Cumulative standard deviation: Difference between revisions
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=={{header| |
=={{header|Nim}}== |
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<lang |
<lang nim>import math, strutils |
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var sdSum, sdSum2, sdN = 0.0 |
var sdSum, sdSum2, sdN = 0.0 |
Revision as of 23:46, 3 January 2015
You are encouraged to solve this task according to the task description, using any language you may know.
Write a stateful function, class, generator or coroutine that takes a series of floating point numbers, one at a time, and returns the running standard deviation of the series. The task implementation should use the most natural programming style of those listed for the function in the implementation language; the task must state which is being used. Do not apply Bessel's correction; the returned standard deviation should always be computed as if the sample seen so far is the entire population.
Use this to compute the standard deviation of this demonstration set, , which is .
See also:
360 Assembly
For maximum compatibility, this program uses only the basic instruction set. Part of the code length is due to the square root algorithm and to the nice output. <lang 360asm>******** Standard deviation of a population STDDEV CSECT
USING STDDEV,R13
SAVEAREA B STM-SAVEAREA(R15)
DC 17F'0' DC CL8'STDDEV'
STM STM R14,R12,12(R13)
ST R13,4(R15) ST R15,8(R13) LR R13,R15 SR R8,R8 s=0 SR R9,R9 ss=0 SR R4,R4 i=0 LA R6,1 LH R7,N
LOOPI BXH R4,R6,ENDLOOPI
LR R1,R4 i BCTR R1,0 SLA R1,1 LH R5,T(R1) ST R5,WW ww=t(i) MH R5,=H'1000' w=ww*1000 AR R8,R5 s=s+w LR R15,R5 MR R14,R5 w*w AR R9,R15 ss=ss+w*w LR R14,R8 s SRDA R14,32 DR R14,R4 /i ST R15,AVG avg=s/i LR R14,R9 ss SRDA R14,32 DR R14,R4 ss/i LR R2,R15 ss/i LR R15,R8 s MR R14,R8 s*s LR R3,R15 LR R15,R4 i MR R14,R4 i*i LR R1,R15 LA R14,0 LR R15,R3 DR R14,R1 (s*s)/(i*i) SR R2,R15 LR R10,R2 std=ss/i-(s*s)/(i*i) LR R11,R10 std SRA R11,1 x=std/2 LR R12,R10 px=std
LOOPWHIL EQU *
CR R12,R11 while px<>=x BE ENDWHILE LR R12,R11 px=x LR R15,R10 std LA R14,0 DR R14,R12 /px LR R1,R12 px AR R1,R15 px+std/px SRA R1,1 /2 LR R11,R1 x=(px+std/px)/2 B LOOPWHIL
ENDWHILE EQU *
LR R10,R11 CVD R4,P8 i MVC C17,MASK17 ED C17,P8 MVC BUF+2(1),C17+15 L R1,WW CVD R1,P8 MVC C17,MASK17 ED C17,P8 MVC BUF+10(1),C17+15 L R1,AVG CVD R1,P8 MVC C18,MASK18 ED C18,P8 MVC BUF+17(5),C18+12 CVD R10,P8 std MVC C18,MASK18 ED C18,P8 MVC BUF+31(5),C18+12 WTO MF=(E,WTOMSG) B LOOPI
ENDLOOPI EQU *
L R13,4(0,R13) LM R14,R12,12(R13) XR R15,R15 BR R14 DS 0D
N DC H'8' T DC H'2',H'4',H'4',H'4',H'5',H'5',H'7',H'9' WW DS F AVG DS F P8 DS PL8 MASK17 DC C' ',13X'20',X'2120',C'-' MASK18 DC C' ',10X'20',X'2120',C'.',3X'20',C'-' C17 DS CL17 C18 DS CL18 WTOMSG DS 0F
DC H'80',XL2'0000'
BUF DC CL80'N=1 ITEM=1 AVG=1.234 STDDEV=1.234 '
YREGS END STDDEV</lang>
- Output:
N=1 ITEM=2 AVG=2.000 STDDEV=0.000 N=2 ITEM=4 AVG=3.000 STDDEV=1.000 N=3 ITEM=4 AVG=3.333 STDDEV=0.942 N=4 ITEM=4 AVG=3.500 STDDEV=0.866 N=5 ITEM=5 AVG=3.800 STDDEV=0.979 N=6 ITEM=5 AVG=4.000 STDDEV=1.000 N=7 ITEM=7 AVG=4.428 STDDEV=1.399 N=8 ITEM=9 AVG=5.000 STDDEV=2.000
Ada
<lang ada>with Ada.Numerics.Elementary_Functions; use Ada.Numerics.Elementary_Functions; with Ada.Text_IO; use Ada.Text_IO;
procedure Test_Deviation is
type Sample is record N : Natural := 0; Mean : Float := 0.0; Squares : Float := 0.0; end record; procedure Add (Data : in out Sample; Point : Float) is begin Data.N := Data.N + 1; Data.Mean := Data.Mean + Point; Data.Squares := Data.Squares + Point ** 2; end Add; function Deviation (Data : Sample) return Float is begin return Sqrt (Data.Squares / Float (Data.N) - (Data.Mean / Float (Data.N)) ** 2); end Deviation;
Data : Sample; Test : array (1..8) of Float := (2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0);
begin
for Item in Test'Range loop Add (Data, Test (Item)); end loop; Put_Line ("Deviation" & Float'Image (Deviation (Data)));
end Test_Deviation;</lang>
- Output:
Deviation 2.00000E+00
ALGOL 68
Note: the use of a UNION to mimic C's enumerated types is "experimental" and probably not typical of "production code". However it is a example of ALGOL 68s conformity CASE clause useful for classroom dissection. <lang Algol68>MODE VALUE = STRUCT(CHAR value),
STDDEV = STRUCT(CHAR stddev), MEAN = STRUCT(CHAR mean), VAR = STRUCT(CHAR var), COUNT = STRUCT(CHAR count), RESET = STRUCT(CHAR reset);
MODE ACTION = UNION ( VALUE, STDDEV, MEAN, VAR, COUNT, RESET );
LONG REAL sum := 0; LONG REAL sum2 := 0; INT num := 0;
PROC stat object = (LONG REAL v, ACTION action)LONG REAL: (
LONG REAL m; CASE action IN (VALUE):( num +:= 1; sum +:= v; sum2 +:= v*v; stat object(0, LOC STDDEV) ), (STDDEV): long sqrt(stat object(0, LOC VAR)), (MEAN): IF num>0 THEN sum/LONG REAL(num) ELSE 0 FI, (VAR):( m := stat object(0, LOC MEAN); IF num>0 THEN sum2/LONG REAL(num)-m*m ELSE 0 FI ), (COUNT): num, (RESET): sum := sum2 := num := 0 ESAC
);
[]LONG REAL v = ( 2,4,4,4,5,5,7,9 );
main: (
LONG REAL sd; FOR i FROM LWB v TO UPB v DO sd := stat object(v[i], LOC VALUE) OD; printf(($"standard dev := "g(0,6)l$, sd))
)</lang>
- Output:
standard dev := 2.000000
A code sample in an object oriented style: <lang Algol68>MODE STAT = STRUCT(
LONG REAL sum, LONG REAL sum2, INT num
);
OP INIT = (REF STAT new)REF STAT:
(init OF class stat)(new);
MODE CLASSSTAT = STRUCT(
PROC (REF STAT, LONG REAL #value#)VOID plusab, PROC (REF STAT)LONG REAL stddev, mean, variance, count, PROC (REF STAT)REF STAT init
);
CLASSSTAT class stat;
plusab OF class stat := (REF STAT self, LONG REAL value)VOID:(
num OF self +:= 1; sum OF self +:= value; sum2 OF self +:= value*value );
OP +:= = (REF STAT lhs, LONG REAL rhs)VOID: # some syntatic sugar #
(plusab OF class stat)(lhs, rhs);
stddev OF class stat := (REF STAT self)LONG REAL:
long sqrt((variance OF class stat)(self));
OP STDDEV = ([]LONG REAL value)LONG REAL: ( # more syntatic sugar #
REF STAT stat = INIT LOC STAT; FOR i FROM LWB value TO UPB value DO stat +:= value[i] OD; (stddev OF class stat)(stat)
);
mean OF class stat := (REF STAT self)LONG REAL:
sum OF self/LONG REAL(num OF self);
variance OF class stat := (REF STAT self)LONG REAL:(
LONG REAL m = (mean OF class stat)(self); sum2 OF self/LONG REAL(num OF self)-m*m );
count OF class stat := (REF STAT self)LONG REAL:
num OF self;
init OF class stat := (REF STAT self)REF STAT:(
sum OF self := sum2 OF self := num OF self := 0; self );
[]LONG REAL value = ( 2,4,4,4,5,5,7,9 );
main: (
printf(($"standard deviation operator = "g(0,6)l$, STDDEV value));
REF STAT stat = INIT LOC STAT;
FOR i FROM LWB value TO UPB value DO stat +:= value[i] OD; printf(($"standard deviation = "g(0,6)l$, (stddev OF class stat)(stat))); printf(($"mean = "g(0,6)l$, (mean OF class stat)(stat))); printf(($"variance = "g(0,6)l$, (variance OF class stat)(stat))); printf(($"count = "g(0,6)l$, (count OF class stat)(stat)))
)</lang>
- Output:
standard deviation operator = 2.000000 standard deviation = 2.000000 mean = 5.000000 variance = 4.000000 count = 8.000000
