 
-- Populate table
INSERT INTO benford (num)
SELECT
ult + penult
FROM
(SELECT MAX(num) AS ult FROM benford),
(SELECT MAX(num) AS penult FROM benford WHERE num NOT IN (SELECT MAX(num) FROM benford))
 
-- Repeat as many times as desired
-- in Oracle SQL*Plus, press "Slash, Enter" a lot of times
-- or wrap this in a loop, but that will require something db-specific...
 
-- Do sums
SELECT
digit,
COUNT(digit) / numbers AS actual,
log(10, 1 + 1 / digit) AS expected
FROM
(
SELECT
FLOOR(num/POWER(10,LENGTH(num)-1)) AS digit
FROM
benford
),
(
SELECT
COUNT(*) AS numbers
FROM
benford
)
GROUP BY digit, numbers
ORDER BY digit;
 
-- Tidy up
DROP TABLE benford;
Output:

I only loaded the first 100 Fibonacci numbers before my fingers were sore from repeating the data load. 8~)

     DIGIT     ACTUAL   EXPECTED
---------- ---------- ----------
         1         .3 .301029996
         2        .18 .176091259
         3        .13 .124938737
         4        .09 .096910013
         5        .08 .079181246
         6        .06  .06694679
         7        .05 .057991947
         8        .07 .051152522
         9        .04 .045757491

9 rows selected.

Stata[edit]

clear
set obs 1000
scalar phi=(1+sqrt(5))/2
gen fib=(phi^_n-(-1/phi)^_n)/sqrt(5)
gen k=real(substr(string(fib),1,1))
hist k, discrete // show a histogram
qui tabulate k, matcell(f) // compute frequencies
 
mata
f=st_matrix("f")
p=log10(1:+1:/(1::9))*sum(f)
// print observed vs predicted probabilities
f,p
1 2
+-----------------------------+
1 | 297 301.0299957 |
2 | 178 176.0912591 |
3 | 127 124.9387366 |
4 | 96 96.91001301 |
5 | 80 79.18124605 |
6 | 67 66.94678963 |
7 | 57 57.99194698 |
8 | 53 51.15252245 |
9 | 45 45.75749056 |
+-----------------------------+

Assuming the data are random, one can also do a goodness of fit chi-square test:

// chi-square statistic
chisq=sum((f-p):^2:/p)
chisq
.2219340262
// p-value
chi2tail(8,chisq)
.9999942179
end

The p-value is very close to 1, showing that the observed distribution is very close to the Benford law.

The fit is not as good with the sequence (2+sqrt(2))^n:

clear
set obs 500
scalar s=2+sqrt(2)
gen a=s^_n
gen k=real(substr(string(a),1,1))
hist k, discrete
qui tabulate k, matcell(f)
 
mata
f=st_matrix("f")
p=log10(1:+1:/(1::9))*sum(f)
f,p
1 2
+-----------------------------+
1 | 134 150.5149978 |
2 | 99 88.04562953 |
3 | 68 62.4693683 |
4 | 34 48.4550065 |
5 | 33 39.59062302 |
6 | 33 33.47339482 |
7 | 33 28.99597349 |
8 | 33 25.57626122 |
9 | 33 22.87874528 |
+-----------------------------+
 
chisq=sum((f-p):^2:/p)
chisq
16.26588528
 
chi2tail(8,chisq)
.0387287805
end

Now the p-value is less than the usual 5% risk, and one would reject the hypothesis that the data follow the Benford law.

Swift[edit]

import Foundation
 
/* Reads from a file and returns the content as a String */
func readFromFile(fileName file:String) -> String{
 
var ret:String = ""
 
let path = Foundation.URL(string: "file://"+file)
 
do {
ret = try String(contentsOf: path!, encoding: String.Encoding.utf8)
}
catch {
print("Could not read from file!")
exit(-1)
}
 
return ret
}
 
/* Calculates the probability following Benford's law */
func benford(digit z:Int) -> Double {
 
if z<=0 || z>9 {
perror("Argument must be between 1 and 9.")
return 0
}
 
return log10(Double(1)+Double(1)/Double(z))
}
 
// get CLI input
if CommandLine.arguments.count < 2 {
print("Usage: Benford [FILE]")
exit(-1)
}
 
let pathToFile = CommandLine.arguments[1]
 
// Read from given file and parse into lines
let content = readFromFile(fileName: pathToFile)
let lines = content.components(separatedBy: "\n")
 
var digitCount:UInt64 = 0
var countDigit:[UInt64] = [0,0,0,0,0,0,0,0,0]
 
