Cumulative standard deviation
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 co-routine 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 .
Related tasks:
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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.Numerics.Elementary_Functions; use Ada.Numerics.Elementary_Functions; with Ada.Text_IO; use Ada.Text_IO; with Ada.Float_Text_IO; use Ada.Float_Text_IO; with Ada.Integer_Text_IO; use Ada.Integer_Text_IO;
procedure Test_Deviation is
type Sample is record
N : Natural := 0;
Sum : Float := 0.0;
SumOfSquares : Float := 0.0;
end record;
procedure Add (Data : in out Sample; Point : Float) is
begin
Data.N := Data.N + 1;
Data.Sum := Data.Sum + Point;
Data.SumOfSquares := Data.SumOfSquares + Point ** 2;
end Add;
function Deviation (Data : Sample) return Float is
begin
return Sqrt (Data.SumOfSquares / Float (Data.N) - (Data.Sum / Float (Data.N)) ** 2);
end Deviation;
Data : Sample; Test : array (1..8) of Integer := (2, 4, 4, 4, 5, 5, 7, 9);
begin
for Index in Test'Range loop
Add (Data, Float(Test(Index)));
Put("N="); Put(Item => Index, Width => 1);
Put(" ITEM="); Put(Item => Test(Index), Width => 1);
Put(" AVG="); Put(Item => Float(Data.Sum)/Float(Index), Fore => 1, Aft => 3, Exp => 0);
Put(" STDDEV="); Put(Item => Deviation (Data), Fore => 1, Aft => 3, Exp => 0);
New_line;
end loop;
end Test_Deviation; </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=0.943 N=4 ITEM=4 AVG=3.500 STDDEV=0.866 N=5 ITEM=5 AVG=3.800 STDDEV=0.980 N=6 ITEM=5 AVG=4.000 STDDEV=1.000 N=7 ITEM=7 AVG=4.429 STDDEV=1.400 N=8 ITEM=9 AVG=5.000 STDDEV=2.000
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); printf(($"value: "g(0,6)," standard dev := "g(0,6)l$, v[i], sd)) OD
)</lang>
- Output:
value: 2.000000 standard dev := .000000 value: 4.000000 standard dev := 1.000000 value: 4.000000 standard dev := .942809 value: 4.000000 standard dev := .866025 value: 5.000000 standard dev := .979796 value: 5.000000 standard dev := 1.000000 value: 7.000000 standard dev := 1.399708 value: 9.000000 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]; printf(($"value: "g(0,6)," standard dev := "g(0,6)l$, value[i], (stddev OF class stat)(stat))) 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:
value: 2.000000 standard dev := .000000 value: 4.000000 standard dev := 1.000000 value: 4.000000 standard dev := .942809 value: 4.000000 standard dev := .866025 value: 5.000000 standard dev := .979796 value: 5.000000 standard dev := 1.000000 value: 7.000000 standard dev := 1.399708 value: 9.000000 standard dev := 2.000000
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
ALGOL W
This is an Algol W version of the third, "unpackaged" Algol 68 sample, which was itself translated from Python. <lang algolw>begin
long real sum, sum2;
integer n;
long real procedure sd (long real value x) ;
begin
sum := sum + x;
sum2 := sum2 + (x*x);
n := n + 1;
if n = 0 then 0 else longsqrt(sum2/n - sum*sum/n/n)
end sd;
sum := sum2 := n := 0;
r_format := "A"; r_w := 14; r_d := 6; % set output to fixed point format %
for i := 2,4,4,4,5,5,7,9
do begin
long real val;
val := i;
write(val, sd(val))
end for_i
end.</lang>
- Output:
2.000000 0.000000
4.000000 1.000000
4.000000 0.942809
4.000000 0.866025
5.000000 0.979795
5.000000 1.000000
7.000000 1.399708
9.000000 2.000000
AppleScript
Accumulation across a fold