A simple - but "unpackaged" - code example, useful if the standard deviation is required on only one set of concurrent data: <lang Algol68>LONG REAL sum, sum2; INT n;
PROC sd = (LONG REAL x)LONG REAL:(
sum +:= x; sum2 +:= x*x; n +:= 1; IF n = 0 THEN 0 ELSE long sqrt(sum2/n - sum*sum/n/n) FI
);
sum := sum2 := n := 0; []LONG REAL values = (2,4,4,4,5,5,7,9); FOR i TO UPB values DO
LONG REAL value = values[i]; printf(($2(xg(0,6))l$, value, sd(value)))
OD</lang>
- Output:
2.000000 .000000 4.000000 1.000000 4.000000 .942809 4.000000 .866025 5.000000 .979796 5.000000 1.000000 7.000000 1.399708 9.000000 2.000000
AutoHotkey
ahk forum: discussion <lang AutoHotkey>std(2),std(4),std(4),std(4),std(5),std(5),std(7) MsgBox % std(9) ; 2
std(x="") {
Static sum:=0, sqr:=0, n:=0 If (x="") ; blank parameter: reset sum := 0, sqr := 0, n := 0 Else sum += x, sqr += x*x, n++ ; update state Return sqrt((sqr-sum*sum/n)/n)
}</lang>
AWK
<lang AWK>
- syntax: GAWK -f STANDARD_DEVIATION.AWK
BEGIN {
n = split("2,4,4,4,5,5,7,9",arr,",") for (i=1; i<=n; i++) { temp[i] = arr[i] printf("%g %g\n",arr[i],stdev(temp)) } exit(0)
} function stdev(arr, i,n,s1,s2,variance,x) {
for (i in arr) { n++ x = arr[i] s1 += x ^ 2 s2 += x } variance = ((n * s1) - (s2 ^ 2)) / (n ^ 2) return(sqrt(variance))
} </lang>
- Output:
2 0 4 1 4 0.942809 4 0.866025 5 0.979796 5 1 7 1.39971 9 2
Axiom
We implement a domain with dependent type T with the operation + and identity 0:<lang Axiom>)abbrev package TESTD TestDomain TestDomain(T : Join(Field,RadicalCategory)): Exports == Implementation where
R ==> Record(n : Integer, sum : T, ssq : T) Exports == AbelianMonoid with _+ : (%,T) -> % _+ : (T,%) -> % sd : % -> T Implementation == R add Rep := R -- similar representation and implementation obj : % 0 == [0,0,0] obj + (obj2:%) == [obj.n + obj2.n, obj.sum + obj2.sum, obj.ssq + obj2.ssq] obj + (x:T) == obj + [1, x, x*x] (x:T) + obj == obj + x sd obj == mean : T := obj.sum / (obj.n::T) sqrt(obj.ssq / (obj.n::T) - mean*mean)</lang>This can be called using:<lang Axiom>T ==> Expression Integer
D ==> TestDomain(T) items := [2,4,4,4,5,5,7,9+x] :: List T; map(sd, scan(+, items, 0$D))
+---------------+ +-+ +-+ +-+ +-+ | 2 2\|2 \|3 2\|6 4\|6 \|7x + 64x + 256 (1) [0,1,-----,----,-----,1,-----,------------------] 3 2 5 7 8 Type: List(Expression(Integer))
eval subst(last %,x=0)
(2) 2 Type: Expression(Integer)</lang>
BBC BASIC
Uses the MOD(array()) and SUM(array()) functions. <lang bbcbasic> MAXITEMS = 100
FOR i% = 1 TO 8 READ n PRINT "Value = "; n ", running SD = " FNrunningsd(n) NEXT END DATA 2,4,4,4,5,5,7,9 DEF FNrunningsd(n) PRIVATE list(), i% DIM list(MAXITEMS) i% += 1 list(i%) = n = SQR(MOD(list())^2/i% - (SUM(list())/i%)^2)</lang>
- Output:
Value = 2, running SD = 0 Value = 4, running SD = 1 Value = 4, running SD = 0.942809043 Value = 4, running SD = 0.866025404 Value = 5, running SD = 0.979795901 Value = 5, running SD = 1 Value = 7, running SD = 1.39970842 Value = 9, running SD = 2
C
<lang c>#include <stdio.h>
- include <stdlib.h>
- include <math.h>
typedef enum Action { STDDEV, MEAN, VAR, COUNT } Action;
typedef struct stat_obj_struct {
double sum, sum2; size_t num; Action action;
} sStatObject, *StatObject;
StatObject NewStatObject( Action action ) {
StatObject so;
so = malloc(sizeof(sStatObject)); so->sum = 0.0; so->sum2 = 0.0; so->num = 0; so->action = action; return so;
}
- define FREE_STAT_OBJECT(so) \
free(so); so = NULL
double stat_obj_value(StatObject so, Action action) {
double num, mean, var, stddev; if (so->num == 0.0) return 0.0; num = so->num; if (action==COUNT) return num; mean = so->sum/num; if (action==MEAN) return mean; var = so->sum2/num - mean*mean; if (action==VAR) return var; stddev = sqrt(var); if (action==STDDEV) return stddev; return 0;
}
double stat_object_add(StatObject so, double v) {
so->num++; so->sum += v; so->sum2 += v*v; return stat_obj_value(so, so->action);
}</lang>
<lang c>double v[] = { 2,4,4,4,5,5,7,9 };
int main() {
int i; StatObject so = NewStatObject( STDDEV );
for(i=0; i < sizeof(v)/sizeof(double) ; i++) printf("val: %lf std dev: %lf\n", v[i], stat_object_add(so, v[i]));
FREE_STAT_OBJECT(so); return 0;
}</lang>
C++
<lang cpp>#include <algorithm>
- include <iostream>
- include <iterator>
- include <cmath>
- include <vector>
- include <iterator>
- include <numeric>
template <typename Iterator> double standard_dev( Iterator begin , Iterator end ) {
double mean = std::accumulate( begin , end , 0 ) / std::distance( begin , end ) ; std::vector<double> squares ; for( Iterator vdi = begin ; vdi != end ; vdi++ ) squares.push_back( std::pow( *vdi - mean , 2 ) ) ; return std::sqrt( std::accumulate( squares.begin( ) , squares.end( ) , 0 ) / squares.size( ) ) ;
}
int main( ) {
double demoset[] = { 2 , 4 , 4 , 4 , 5 , 5 , 7 , 9 } ; int demosize = sizeof demoset / sizeof *demoset ; std::cout << "The standard deviation of\n" ; std::copy( demoset , demoset + demosize , std::ostream_iterator<double>( std::cout, " " ) ) ; std::cout << "\nis " << standard_dev( demoset , demoset + demosize ) << " !\n" ; return 0 ;
}</lang>
C#
<lang csharp>using System; using System.Collections.Generic; using System.Linq;
namespace standardDeviation {
class Program { static void Main(string[] args) { List<double> nums = new List<double> { 2, 4, 4, 4, 5, 5, 7, 9 }; for (int i = 1; i <= nums.Count; i++) Console.WriteLine(sdev(nums.GetRange(0, i))); }
static double sdev(List<double> nums) { List<double> store = new List<double>(); foreach (double n in nums) store.Add((n - nums.Average()) * (n - nums.Average()));
return Math.Sqrt(store.Sum() / store.Count); } }
}</lang>
0 1 0,942809041582063 0,866025403784439 0,979795897113271 1 1,39970842444753 2
Clojure
<lang lisp> (defn std-dev [samples]
(let [n (count samples)
mean (/ (reduce + samples) n) intermediate (map #(Math/pow (- %1 mean) 2) samples)]
(Math/sqrt (/ (reduce + intermediate) n))))
(println (std-dev [2 4 4 4 5 5 7 9])) ;;2.0
</lang>
COBOL
Using an intrinsic function: <lang cobol>FUNCTION STANDARD-DEVIATION(2, 4, 4, 4, 5, 5, 7, 9)</lang>
How this is implemented in the standard: <lang cobol>FUNCTION SQRT(FUNCTION VARIANCE(2, 4, 4, 4, 5, 5, 7, 9))</lang>
A complete implementation:
<lang cobol> >>SOURCE FREE IDENTIFICATION DIVISION. PROGRAM-ID. std-dev.
ENVIRONMENT DIVISION. CONFIGURATION SECTION. REPOSITORY.
FUNCTION sum-arr .
DATA DIVISION. WORKING-STORAGE SECTION. 78 Arr-Len VALUE 8. 01 arr-area VALUE "0204040405050709".
03 arr PIC 99 OCCURS Arr-Len TIMES.
01 i PIC 99.
01 avg PIC 9(3)V99.
01 std-dev PIC 9(3)V99.
PROCEDURE DIVISION.
DIVIDE FUNCTION sum-arr(arr-area) BY Arr-Len GIVING avg ROUNDED
PERFORM VARYING i FROM 1 BY 1 UNTIL i > Arr-Len COMPUTE arr (i) = (arr (i) - avg) ** 2 END-PERFORM
COMPUTE std-dev = FUNCTION SQRT(FUNCTION sum-arr(arr-area) / Arr-Len) DISPLAY std-dev .
END PROGRAM std-dev.
IDENTIFICATION DIVISION.
FUNCTION-ID. sum-arr.
DATA DIVISION. LOCAL-STORAGE SECTION. 01 i PIC 99.
LINKAGE SECTION. 78 Arr-Len VALUE 8. 01 arr-area.