// check digits line by line
for line in lines {
if line == "" {
continue
}
let charLine = Array(line.characters)
switch(charLine[0]){
case "1":
countDigit[0] += 1
digitCount += 1
break
case "2":
countDigit[1] += 1
digitCount += 1
break
case "3":
countDigit[2] += 1
digitCount += 1
break
case "4":
countDigit[3] += 1
digitCount += 1
break
case "5":
countDigit[4] += 1
digitCount += 1
break
case "6":
countDigit[5] += 1
digitCount += 1
break
case "7":
countDigit[6] += 1
digitCount += 1
break
case "8":
countDigit[7] += 1
digitCount += 1
break
case "9":
countDigit[8] += 1
digitCount += 1
break
default:
break
}
 
}
 
// print result
print("Digit\tBenford [%]\tObserved [%]\tDeviation")
print("~~~~~\t~~~~~~~~~~~~\t~~~~~~~~~~~~\t~~~~~~~~~")
for i in 0..<9 {
let temp:Double = Double(countDigit[i])/Double(digitCount)
let ben = benford(digit: i+1)
print(String(format: "%d\t%.2f\t\t%.2f\t\t%.4f", i+1,ben*100,temp*100,ben-temp))
}
Output:
$ ./Benford
Usage: Benford [FILE]
$ ./Benford Fibonacci.txt
Digit	Benford [%]	Observed [%]	Deviation
~~~~~	~~~~~~~~~~~~	~~~~~~~~~~~~	~~~~~~~~~
1	30.10		30.10		0.0000
2	17.61		17.70		-0.0009
3	12.49		12.50		-0.0001
4	9.69		9.60		0.0009
5	7.92		8.00		-0.0008
6	6.69		6.70		-0.0001
7	5.80		5.60		0.0020
8	5.12		5.30		-0.0018
9	4.58		4.50		0.0008

Tcl[edit]

proc benfordTest {numbers} {
# Count the leading digits (RE matches first digit in each number,
# even if negative)
set accum {1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0}
foreach n $numbers {
if {[regexp {[1-9]} $n digit]} {
dict incr accum $digit
}
}
 
# Print the report
puts " digit | measured | theory"
puts "-------+----------+--------"
dict for {digit count} $accum {
puts [format "%6d | %7.2f%% | %5.2f%%" $digit \
[expr {$count * 100.0 / [llength $numbers]}] \
[expr {log(1+1./$digit)/log(10)*100.0}]]
}
}

Demonstrating with Fibonacci numbers:

proc fibs n {
for {set a 1;set b [set i 0]} {$i < $n} {incr i} {
lappend result [set b [expr {$a + [set a $b]}]]
}
return $result
}
benfordTest [fibs 1000]
Output:
 digit | measured | theory
-------+----------+--------
     1 |   30.10% | 30.10%
     2 |   17.70% | 17.61%
     3 |   12.50% | 12.49%
     4 |    9.60% |  9.69%
     5 |    8.00% |  7.92%
     6 |    6.70% |  6.69%
     7 |    5.60% |  5.80%
     8 |    5.30% |  5.12%
     9 |    4.50% |  4.58%

Visual FoxPro[edit]

 
#DEFINE CTAB CHR(9)
#DEFINE COMMA ","
#DEFINE CRLF CHR(13) + CHR(10)
LOCAL i As Integer, n As Integer, n1 As Integer, rho As Double, c As String
n = 1000
LOCAL ARRAY a[n,2], res[1]
CLOSE DATABASES ALL
CREATE CURSOR fibo(dig C(1))
INDEX ON dig TAG dig COLLATE "Machine"
SET ORDER TO 0
*!* Populate the cursor with the leading digit of the first 1000 Fibonacci numbers
a[1,1] = "1"
a[1,2] = 1
a[2,1] = "1"
a[2,2] = 1
FOR i = 3 TO n
a[i,2] = a[i-2,2] + a[i-1,2]
a[i,1] = LEFT(TRANSFORM(a[i,2]), 1)
ENDFOR
APPEND FROM ARRAY a FIELDS dig
CREATE CURSOR results (digit I, count I, prob B(6), expected B(6))
INSERT INTO results ;
SELECT dig, COUNT(1), COUNT(1)/n, Pr(VAL(dig)) FROM fibo GROUP BY dig ORDER BY dig
n1 = RECCOUNT()