<lang AppleScript>-- stdDevInc :: Accumulator -> Num -> Index -> Accumulator -- stdDevInc :: {sum:, squaresSum:, stages:} -> Real -> Integer -- -> {sum:, squaresSum:, stages:} on stdDevInc(a, n, i)
set sum to (sum of a) + n
set squaresSum to (squaresSum of a) + (n ^ 2)
set stages to (stages of a) & ¬
((squaresSum / i) - ((sum / i) ^ 2)) ^ 0.5
{sum:sum, squaresSum:squaresSum, stages:stages}
end stdDevInc
-- TEST
on run
set lstSample to [2, 4, 4, 4, 5, 5, 7, 9]
stages of foldl(stdDevInc, ¬
{sum:0, squaresSum:0, stages:[]}, lstSample)
--> {0.0, 1.0, 0.942809041582, 0.866025403784, 0.979795897113, 1.0, 1.399708424448, 2.0}
end run
-- GENERIC FUNCTIONS ----------------------------------------------------------------------
-- foldl :: (a -> b -> a) -> a -> [b] -> a on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to lambda(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f)
if class of f is script then
f
else
script
property lambda : f
end script
end if
end mReturn</lang>
- Output:
<lang AppleScrip>{0.0, 1.0, 0.942809041582, 0.866025403784, 0.979795897113, 1.0, 1.399708424448, 2.0}</lang>
AutoHotkey
<lang AutoHotkey>Data := [2,4,4,4,5,5,7,9]
for k, v in Data {
FileAppend, % "#" a_index " value = " v " stddev = " stddev(v) "`n", * ; send to stdout
} return
stddev(x) { static n, sum, sum2 n++ sum += x sum2 += x*x
return sqrt((sum2/n) - (((sum*sum)/n)/n)) }</lang>
- Output:
#1 value = 2 stddev 0 0.000000 #2 value = 4 stddev 0 1.000000 #3 value = 4 stddev 0 0.942809 #4 value = 4 stddev 0 0.866025 #5 value = 5 stddev 0 0.979796 #6 value = 5 stddev 0 1.000000 #7 value = 7 stddev 0 1.399708 #8 value = 9 stddev 0 2.000000
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 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
C++
No attempt to handle different types -- standard deviation is intrinsically a real number. <lang cpp>
- include <assert.h>
- include <cmath>
- include <vector>
- include <iostream>
template<int N> struct MomentsAccumulator_ { std::vector<double> m_; MomentsAccumulator_() : m_(N + 1, 0.0) {} void operator()(double v) { double inc = 1.0; for (auto& mi : m_) { mi += inc; inc *= v; } } };
double Stdev(const std::vector<double>& moments) { assert(moments.size() > 2); assert(moments[0] > 0.0); const double mean = moments[1] / moments[0]; const double meanSquare = moments[2] / moments[0]; return sqrt(meanSquare - mean * mean); }
int main(void) { std::vector<int> data({ 2, 4, 4, 4, 5, 5, 7, 9 }); MomentsAccumulator_<2> accum; for (auto d : data) { accum(d); std::cout << "Running stdev: " << Stdev(accum.m_) << "\n"; } } </lang>
Clojure
<lang lisp> (defn stateful-std-deviation[x]
(letfn [(std-dev[x]
(let [v (deref (find-var (symbol (str *ns* "/v"))))]
(swap! v conj x)
(let [m (/ (reduce + @v) (count @v))]
(Math/sqrt (/ (reduce + (map #(* (- m %) (- m %)) @v)) (count @v))))))]
(when (nil? (resolve 'v))
(intern *ns* 'v (atom [])))
(std-dev x)))
</lang>
COBOL
<lang cobol>IDENTIFICATION DIVISION. PROGRAM-ID. run-stddev. environment division. input-output section. file-control.
select input-file assign to "input.txt" organization is line sequential.
data division. file section. fd input-file.
01 inp-record. 03 inp-fld pic 9(03).
working-storage section. 01 filler pic 9(01) value 0.
88 no-more-input value 1.
01 ws-tb-data.
03 ws-tb-size pic 9(03).
03 ws-tb-table.
05 ws-tb-fld pic s9(05)v9999 comp-3 occurs 0 to 100 times
depending on ws-tb-size.