03 arr PIC 99 OCCURS Arr-Len TIMES.
01 arr-sum PIC 99.
PROCEDURE DIVISION USING arr-area RETURNING arr-sum.
INITIALIZE arr-sum *> Without this, arr-sum is initialised incorrectly. PERFORM VARYING i FROM 1 BY 1 UNTIL i > Arr-Len ADD arr (i) TO arr-sum END-PERFORM .
END FUNCTION sum-arr.</lang>
CoffeeScript
Uses a class instance to maintain state.
<lang coffeescript> class StandardDeviation
constructor: -> @sum = 0 @sumOfSquares = 0 @values = 0 @deviation = 0
include: ( n ) -> @values += 1 @sum += n @sumOfSquares += n * n mean = @sum / @values mean *= mean @deviation = Math.sqrt @sumOfSquares / @values - mean
dev = new StandardDeviation values = [ 2, 4, 4, 4, 5, 5, 7, 9 ] tmp = []
for value in values
tmp.push value dev.include value console.log """ Values: #{ tmp } Standard deviation: #{ dev.deviation }
"""
</lang>
- Output:
Values: 2 Standard deviation: 0 Values: 2,4 Standard deviation: 1 Values: 2,4,4 Standard deviation: 0.9428090415820626 Values: 2,4,4,4 Standard deviation: 0.8660254037844386 Values: 2,4,4,4,5 Standard deviation: 0.9797958971132716 Values: 2,4,4,4,5,5 Standard deviation: 1 Values: 2,4,4,4,5,5,7 Standard deviation: 1.3997084244475297 Values: 2,4,4,4,5,5,7,9 Standard deviation: 2
Common Lisp
<lang lisp>(defun std-dev (samples)
(let* ((n (length samples))
(mean (/ (reduce #'+ samples) n)) (tmp (mapcar (lambda (x) (expt (- x mean) 2)) samples)))
(sqrt (/ (reduce #'+ tmp) n))))
(format t "~a" (std-dev '(2 4 4 4 5 5 7 9))) </lang>
Based on some googled web site; written ages ago.
<lang lisp>(defun arithmetic-average (samples)
(/ (reduce #'+ samples) (length samples)))
(defun standard-deviation (samples)
(let ((mean (arithmetic-average samples))) (sqrt (* (/ 1.0d0 (length samples)) (reduce #'+ samples :key (lambda (x) (expt (- x mean) 2)))))))
(defun make-deviator ()
(let ((numbers '())) (lambda (x) (push x numbers) (standard-deviation numbers))))</lang>
<lang lisp>CL-USER> (loop with deviator = (make-deviator)
for i in '(2 4 4 4 5 5 7 9) collect (list i (funcall deviator i))) ==>
((2 0.0d0)
(4 1.0d0) (4 0.9428090415820634d0) (4 0.8660254037844386d0) (5 0.9797958971132713d0) (5 1.0d0) (7 1.3997084244475304d0) (9 2.0d0))</lang>
Since we don't care about the sample values once std dev is computed, we only need to keep track of their sum and square sums, hence:<lang lisp>(defun running-stddev ()
(let ((sum 0) (sq 0) (n 0)) (lambda (x) (incf sum x) (incf sq (* x x)) (incf n) (/ (sqrt (- (* n sq) (* sum sum))) n))))
(loop with f = (running-stddev) for i in '(2 4 4 4 5 5 7 9) do (format t "~a ~a~%" i (funcall f i)))</lang>
Component Pascal
BlackBox Component Builder <lang oberon2> MODULE StandardDeviation; IMPORT StdLog, Args,Strings,Math;
PROCEDURE Mean(x: ARRAY OF REAL; n: INTEGER; OUT mean: REAL); VAR i: INTEGER; total: REAL; BEGIN total := 0.0; FOR i := 0 TO n - 1 DO total := total + x[i] END; mean := total /n END Mean;
PROCEDURE SDeviation(x : ARRAY OF REAL;n: INTEGER): REAL; VAR i: INTEGER; mean,sum: REAL; BEGIN Mean(x,n,mean); sum := 0.0; FOR i := 0 TO n - 1 DO sum:= sum + ((x[i] - mean) * (x[i] - mean)); END; RETURN Math.Sqrt(sum/n); END SDeviation;
PROCEDURE Do*;
VAR
p: Args.Params;
x: POINTER TO ARRAY OF REAL;
i,done: INTEGER;
BEGIN
Args.Get(p);
IF p.argc > 0 THEN
NEW(x,p.argc);
FOR i := 0 TO p.argc - 1 DO x[i] := 0.0 END;
FOR i := 0 TO p.argc - 1 DO
Strings.StringToReal(p.args[i],x[i],done);
StdLog.Int(i + 1);StdLog.String(" :> ");StdLog.Real(SDeviation(x,i + 1));StdLog.Ln
END
END
END Do;
END StandardDeviation.
</lang>
Execute: ^Q StandardDeviation.Do 2 4 4 4 5 5 7 9 ~
- Output:
1 :> 0.0 2 :> 1.0 3 :> 0.9428090415820634 4 :> 0.8660254037844386 5 :> 0.9797958971132712 6 :> 1.0 7 :> 1.39970842444753 8 :> 2.0
D
<lang d>import std.stdio, std.math;
struct StdDev {
real sum = 0.0, sqSum = 0.0; long nvalues;
void addNumber(in real input) pure nothrow { nvalues++; sum += input; sqSum += input ^^ 2; }
real getStdDev() const pure nothrow { if (nvalues == 0) return 0.0; immutable real mean = sum / nvalues; return sqrt(sqSum / nvalues - mean ^^ 2); }
}
void main() {
StdDev stdev;
foreach (el; [2.0, 4, 4, 4, 5, 5, 7, 9]) { stdev.addNumber(el); writefln("%e", stdev.getStdDev()); }
}</lang>
- Output:
0.000000e+00 1.000000e+00 9.428090e-01 8.660254e-01 9.797959e-01 1.000000e+00 1.399708e+00 2.000000e+00
Delphi
Delphi has 2 functions in Math unit for standard deviation: StdDev (with Bessel correction) and PopnStdDev (without Bessel correction). The task assumes the second function:
<lang Delphi>program StdDevTest;
{$APPTYPE CONSOLE}
uses
Math;
begin
Writeln(PopnStdDev([2,4,4,4,5,5,7,9])); Readln;
end.</lang>
E
This implementation produces two (function) objects sharing state. It is idiomatic in E to separate input from output (read from write) rather than combining them into one object.
The algorithm is
and the results were checked against #Python.
<lang e>def makeRunningStdDev() {
var sum := 0.0 var sumSquares := 0.0 var count := 0.0 def insert(v) { sum += v sumSquares += v ** 2 count += 1 } /** Returns the standard deviation of the inputs so far, or null if there have been no inputs. */ def stddev() { if (count > 0) { def meanSquares := sumSquares/count def mean := sum/count def variance := meanSquares - mean**2 return variance.sqrt() } } return [insert, stddev]
}</lang>
<lang e>? def [insert, stddev] := makeRunningStdDev()
- value: <insert>, <stddev>
? [stddev()]
- value: [null]
? for value in [2,4,4,4,5,5,7,9] { > insert(value) > println(stddev()) > } 0.0 1.0 0.9428090415820626 0.8660254037844386 0.9797958971132716 1.0 1.3997084244475297 2.0</lang>
Emacs Lisp
This implementation uses a temporary buffer (the central data structure of emacs) to have simple local variables.
<lang lisp>(defun running-std (x)
; ensure that we have a float to avoid potential integer math errors. (setq x (float x)) ; define variables to use (defvar running-sum 0 "the running sum of all known values") (defvar running-len 0 "the running number of all known values") (defvar running-squared-sum 0 "the running squared sum of all known values") ; and make them local to this buffer (make-local-variable 'running-sum) (make-local-variable 'running-len) (make-local-variable 'running-squared-sum) ; now process the new value (setq running-sum (+ running-sum x)) (setq running-len (1+ running-len)) (setq running-squared-sum (+ running-squared-sum (* x x))) ; and calculate the new standard deviation (sqrt (- (/ running-squared-sum running-len) (/ (* running-sum running-sum) (* running-len running-len )))))</lang>
<lang lisp>(with-temp-buffer
(loop for i in '(2 4 4 4 5 5 7 9) do (insert (number-to-string (running-std i))) (newline)) (message (buffer-substring (point-min) (1- (point-max)))))
"0.0 1.0 0.9428090415820636 0.8660254037844386 0.9797958971132716 1.0 1.399708424447531 2.0"</lang>
Erlang
<lang Erlang> -module( standard_deviation ).
-export( [add_sample/2, create/0, destroy/1, get/1, task/0] ).