*!* Correlation coefficient
SELECT (n1*SUM(prob*expected) - SUM(prob)*SUM(expected))/;
(SQRT(n1*SUM(prob*prob) - SUM(prob)*SUM(prob))*SQRT(n1*SUM(expected*expected) - SUM(expected)*SUM(expected))) ;
FROM results INTO ARRAY res
rho = CAST(res[1] As B(6))
SET SAFETY OFF
COPY TO benford.txt TYPE CSV
c = FILETOSTR("benford.txt")
*!* Replace commas with tabs
c = STRTRAN(c, COMMA, CTAB) + CRLF + "Correlation Coefficient: " + TRANSFORM(rho)
STRTOFILE(c, "benford.txt", 0)
SET SAFETY ON
 
FUNCTION Pr(d As Integer) As Double
RETURN LOG10(1 + 1/d)
ENDFUNC
 
Output:
digit	count	prob	expected
1		301	0.301000	0.301000
2		177	0.177000	0.176100
3		125	0.125000	0.124900
4		 96	0.096000	0.096900
5		 80	0.080000	0.079200
6		 67	0.067000	0.066900
7		 56	0.056000	0.058000
8		 53	0.053000	0.051200
9		 45	0.045000	0.045800

Correlation Coefficient: 0.999908

zkl[edit]

Translation of: Go
show(  // use list (fib(1)...fib(1000)) --> (1..4.34666e+208)
(0).pump(1000,List,fcn(ab){ab.append(ab.sum(0.0)).pop(0)}.fp(L(1,1))),
"First 1000 Fibonacci numbers");
 
fcn show(data,title){
f:=(0).pump(9,List,Ref.fp(0)); // (Ref(0),Ref(0)...
foreach v in (data){ // eg 1.49707e+207 ("g" format) --> "1" (first digit)
f[v.toString()[0].toInt()-1].inc(); }
println(title);
println("Digit Observed Predicted");
foreach i,n in ([1..].zip(f)){ // -->(1,Ref)...(9,Ref)
println("  %d  %9.3f  %8.3f".fmt(i,n.value.toFloat()/data.len(),
(1.0+1.0/i).log10()))
}
}
Translation of: CoffeeScript
var BN=Import("zklBigNum");
 
fcn fibgen(a,b) { return(a,self.fcn.fp(b,a+b)) } //-->L(fib,fcn)
 
benford := [0..9].pump(List,Ref.fp(0)).copy(); //L(Ref(0),...)
 
const N=1000;
 
[1..N].reduce('wrap(fiber,_){
n,f:=fiber;
benford[n.toString()[0]].inc(); // first digit of fib
f() // next (fib,fcn) pair
},fibgen(BN(1),BN(1)));
 
// de-ref Refs ie convert to int to float, divide by N
actual  := benford.apply(T("value","toFloat",'/(N)));
expected := [1..9].apply(fcn(x){(1.0 + 1.0/x).log10()});
 
println("Leading digital distribution of the first 1,000 Fibonacci numbers");
println("Digit\tActual\tExpected");
foreach i in ([1..9]){ println("%d\t%.3f\t%.3f".fmt(i,actual[i], expected[i-1])); }
Output:
First 1000 Fibonacci numbers
Digit  Observed  Predicted
  1      0.301     0.301
  2      0.177     0.176
  3      0.125     0.125
  4      0.096     0.097
  5      0.080     0.079
  6      0.067     0.067
  7      0.056     0.058
  8      0.053     0.051
  9      0.045     0.046

ZX Spectrum Basic[edit]

Translation of: Liberty BASIC
10 RANDOMIZE 
20 DIM b(9)
30 LET n=100
40 FOR i=1 TO n
50 GO SUB 1000
60 LET n$=STR$ fiboI
70 LET d=VAL n$(1)
80 LET b(d)=b(d)+1
90 NEXT i
100 PRINT "Digit";TAB 6;"Actual freq";TAB 18;"Expected freq"
110 FOR i=1 TO 9
120 LET pdi=(LN (i+1)/LN 10)-(LN i/LN 10)
130 PRINT i;TAB 6;b(i)/n;TAB 18;pdi
140 NEXT i
150 STOP
1000 REM Fibonacci
1010 LET fiboI=0: LET b=1
1020 FOR j=1 TO i
1030 LET temp=fiboI+b
1040 LET fiboI=b
1050 LET b=temp
1060 NEXT j
1070 RETURN
 

The results obtained are adjusted fairly well, except for the number 8. This occurs with Sinclair BASIC, Sam BASIC and SpecBAS fits.