-compile({no_auto_import,[get/1]}).
add_sample( Pid, N ) -> Pid ! {add, N}.
create() -> erlang:spawn_link( fun() -> loop( [] ) end ).
destroy( Pid ) -> Pid ! stop.
get( Pid ) -> Pid ! {get, erlang:self()}, receive {get, Value, Pid} -> Value end.
task() -> Pid = create(), [add_print(Pid, X, add_sample(Pid, X)) || X <- [2,4,4,4,5,5,7,9]], destroy( Pid ).
add_print( Pid, N, _Add ) -> io:fwrite( "Standard deviation ~p when adding ~p~n", [get(Pid), N] ).
loop( Ns ) -> receive {add, N} -> loop( [N | Ns] ); {get, Pid} -> Pid ! {get, loop_calculate( Ns ), erlang:self()}, loop( Ns ); stop -> ok end.
loop_calculate( Ns ) -> Average = loop_calculate_average( Ns ), math:sqrt( loop_calculate_average([math:pow(X - Average, 2) || X <- Ns]) ).
loop_calculate_average( Ns ) -> lists:sum( Ns ) / erlang:length( Ns ). </lang>
- Output:
9> standard_deviation:task(). Standard deviation 0.0 when adding 2 Standard deviation 1.0 when adding 4 Standard deviation 0.9428090415820634 when adding 4 Standard deviation 0.8660254037844386 when adding 4 Standard deviation 0.9797958971132712 when adding 5 Standard deviation 1.0 when adding 5 Standard deviation 1.3997084244475302 when adding 7 Standard deviation 2.0 when adding 9
Factor
<lang factor>USING: accessors io kernel math math.functions math.parser sequences ; IN: standard-deviator
TUPLE: standard-deviator sum sum^2 n ;
- <standard-deviator> ( -- standard-deviator )
0.0 0.0 0 standard-deviator boa ;
- current-std ( standard-deviator -- std )
[ [ sum^2>> ] [ n>> ] bi / ] [ [ sum>> ] [ n>> ] bi / sq ] bi - sqrt ;
- add-value ( value standard-deviator -- )
[ nip [ 1 + ] change-n drop ] [ [ + ] change-sum drop ] [ [ [ sq ] dip + ] change-sum^2 drop ] 2tri ;
- main ( -- )
{ 2 4 4 4 5 5 7 9 } <standard-deviator> [ [ add-value ] curry each ] keep current-std number>string print ;</lang>
Forth
<lang forth>: f+! ( x addr -- ) dup f@ f+ f! ;
- st-count ( stats -- n ) f@ ;
- st-sum ( stats -- sum ) float+ f@ ;
- st-sumsq ( stats -- sum*sum ) 2 floats + f@ ;
- st-add ( fnum stats -- )
1e dup f+! float+ fdup dup f+! float+ fdup f* f+! ;
- st-mean ( stats -- mean )
dup st-sum st-count f/ ;
- st-variance ( stats -- var )
dup st-sumsq dup st-mean fdup f* dup st-count f* f- st-count f/ ;
- st-stddev ( stats -- stddev )
st-variance fsqrt ;</lang>
This variation is more numerically stable when there are large numbers of samples or large sample ranges. <lang forth>: st-count ( stats -- n ) f@ ;
- st-mean ( stats -- mean ) float+ f@ ;
- st-nvar ( stats -- n*var ) 2 floats + f@ ;
- st-variance ( stats -- var ) dup st-nvar st-count f/ ;
- st-stddev ( stats -- stddev ) st-variance fsqrt ;
- st-add ( x stats -- )
1e dup f+! \ update count fdup dup st-mean f- fswap ( delta x ) fover dup st-count f/ ( delta x delta/n ) float+ dup f+! \ update mean ( delta x ) dup f@ f- f* float+ f+! ; \ update nvar</lang>
<lang forth>create stats 0e f, 0e f, 0e f,
2e stats st-add 4e stats st-add 4e stats st-add 4e stats st-add 5e stats st-add 5e stats st-add 7e stats st-add 9e stats st-add
stats st-stddev f. \ 2.</lang>
Fortran
This one imitates C and suffers the same problems: the function is not thread-safe and must be used to compute the stddev for one set per time.
<lang fortran>program Test_Stddev
implicit none
real, dimension(8) :: v = (/ 2,4,4,4,5,5,7,9 /) integer :: i real :: sd
do i = 1, size(v) sd = stat_object(v(i)) end do
print *, "std dev = ", sd
contains
recursive function stat_object(a, cmd) result(stddev) real :: stddev real, intent(in) :: a character(len=*), intent(in), optional :: cmd
real, save :: summa = 0.0, summa2 = 0.0 integer, save :: num = 0
real :: m
if ( .not. present(cmd) ) then num = num + 1 summa = summa + a summa2 = summa2 + a*a stddev = stat_object(0.0, "stddev") else select case(cmd) case("stddev") stddev = sqrt(stat_object(0.0, "variance")) case("variance") m = stat_object(0.0, "mean") if ( num > 0 ) then stddev = summa2/real(num) - m*m else stddev = 0.0 end if case("count") stddev = real(num) case("mean") if ( num > 0 ) then stddev = summa/real(num) else stddev = 0.0 end if case("reset") summa = 0.0 summa2 = 0.0 num = 0 case default stddev = 0.0 end select end if
end function stat_object
end program Test_Stddev</lang>
Using built-in array awareness
This uses Fortran's built-in array features (which aren't available in C)
<lang fortran> program stats
implicit none
integer, parameter :: N = 8 integer :: data(N) real(8) :: mean real(8) :: std_dev1, std_dev2
! Set the data data = [2,4,4,4,5,5,7,9] ! Strictly this is a Fortran 2003 construct
! Use intrinsic function 'sum' to calculate the mean mean = sum(data)/N
! Method1: ! Calculate the standard deviation directly from the definition std_dev1 = sqrt(sum((data - mean)**2)/N)
! Method 2: ! Use the alternative version that is less susceptible to rounding error std_dev2 = sqrt(sum(data**2)/N - mean**2)
write(*,'(a,8i2)') 'Data = ',data write(*,'(a,f3.1)') 'Mean = ',mean write(*,'(a,f3.1)') 'Standard deviation (method 1) = ',std_dev1 write(*,'(a,f3.1)') 'Standard deviation (method 2) = ',std_dev2
end program stats </lang>
Go
Algorithm to reduce rounding errors from WP article.
State maintained with a closure. <lang go>package main
import (
"fmt" "math"
)
func newRsdv() func(float64) float64 {
var n, a, q float64 return func(x float64) float64 { n++ a1 := a+(x-a)/n q, a = q+(x-a)*(x-a1), a1 return math.Sqrt(q/n) }
}
func main() {
r := newRsdv() for _, x := range []float64{2,4,4,4,5,5,7,9} { fmt.Println(r(x)) }
}</lang>
- Output:
0 1 0.9428090415820634 0.8660254037844386 0.9797958971132713 1 1.3997084244475304 2
Groovy
Solution: <lang groovy>def sum = 0 def sumSq = 0 def count = 0 [2,4,4,4,5,5,7,9].each {
sum += it sumSq += it*it count++ println "running std.dev.: ${(sumSq/count - (sum/count)**2)**0.5}"
}</lang>
- Output:
running std.dev.: 0 running std.dev.: 1 running std.dev.: 0.9428090416999145 running std.dev.: 0.8660254037844386 running std.dev.: 0.9797958971132712 running std.dev.: 1 running std.dev.: 1.3997084243469262 running std.dev.: 2
Haskell
We store the state in the ST
monad using an STRef
.
<lang haskell>import Data.List (genericLength) import Data.STRef import Control.Monad.ST
sd :: RealFloat a => [a] -> a sd l = sqrt $ sum (map ((^2) . subtract mean) l) / n
where n = genericLength l mean = sum l / n
sdAccum :: RealFloat a => ST s (a -> ST s a) sdAccum = do
accum <- newSTRef [] return $ \x -> do modifySTRef accum (x:) list <- readSTRef accum return $ sd list
main = mapM_ print results
where results = runST $ do runningSD <- sdAccum mapM runningSD [2, 4, 4, 4, 5, 5, 7, 9]</lang>
Haxe
<lang haxe>using Lambda;
class Main { static function main():Void { var nums = [2, 4, 4, 4, 5, 5, 7, 9]; for (i in 1...nums.length+1) Sys.println(sdev(nums.slice(0, i))); }
static function average<T:Float>(nums:Array<T>):Float { return nums.fold(function(n, t) return n + t, 0) / nums.length; }
static function sdev<T:Float>(nums:Array<T>):Float { var store = []; var avg = average(nums); for (n in nums) { store.push((n - avg) * (n - avg)); }
return Math.sqrt(average(store)); } }</lang>
0 1 0.942809041582063 0.866025403784439 0.979795897113271 1 1.39970842444753 2
HicEst
<lang HicEst>REAL :: n=8, set(n), sum=0, sum2=0
set = (2,4,4,4,5,5,7,9)
DO k = 1, n
WRITE() 'Adding ' // set(k) // 'stdev = ' // stdev(set(k))
ENDDO
END ! end of "main"
FUNCTION stdev(x)
USE : sum, sum2, k sum = sum + x sum2 = sum2 + x*x stdev = ( sum2/k - (sum/k)^2) ^ 0.5 END</lang>
Adding 2 stdev = 0 Adding 4 stdev = 1 Adding 4 stdev = 0.9428090416 Adding 4 stdev = 0.8660254038 Adding 5 stdev = 0.9797958971 Adding 5 stdev = 1 Adding 7 stdev = 1.399708424 Adding 9 stdev = 2
Icon and Unicon
<lang Icon>rocedure main()
stddev() # reset state / empty every s := stddev(![2,4,4,4,5,5,7,9]) do
write("stddev (so far) := ",s)
end
procedure stddev(x) /: running standard deviation static X,sumX,sum2X
if /x then { # reset state X := [] sumX := sum2X := 0. } else { # accumulate put(X,x) sumX +:= x sum2X +:= x^2 return sqrt( (sum2X / *X) - (sumX / *X)^2 ) }
end</lang>
- Output:
stddev (so far) := 0.0 stddev (so far) := 1.0 stddev (so far) := 0.9428090415820626 stddev (so far) := 0.8660254037844386 stddev (so far) := 0.9797958971132716 stddev (so far) := 1.0 stddev (so far) := 1.39970842444753 stddev (so far) := 2.0
J
J is block-oriented; it expresses algorithms with the semantics of all the data being available at once. It does not have native lexical closure or coroutine semantics. It is possible to implement these semantics in J; see Moving Average for an example. We will not reprise that here. <lang j> mean=: +/ % #
dev=: - mean stddevP=: [: %:@mean *:@dev NB. A) 3 equivalent defs for stddevP stddevP=: [: mean&.:*: dev NB. B) uses Under (&.:) to apply inverse of *: after mean stddevP=: %:@(mean@:*: - *:@mean) NB. C) sqrt of ((mean of squares) - (square of mean))
stddevP\ 2 4 4 4 5 5 7 9
0 1 0.942809 0.866025 0.979796 1 1.39971 2</lang>
Alternatives:
Using verbose names for J primitives.
<lang j> of =: @:
sqrt =: %: sum =: +/ squares=: *: data =: ] mean =: sum % #
stddevP=: sqrt of mean of squares of (data-mean)
stddevP\ 2 4 4 4 5 5 7 9
0 1 0.942809 0.866025 0.979796 1 1.39971 2</lang>
Or we could take a cue from the R implementation and reverse the Bessel correction to stddev:
<lang j> require'stats'
(%:@:(%~<:)@:# * stddev)\ 2 4 4 4 5 5 7 9
0 1 0.942809 0.866025 0.979796 1 1.39971 2</lang>
Java
<lang java>public class StdDev {
int n = 0; double sum = 0; double sum2 = 0;
public double sd(double x) {
n++; sum += x; sum2 += x*x;
return Math.sqrt(sum2/n - sum*sum/n/n);
}
public static void main(String[] args) { double[] testData = {2,4,4,4,5,5,7,9}; StdDev sd = new StdDev();
for (double x : testData) { System.out.println(sd.sd(x)); } }
}</lang>
JavaScript
Uses a closure. <lang javascript>function running_stddev() {
var n = 0; var sum = 0.0; var sum_sq = 0.0; return function(num) { n++; sum += num; sum_sq += num*num; return Math.sqrt( (sum_sq / n) - Math.pow(sum / n, 2) ); }
}
var sd = running_stddev(); var nums = [2,4,4,4,5,5,7,9]; var stddev = []; for (var i in nums)
stddev.push( sd(nums[i]) );
// using WSH WScript.Echo(stddev.join(', ');</lang>
- Output:
0, 1, 0.942809041582063, 0.866025403784439, 0.979795897113273, 1, 1.39970842444753, 2
jq
Observations from a file or array
We first define a filter, "simulate", that, if given a file of observations, will emit the standard deviations of the observations seen so far. The current state is stored in a JSON object, with this structure:
{ "n": _, "ssd": _, "mean": _ }
where "n" is the number of observations seen, "mean" is their average, and "ssd" is the sum of squared deviations about that mean.
The challenge here is to ensure accuracy for very large n, without sacrificing efficiency. The key idea in that regard is that if m is the mean of a series of n observations, x, we then have for any a:
SIGMA( (x - a)^2 ) == SIGMA( (x-m)^2 ) + n * (a-m)^2 == SSD + n*(a-m)^2 where SSD is the sum of squared deviations about the mean.
<lang jq># Compute the standard deviation of the observations
- seen so far, given the current state as input:
def standard_deviation: .ssd / .n | sqrt;
def update_state(observation):
def sq: .*.; ((.mean * .n + observation) / (.n + 1)) as $newmean | (.ssd + .n * ((.mean - $newmean) | sq)) as $ssd | { "n": (.n + 1), "ssd": ($ssd + ((observation - $newmean) | sq)), "mean": $newmean }
def initial_state: { "n": 0, "ssd": 0, "mean": 0 };
- Given an array of observations presented as input:
def simulate:
def _simulate(i; observations): if (observations|length) <= i then empty else update_state(observations[i]) | standard_deviation, _simulate(i+1; observations) end ; . as $in | initial_state | _simulate(0; $in);
- Begin:
simulate</lang> Example 1
# observations.txt 2 4 4 4 5 5 7 9
- Output:
<lang sh> $ jq -s -f Dynamic_standard_deviation.jq observations.txt 0 1 0.9428090415820634 0.8660254037844386 0.9797958971132711 0.9999999999999999 1.3997084244475302 1.9999999999999998 </lang>
Observations from a stream
Recent versions of jq (after 1.4) support retention of state while processing a stream. This means that any generator (including generators that produce items indefinitely) can be used as the source of observations, without first having to capture all the observations, e.g. in a file or array. <lang jq># requires jq version > 1.4 def simulate(stream):
foreach stream as $observation (initial_state; update_state($observation); standard_deviation);</lang>
Example 2:
simulate( range(0;10) )
- Output:
0 0.5 0.816496580927726 1.118033988749895 1.4142135623730951 1.707825127659933 2 2.29128784747792 2.581988897471611 2.8722813232690143
Observations from an external stream
The following illustrates how jq can be used to process observations from an external (potentially unbounded) stream, one at a time. Here we use bash to manage the calls to jq.
The definitions of the filters update_state/1 and initial_state/0 are as above but are repeated so that this script is self-contained. <lang sh>#!/bin/bash
- jq is assumed to be on PATH
PROGRAM=' def standard_deviation: .ssd / .n | sqrt;
def update_state(observation):
def sq: .*.; ((.mean * .n + observation) / (.n + 1)) as $newmean | (.ssd + .n * ((.mean - $newmean) | sq)) as $ssd | { "n": (.n + 1), "ssd": ($ssd + ((observation - $newmean) | sq)), "mean": $newmean }
def initial_state: { "n": 0, "ssd": 0, "mean": 0 };
- Input should be [observation, null] or [observation, state]
def standard_deviations:
. as $in | if type == "array" then (if .[1] == null then initial_state else .[1] end) as $state | $state | update_state($in[0]) | standard_deviation, . else empty end
standard_deviations ' state=null while read -p "Next observation: " observation do
result=$(echo "[ $observation, $state ]" | jq -c "$PROGRAM") sed -n 1p <<< "$result" state=$(sed -n 2p <<< "$result")
done</lang> Example 3 <lang sh>$ ./standard_deviation_server.sh Next observation: 10 0 Next observation: 20 5 Next observation: 0 8.16496580927726 </lang>
Liberty BASIC
Using a global array to maintain the state. Implements definition explicitly. <lang lb>
dim SD.storage$( 100) ' can call up to 100 versions, using ID to identify.. arrays are global. ' holds (space-separated) number of data items so far, current sum.of.values and current sum.of.squares
for i =1 to 8 read x print "New data "; x; " so S.D. now = "; using( "###.######", standard.deviation( 1, x)) next i
end
function standard.deviation( ID, in)
if SD.storage$( ID) ="" then SD.storage$( ID) ="0 0 0" num.so.far =val( word$( SD.storage$( ID), 1)) sum.vals =val( word$( SD.storage$( ID), 2)) sum.sqs =val( word$( SD.storage$( ID), 3)) num.so.far =num.so.far +1 sum.vals =sum.vals +in sum.sqs =sum.sqs +in^2
' standard deviation = square root of (the average of the squares less the square of the average) standard.deviation =( ( sum.sqs /num.so.far) - ( sum.vals /num.so.far)^2)^0.5
SD.storage$( ID) =str$( num.so.far) +" " +str$( sum.vals) +" " +str$( sum.sqs)
end function
Data 2, 4, 4, 4, 5, 5, 7, 9
</lang>
New data 2 so S.D. now = 0.000000 New data 4 so S.D. now = 1.000000 New data 4 so S.D. now = 0.942809 New data 4 so S.D. now = 0.866025 New data 5 so S.D. now = 0.979796 New data 5 so S.D. now = 1.000000 New data 7 so S.D. now = 1.399708 New data 9 so S.D. now = 2.000000
Lua
Uses a closure. Translation of JavaScript. <lang lua>function stdev()
local sum, sumsq, k = 0,0,0 return function(n) sum, sumsq, k = sum + n, sumsq + n^2, k+1 return math.sqrt((sumsq / k) - (sum/k)^2) end
end
ldev = stdev() for i, v in ipairs{2,4,4,4,5,5,7,9} do
print(ldev(v))
end</lang>
Mathematica
<lang Mathematica>runningSTDDev[n_] := (If[Not[ValueQ[$Data]], $Data = {}];
StandardDeviation[AppendTo[$Data, n]])</lang>
MATLAB / Octave
The simple form is, computing only the standand deviation of the whole data set:
<lang Matlab> x = [2,4,4,4,5,5,7,9];
n = length (x);
m = mean (x); x2 = mean (x .* x); dev= sqrt (x2 - m * m) dev = 2 </lang>
When the intermediate results are also needed, one can use this vectorized form:
<lang Matlab> m = cumsum(x) ./ [1:n]; % running mean
x2= cumsum(x.^2) ./ [1:n]; % running squares
dev = sqrt(x2 - m .* m) dev = 0.00000 1.00000 0.94281 0.86603 0.97980 1.00000 1.39971 2.00000
</lang>
МК-61/52
<lang>0 П4 П5 П6 С/П П0 ИП5 + П5 ИП0 x^2 ИП6 + П6 КИП4 ИП6 ИП4 / ИП5 ИП4 / x^2 - КвКор БП 04</lang>
Instruction: В/О С/П number С/П number С/П ...
Nim
<lang nim>import math, strutils
var sdSum, sdSum2, sdN = 0.0 proc sd(x): float =
sdN += 1 sdSum += float(x) sdSum2 += float(x*x) sqrt(sdSum2/sdN - sdSum*sdSum/sdN/sdN)
for value in [2,4,4,4,5,5,7,9]:
echo value, " ", formatFloat(sd(value), precision = 0)</lang>
- Output:
2 0 4 1 4 0.942809 4 0.866025 5 0.979796 5 1 7 1.39971 9 2
Objective-C
<lang objc>#import <Foundation/Foundation.h>
@interface SDAccum : NSObject {
double sum, sum2; unsigned int num;
} -(double)value: (double)v; -(unsigned int)count; -(double)mean; -(double)variance; -(double)stddev; @end
@implementation SDAccum -(double)value: (double)v {
sum += v; sum2 += v*v; num++; return [self stddev];
} -(unsigned int)count {
return num;
} -(double)mean {
return (num>0) ? sum/(double)num : 0.0;
} -(double)variance {
double m = [self mean]; return (num>0) ? (sum2/(double)num - m*m) : 0.0;
} -(double)stddev {
return sqrt([self variance]);
} @end
int main() {
@autoreleasepool {
double v[] = { 2,4,4,4,5,5,7,9 };
SDAccum *sdacc = [[SDAccum alloc] init];
for(int i=0; i < sizeof(v)/sizeof(*v) ; i++) printf("adding %f\tstddev = %f\n", v[i], [sdacc value: v[i]]);
} return 0;
}</lang>
Blocks
<lang objc>#import <Foundation/Foundation.h>
typedef double (^Func)(double); // a block that takes a double and returns a double
Func sdCreator() {
__block int n = 0; __block double sum = 0; __block double sum2 = 0; return ^(double x) { sum += x; sum2 += x*x; n++; return sqrt(sum2/n - sum*sum/n/n); };
}
int main() {
@autoreleasepool {
double v[] = { 2,4,4,4,5,5,7,9 };
Func sdacc = sdCreator();
for(int i=0; i < sizeof(v)/sizeof(*v) ; i++) printf("adding %f\tstddev = %f\n", v[i], sdacc(v[i]));
} return 0;
}</lang>
Objeck
<lang objeck> use Structure;
bundle Default {
class StdDev { nums : FloatVector; New() { nums := FloatVector->New(); } function : Main(args : String[]) ~ Nil { sd := StdDev->New(); test_data := [2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0]; each(i : test_data) { sd->AddNum(test_data[i]); sd->GetSD()->PrintLine(); }; } method : public : AddNum(num : Float) ~ Nil { nums->AddBack(num); } method : public : native : GetSD() ~ Float { sq_diffs := 0.0; avg := nums->Average(); each(i : nums) { num := nums->Get(i); sq_diffs += (num - avg) * (num - avg); }; return (sq_diffs / nums->Size())->SquareRoot(); } }
} </lang>
OCaml
<lang ocaml>let sqr x = x *. x
let stddev l =
let n, sx, sx2 = List.fold_left (fun (n, sx, sx2) x -> succ n, sx +. x, sx2 +. sqr x) (0, 0., 0.) l in sqrt ((sx2 -. sqr sx /. float n) /. float n)
let _ =
let l = [ 2.;4.;4.;4.;5.;5.;7.;9. ] in Printf.printf "List: "; List.iter (Printf.printf "%g ") l; Printf.printf "\nStandard deviation: %g\n" (stddev l)</lang>
- Output:
List: 2 4 4 4 5 5 7 9 Standard deviation: 2
ooRexx
<lang rexx>sdacc = .SDAccum~new x = .array~of(2,4,4,4,5,5,7,9) sd = 0 do i = 1 to x~size
sd = sdacc~value(x[i])
end
say "std dev = "sd
- class SDAccum
- method sum attribute
- method sum2 attribute
- method count attribute
- method init
self~sum = 0.0 self~sum2 = 0.0 self~count = 0
- method value
expose sum sum2 count parse arg x sum = sum + x sum2 = sum2 + x*x count = count + 1 return self~stddev
- method mean
expose sum count return sum/count
- method variance
expose sum2 count m = self~mean return sum2/count - m*m
- method stddev
return self~sqrt(self~variance)
- method sqrt
arg n if n = 0 then return 0 ans = n / 2 prev = n do until prev = ans prev = ans ans = ( prev + ( n / prev ) ) / 2 end return ans</lang>
PARI/GP
Uses the Cramer-Young updating algorithm. For demonstration it displays the mean and variance at each step. <lang parigp>newpoint(x)={
myT=x; myS=0; myN=1; [myT,myS]/myN
}; addpoint(x)={
myT+=x; myN++; myS+=(myN*x-myT)^2/myN/(myN-1); [myT,myS]/myN
}; addpoints(v)={
print(newpoint(v[1])); for(i=2,#v,print(addpoint(v[i]))); print("Mean: ",myT/myN); print("Standard deviation: ",sqrt(myS/myN))
}; addpoints([2,4,4,4,5,5,7,9])</lang>
Pascal
<lang pascal>program stddev; uses math; const
n=8;
var
arr: array[1..n] of real =(2,4,4,4,5,5,7,9);
function stddev(n: integer): real; var
i: integer; s1,s2,variance,x: real;
begin
for i:=1 to n do begin x:=arr[i]; s1:=s1+power(x,2); s2:=s2+x end; variance:=((n*s1)-(power(s2,2)))/(power(n,2)); stddev:=sqrt(variance)
end; var
i: integer;
begin
for i:=1 to n do begin writeln(i,' item=',arr[i]:2:0,' stddev=',stddev(i):18:15) end
end.</lang>
- Output:
1 item= 2 stddev= 0.000000000000000 2 item= 4 stddev= 1.000000000000000 3 item= 4 stddev= 0.942809041582064 4 item= 4 stddev= 0.866025403784439 5 item= 5 stddev= 0.979795897113271 6 item= 5 stddev= 1.000000000000000 7 item= 7 stddev= 1.399708424447530 8 item= 9 stddev= 2.000000000000000
Perl
<lang perl>{
package SDAccum; sub new {
my $class = shift; my $self = {}; $self->{sum} = 0.0; $self->{sum2} = 0.0; $self->{num} = 0; bless $self, $class; return $self;
} sub count {
my $self = shift; return $self->{num};
} sub mean {
my $self = shift; return ($self->{num}>0) ? $self->{sum}/$self->{num} : 0.0;
} sub variance {
my $self = shift; my $m = $self->mean; return ($self->{num}>0) ? $self->{sum2}/$self->{num} - $m * $m : 0.0;
} sub stddev {
my $self = shift; return sqrt($self->variance);
} sub value {
my $self = shift; my $v = shift; $self->{sum} += $v; $self->{sum2} += $v * $v; $self->{num}++; return $self->stddev;
}
}</lang>
<lang perl>my $sdacc = SDAccum->new; my $sd;
foreach my $v ( 2,4,4,4,5,5,7,9 ) {
$sd = $sdacc->value($v);
} print "std dev = $sd\n";</lang>
A much shorter version using a closure and a property of the variance:
<lang perl># <(x - <x>)²> = <x²> - <x>² {
my $num, $sum, $sum2; sub stddev {
my $x = shift; $num++; return sqrt( ($sum2 += $x**2) / $num - (($sum += $x) / $num)**2 );
}
}
print stddev($_), "\n" for qw(2 4 4 4 5 5 7 9);</lang>
- Output:
0 1 0.942809041582063 0.866025403784439 0.979795897113272 1 1.39970842444753 2
Perl 6
Using a closure: <lang perl6>sub sd (@a) {
my $mean = @a R/ [+] @a; sqrt @a R/ [+] map (* - $mean)**2, @a;
} sub sdaccum {
my @a; return { push @a, $^x; sd @a; };
} my &f = sdaccum; say f $_ for 2, 4, 4, 4, 5, 5, 7, 9;</lang>
Using a state variable: <lang perl6># remember that <(x-<x>)²> = <x²> - <x>² sub stddev($x) {
sqrt (.[2] += $x**2) / ++.[0] - ((.[1] += $x) / .[0])**2 given state @;
}
say stddev $_ for <2 4 4 4 5 5 7 9>;</lang>
- Output:
0 1 0.942809041582063 0.866025403784439 0.979795897113271 1 1.39970842444753 2
PL/I
<lang pli>*process source attributes xref;
stddev: proc options(main); declare a(10) float init(1,2,3,4,5,6,7,8,9,10); declare stdev float; declare i fixed binary; stdev=std_dev(a); put skip list('Standard deviation', stdev); std_dev: procedure(a) returns(float); declare a(*) float, n fixed binary; n=hbound(a,1); begin; declare b(n) float, average float; declare i fixed binary; do i=1 to n; b(i)=a(i); end; average=sum(a)/n; put skip data(average); return( sqrt(sum(b**2)/n - average**2) ); end; end std_dev; end;</lang>
- Output:
AVERAGE= 5.50000E+0000; Standard deviation 2.87228E+0000
PicoLisp
<lang PicoLisp>(scl 2)
(de stdDev ()
(curry ((Data)) (N) (push 'Data N) (let (Len (length Data) M (*/ (apply + Data) Len)) (sqrt (*/ (sum '((N) (*/ (- N M) (- N M) 1.0)) Data ) 1.0 Len ) T ) ) ) )
(let Fun (stdDev)
(for N (2.0 4.0 4.0 4.0 5.0 5.0 7.0 9.0) (prinl (format N *Scl) " -> " (format (Fun N) *Scl)) ) )</lang>
- Output:
2.00 -> 0.00 4.00 -> 1.00 4.00 -> 0.94 4.00 -> 0.87 5.00 -> 0.98 5.00 -> 1.00 7.00 -> 1.40 9.00 -> 2.00
PowerShell
This implementation takes the form of an advanced function which can act like a cmdlet and receive input from the pipeline. <lang powershell>function Get-StandardDeviation {
begin { $avg = 0 $nums = @() } process { $nums += $_ $avg = ($nums | Measure-Object -Average).Average $sum = 0; $nums | ForEach-Object { $sum += ($avg - $_) * ($avg - $_) } [Math]::Sqrt($sum / $nums.Length) }
}</lang> Usage as follows:
PS> 2,4,4,4,5,5,7,9 | Get-StandardDeviation 0 1 0.942809041582063 0.866025403784439 0.979795897113271 1 1.39970842444753 2
PureBasic
<lang PureBasic>;Define our Standard deviation function Declare.d Standard_deviation(x)
- Main program
If OpenConsole()
Define i, x Restore MyList For i=1 To 8 Read.i x PrintN(StrD(Standard_deviation(x))) Next i Print(#CRLF$+"Press ENTER to exit"): Input()
EndIf
- Calculation procedure, with memory
Procedure.d Standard_deviation(In)
Static in_summa, antal Static in_kvadrater.q in_summa+in in_kvadrater+in*in antal+1 ProcedureReturn Pow((in_kvadrater/antal)-Pow(in_summa/antal,2),0.50)
EndProcedure
- data section
DataSection MyList:
Data.i 2,4,4,4,5,5,7,9
EndDataSection</lang>
- Output:
0.0000000000 1.0000000000 0.9428090416 0.8660254038 0.9797958971 1.0000000000 1.3997084244 2.0000000000
Python
Using a function with attached properties
The program should work with Python 2.x and 3.x, although the output would not be a tuple in 3.x <lang python>>>> from math import sqrt >>> def sd(x):
sd.sum += x sd.sum2 += x*x sd.n += 1.0 sum, sum2, n = sd.sum, sd.sum2, sd.n return sqrt(sum2/n - sum*sum/n/n)
>>> sd.sum = sd.sum2 = sd.n = 0 >>> for value in (2,4,4,4,5,5,7,9):
print (value, sd(value))
(2, 0.0)
(4, 1.0)
(4, 0.94280904158206258)
(4, 0.8660254037844386)
(5, 0.97979589711327075)
(5, 1.0)
(7, 1.3997084244475311)
(9, 2.0)
>>></lang>
Using a class instance
<lang python>>>> class SD(object): # Plain () for python 3.x def __init__(self): self.sum, self.sum2, self.n = (0,0,0) def sd(self, x): self.sum += x self.sum2 += x*x self.n += 1.0 sum, sum2, n = self.sum, self.sum2, self.n return sqrt(sum2/n - sum*sum/n/n)
>>> sd_inst = SD() >>> for value in (2,4,4,4,5,5,7,9): print (value, sd_inst.sd(value))</lang>
Using a Closure
<lang python>>>> from math import sqrt >>> def sdcreator(): sum = sum2 = n = 0 def sd(x): nonlocal sum, sum2, n
sum += x sum2 += x*x n += 1.0 return sqrt(sum2/n - sum*sum/n/n) return sd
>>> sd = sdcreator() >>> for value in (2,4,4,4,5,5,7,9): print (value, sd(value))
2 0.0
4 1.0
4 0.942809041582
4 0.866025403784
5 0.979795897113
5 1.0
7 1.39970842445
9 2.0</lang>
Using an extended generator
<lang python>>>> from math import sqrt >>> def sdcreator(): sum = sum2 = n = 0 while True: x = yield sqrt(sum2/n - sum*sum/n/n) if n else None
sum += x sum2 += x*x n += 1.0
>>> sd = sdcreator() >>> sd.send(None) >>> for value in (2,4,4,4,5,5,7,9): print (value, sd.send(value))
2 0.0
4 1.0
4 0.942809041582
4 0.866025403784
5 0.979795897113
5 1.0
7 1.39970842445
9 2.0</lang>
R
Built-in Std Dev fn
<lang rsplus>#The built-in standard deviation function applies the Bessel correction. To reverse this, we can apply an uncorrection.
- If na.rm is true, missing data points (NA values) are removed.
reverseBesselCorrection <- function(x, na.rm=FALSE) { if(na.rm) x <- x[!is.na(x)] len <- length(x) if(len < 2) stop("2 or more data points required") sqrt((len-1)/len) } testdata <- c(2,4,4,4,5,5,7,9) reverseBesselCorrection(testdata)*sd(testdata) #2</lang>
From scratch
<lang rsplus>#Again, if na.rm is true, missing data points (NA values) are removed.
uncorrectedsd <- function(x, na.rm=FALSE) { len <- length(x) if(len < 2) stop("2 or more data points required") mu <- mean(x, na.rm=na.rm) ssq <- sum((x - mu)^2, na.rm=na.rm) usd <- sqrt(ssq/len) usd } uncorrectedsd(testdata) #2</lang>
Racket
<lang racket>
- lang racket
(require math) (define running-stddev
(let ([ns '()]) (λ(n) (set! ns (cons n ns)) (stddev ns))))
- run it on each number, return the last result
(last (map running-stddev '(2 4 4 4 5 5 7 9))) </lang>
REXX
Uses running sums. <lang rexx>/*REXX pgm finds & displays the standard deviation of a given set of #s.*/ parse arg # /*any optional args on the C.L. ?*/ if #= then #=2 4 4 4 5 5 7 9 /*None given? Then use default.*/ w=words(#); s=0; ss=0 /*define: #items; a couple sums.*/
do j=1 for w; _=word(#,j); s=s+_; ss=ss+_*_ say ' item' right(j,length(w))":" right(_,4), ' average=' left(s/j,12), ' standard deviation=' left(sqrt( ss/j - (s/j)**2 ),15) end /*j*/
exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────SQRT subroutine─────────────────────*/ sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); numeric digits 11 numeric form; m.=11; p=d+d%4+2; parse value format(x,2,1,,0) 'E0' with g 'E' _ . g=g*.5'E'_%2; do j=0 while p>9; m.j=p; p=p%2+1; end
do k=j+5 to 0 by -1; if m.k>11 then numeric digits m.k; g=.5*(g+x/g); end
numeric digits d; return g/1</lang>
- Output:
using the default input
item 1: 2 average= 2 standard deviation= 0 item 2: 4 average= 3 standard deviation= 1 item 3: 4 average= 3.33333333 standard deviation= 0.942809047 item 4: 4 average= 3.5 standard deviation= 0.866025404 item 5: 5 average= 3.8 standard deviation= 0.979795897 item 6: 5 average= 4 standard deviation= 1 item 7: 7 average= 4.42857143 standard deviation= 1.39970843 item 8: 9 average= 5 standard deviation= 2
Ruby
Object
Uses an object to keep state.
"Simplification of the formula [...] for standard deviation [...] can be memorized as taking the square root of (the average of the squares less the square of the average)." c.f. wikipedia.
<lang ruby>class StdDevAccumulator
def initialize @n, @sum, @sumofsquares = 0, 0.0, 0.0 end def <<(num) # return self to make this possible: sd << 1 << 2 << 3 # => 0.816496580927726 @n += 1 @sum += num @sumofsquares += num**2 self end def stddev Math.sqrt( (@sumofsquares / @n) - (@sum / @n)**2 ) end def to_s stddev.to_s end
end
sd = StdDevAccumulator.new i = 0 [2,4,4,4,5,5,7,9].each {|n| puts "adding #{n}: stddev of #{i+=1} samples is #{sd << n}" }</lang>
adding 2: stddev of 1 samples is 0.0 adding 4: stddev of 2 samples is 1.0 adding 4: stddev of 3 samples is 0.942809041582063 adding 4: stddev of 4 samples is 0.866025403784439 adding 5: stddev of 5 samples is 0.979795897113272 adding 5: stddev of 6 samples is 1.0 adding 7: stddev of 7 samples is 1.39970842444753 adding 9: stddev of 8 samples is 2.0
Closure
<lang ruby>def sdaccum
n, sum, sum2 = 0, 0.0, 0.0 lambda do |num| n += 1 sum += num sum2 += num**2 Math.sqrt( (sum2 / n) - (sum / n)**2 ) end
end
sd = sdaccum [2,4,4,4,5,5,7,9].each {|n| print sd.call(n), ", "}</lang>
0.0, 1.0, 0.942809041582063, 0.866025403784439, 0.979795897113272, 1.0, 1.39970842444753, 2.0,
Run BASIC
<lang runbasic>dim sdSave$(100) 'can call up to 100 versions
'holds (space-separated) number of data , sum of values and sum of squares
sd$ = "2,4,4,4,5,5,7,9"
for num = 1 to 8
stdData = val(word$(sd$,num,",")) sumVal = sumVal + stdData sumSqs = sumSqs + stdData^2 ' standard deviation = square root of (the average of the squares less the square of the average) standDev =((sumSqs / num) - (sumVal /num) ^ 2) ^ 0.5 sdSave$(num) = str$(num);" ";str$(sumVal);" ";str$(sumSqs) print num;" value in = ";stdData; " Stand Dev = "; using("###.######", standDev)
next num</lang>
1 value in = 2 Stand Dev = 0.000000 2 value in = 4 Stand Dev = 1.000000 3 value in = 4 Stand Dev = 0.942809 4 value in = 4 Stand Dev = 0.866025 5 value in = 5 Stand Dev = 0.979796 6 value in = 5 Stand Dev = 1.000000 7 value in = 7 Stand Dev = 1.399708 8 value in = 9 Stand Dev = 2.000000
SAS
<lang SAS>
- --Load the test data;
data test1;
input x @@; obs=_n_;
datalines; 2 4 4 4 5 5 7 9
run;
- --Create a dataset with the cummulative data for each set of data for which the SD should be calculated;
data test2 (drop=i obs);
set test1; y=x; do i=1 to n; set test1 (rename=(obs=setid)) nobs=n point=i; if obs<=setid then output; end;
proc sort;
by setid;
run;
- --Calulate the standards deviation (and mean) using PROC MEANS;
proc means data=test2 vardef=n noprint; *--use vardef=n option to calculate the population SD;
by setid; var y; output out=stat1 n=n mean=mean std=sd;
run;
- --Output the calculated standard deviations;
proc print data=stat1 noobs;
var n sd /*mean*/;
run; </lang>
- Output:
N SD 1 0.00000 2 1.00000 3 0.94281 4 0.86603 5 0.97980 6 1.00000 7 1.39971 8 2.00000
Scala
<lang Scala>import scala.math._ import Numeric.Implicits._
object StddevCalc extends App {
def avg[T](ts: Iterable[T])(implicit num: Fractional[T]): T = { num.div(ts.sum, num.fromInt(ts.size)) // Leaving with type of function T }
def calcAvgAndStddev[T](ts: Iterable[T])(implicit num: Fractional[T]): (T, Double) = { val mean = avg(ts) // Leave val type of T val stdDev = // Root of mean diffs sqrt(num.toDouble(ts.foldLeft(num.zero)((b, a) => num.plus(b, num.times(num.minus(a, mean), num.minus(a, mean))))) / ts.size) (mean, stdDev) }
def calcAvgAndStddev(ts: Iterable[BigDecimal]): (Double, Double) = // Overloaded for BigDecimal calcAvgAndStddev(ts.map(_.toDouble))
println(calcAvgAndStddev(List(2.0E0, 4.0, 4, 4, 5, 5, 7, 9))) println(calcAvgAndStddev(Set(1.0, 2, 3, 4))) println(calcAvgAndStddev(0.1 to 1.1 by 0.05)) println(calcAvgAndStddev(List(BigDecimal(120), BigDecimal(1200))))
}</lang>
Scheme
<lang scheme> (define ((running-stddev . nums) num)
(set! nums (cons num nums)) (sqrt (- (/ (apply + (map (lambda (i) (* i i)) nums)) (length nums)) (expt (/ (apply + nums) (length nums)) 2))))
</lang>
Scilab
Scilab has the built-in function stdev to compute the standard deviation of a sample so it is straightforward to have the standard deviation of a sample with a correction of the bias. <lang>T=[2,4,4,4,5,5,7,9]; stdev(T)*sqrt((length(T)-1)/length(T))</lang>
- Output:
-->T=[2,4,4,4,5,5,7,9]; -->stdev(T)*sqrt((length(T)-1)/length(T)) ans = 2.
Smalltalk
<lang smalltalk>Object subclass: SDAccum [
|sum sum2 num| SDAccum class >> new [ |o| o := super basicNew. ^ o init. ] init [ sum := 0. sum2 := 0. num := 0 ] value: aValue [ sum := sum + aValue. sum2 := sum2 + ( aValue * aValue ). num := num + 1. ^ self stddev ] count [ ^ num ] mean [ num>0 ifTrue: [^ sum / num] ifFalse: [ ^ 0.0 ] ] variance [ |m| m := self mean. num>0 ifTrue: [^ (sum2/num) - (m*m) ] ifFalse: [ ^ 0.0 ] ] stddev [ ^ (self variance) sqrt ]
].</lang>
<lang smalltalk>|sdacc sd| sdacc := SDAccum new.
- ( 2 4 4 4 5 5 7 9 ) do: [ :v | sd := sdacc value: v ].
('std dev = %1' % { sd }) displayNl.</lang>
Swift
<lang Swift>import Darwin class stdDev{
var n:Double = 0.0 var sum:Double = 0.0 var sum2:Double = 0.0 init(){ let testData:[Double] = [2,4,4,4,5,5,7,9]; for x in testData{ var a:Double = calcSd(x) println("value \(Int(x)) SD = \(a)"); } } func calcSd(x:Double)->Double{ n += 1 sum += x sum2 += x*x return sqrt( sum2 / n - sum*sum / n / n) }
} var aa = stdDev()</lang>
- Output:
value 2 SD = 0.0 value 4 SD = 1.0 value 4 SD = 0.942809041582063 value 4 SD = 0.866025403784439 value 5 SD = 0.979795897113271 value 5 SD = 1.0 value 7 SD = 1.39970842444753 value 9 SD = 2.0
Tcl
With a Class
or
<lang tcl>oo::class create SDAccum {
variable sum sum2 num constructor {} { set sum 0.0 set sum2 0.0 set num 0 } method value {x} { set sum2 [expr {$sum2 + $x**2}] set sum [expr {$sum + $x}] incr num return [my stddev] } method count {} { return $num } method mean {} { expr {$sum / $num} } method variance {} { expr {$sum2/$num - [my mean]**2} } method stddev {} { expr {sqrt([my variance])} }
}
- Demonstration
set sdacc [SDAccum new] foreach val {2 4 4 4 5 5 7 9} {
set sd [$sdacc value $val]
} puts "the standard deviation is: $sd"</lang>
- Output:
the standard deviation is: 2.0
With a Coroutine
<lang tcl># Make a coroutine out of a lambda application coroutine sd apply {{} {
set sum 0.0 set sum2 0.0 set sd {} # Keep processing argument values until told not to... while {[set val [yield $sd]] ne "stop"} { incr n set sum [expr {$sum + $val}] set sum2 [expr {$sum2 + $val**2}] set sd [expr {sqrt($sum2/$n - ($sum/$n)**2)}] }
}}
- Demonstration
foreach val {2 4 4 4 5 5 7 9} {
set sd [sd $val]
} sd stop puts "the standard deviation is: $sd"</lang>
TI-83 BASIC
On the TI-83 family, standard deviation of a population is a builtin function (σx):
• Press [STAT] select [EDIT] followed by [ENTER] • then enter for list L1 in the table : 2, 4, 4, 4, 5, 5, 7, 9 • Or enter {2,4,4,4,5,5,7,9}→L1 • Press [STAT] select [CALC] then [1-Var Stats] select list L1 followed by [ENTER] • Then σx (=2) gives the standard deviation of the population
Visual Basic
Note that the helper function avg
is not named average
to avoid a name conflict with WorksheetFunction.Average
in MS Excel.
<lang vb>Function avg(what() As Variant) As Variant
'treats non-numeric strings as zero Dim L0 As Variant, total As Variant For L0 = LBound(what) To UBound(what) If IsNumeric(what(L0)) Then total = total + what(L0) Next avg = total / (1 + UBound(what) - LBound(what))
End Function
Function standardDeviation(fp As Variant) As Variant
Static list() As Variant Dim av As Variant, tmp As Variant, L0 As Variant
'add to sequence if numeric If IsNumeric(fp) Then On Error GoTo makeArr 'catch undimensioned list ReDim Preserve list(UBound(list) + 1) On Error GoTo 0 list(UBound(list)) = fp End If
'get average av = avg(list())
'the actual work For L0 = 0 To UBound(list) tmp = tmp + ((list(L0) - av) ^ 2) Next tmp = Sqr(tmp / (UBound(list) + 1))
standardDeviation = tmp
Exit Function
makeArr:
If 9 = Err.Number Then ReDim list(0) Else 'something's wrong Err.Raise Err.Number End If Resume Next
End Function
Sub tester()
Dim x As Variant x = Array(2, 4, 4, 4, 5, 5, 7, 9) For L0 = 0 To UBound(x) Debug.Print standardDeviation(x(L0)) Next
End Sub</lang>
- Output:
0 1 0.942809041582063 0.866025403784439 0.979795897113271 1 1.39970842444753 2
XPL0
<lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations int A, I; real N, S, S2; [A:= [2,4,4,4,5,5,7,9]; N:= 0.0; S:= 0.0; S2:= 0.0; for I:= 0 to 8-1 do
[N:= N + 1.0; S:= S + float(A(I)); S2:= S2 + float(sq(A(I))); RlOut(0, sqrt((S2/N) - sq(S/N))); ];
CrLf(0); ]</lang>
- Output:
0.00000 1.00000 0.94281 0.86603 0.97980 1.00000 1.39971 2.00000
zkl
<lang zkl>fcn sdf{ fcn(x,xs){
m:=xs.append(x.toFloat()).sum(0.0)/xs.len(); (xs.reduce('wrap(p,x){(x-m)*(x-m) +p},0.0)/xs.len()).sqrt() }.fp1(L())
}</lang>
- Output:
zkl: T(2,4,4,4,5,5,7,9).pump(Void,sdf()) 2 zkl: sd:=sdf(); sd(2);sd(4);sd(4);sd(4);sd(5);sd(5);sd(7);sd(9) 2
- Programming Tasks
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