01 ws-stddev pic s9(05)v9999 comp-3. PROCEDURE DIVISION.
move 0 to ws-tb-size
open input input-file
read input-file
at end
set no-more-input to true
end-read
perform
test after
until no-more-input
add 1 to ws-tb-size
move inp-fld to ws-tb-fld (ws-tb-size)
call 'stddev' using by reference ws-tb-data
ws-stddev
display 'inp=' inp-fld ' stddev=' ws-stddev
read input-file at end set no-more-input to true end-read
end-perform
close input-file
stop run.
end program run-stddev. IDENTIFICATION DIVISION. PROGRAM-ID. stddev. data division. working-storage section. 01 ws-tbx pic s9(03) comp. 01 ws-tb-work.
03 ws-sum pic s9(05)v9999 comp-3 value +0. 03 ws-sumsq pic s9(05)v9999 comp-3 value +0. 03 ws-avg pic s9(05)v9999 comp-3 value +0.
linkage section. 01 ws-tb-data.
03 ws-tb-size pic 9(03).
03 ws-tb-table.
05 ws-tb-fld pic s9(05)v9999 comp-3 occurs 0 to 100 times
depending on ws-tb-size.
01 ws-stddev pic s9(05)v9999 comp-3. PROCEDURE DIVISION using ws-tb-data ws-stddev.
compute ws-sum = 0
perform test before varying ws-tbx from 1 by +1 until ws-tbx > ws-tb-size
compute ws-sum = ws-sum + ws-tb-fld (ws-tbx)
end-perform
compute ws-avg rounded = ws-sum / ws-tb-size
compute ws-sumsq = 0
perform test before varying ws-tbx from 1 by +1 until ws-tbx > ws-tb-size
compute ws-sumsq = ws-sumsq
+ (ws-tb-fld (ws-tbx) - ws-avg) ** 2.0
end-perform
compute ws-stddev = ( ws-sumsq / ws-tb-size) ** 0.5
goback.
end program stddev. </lang> <lang cobol>sample output: inp=002 stddev=+00000.0000 inp=004 stddev=+00001.0000 inp=004 stddev=+00000.9427 inp=004 stddev=+00000.8660 inp=005 stddev=+00000.9797 inp=005 stddev=+00001.0000 inp=007 stddev=+00001.3996 inp=009 stddev=+00002.0000 </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
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))))
CL-USER> (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))) NIL 2 0.0 4 1.0 4 0.94280905 4 0.8660254 5 0.97979593 5 1.0 7 1.3997085 9 2.0</lang>
In the REPL, one step at a time: <lang lisp>CL-USER> (setf fn (running-stddev))
- <Interpreted Closure (:INTERNAL RUNNING-STDDEV) @ #x21b9a492>
CL-USER> (funcall fn 2) 0.0 CL-USER> (funcall fn 4) 1.0 CL-USER> (funcall fn 4) 0.94280905 CL-USER> (funcall fn 4) 0.8660254 CL-USER> (funcall fn 5) 0.97979593 CL-USER> (funcall fn 5) 1.0 CL-USER> (funcall fn 7) 1.3997085 CL-USER> (funcall fn 9) 2.0 </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
See: #Pascal
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>
Elixir
<lang elixir>defmodule Standard_deviation do
def add_sample( pid, n ), do: send( pid, {:add, n} )
def create, do: spawn_link( fn -> loop( [] ) end )
def destroy( pid ), do: send( pid, :stop )
def get( pid ) do
send( pid, {:get, self()} )
receive do
{ :get, value, _pid } -> value
end
end
def task do
pid = create()
for x <- [2,4,4,4,5,5,7,9], do: add_print( pid, x, add_sample(pid, x) )
destroy( pid )
end
defp add_print( pid, n, _add ) do
IO.puts "Standard deviation #{ get(pid) } when adding #{ n }"
end
defp loop( ns ) do
receive do
{:add, n} -> loop( [n | ns] )
{:get, pid} ->
send( pid, {:get, loop_calculate( ns ), self()} )
loop( ns )
:stop -> :ok
end
end
defp loop_calculate( ns ) do
average = loop_calculate_average( ns )
:math.sqrt( loop_calculate_average( for x <- ns, do: :math.pow(x - average, 2) ) )
end
defp loop_calculate_average( ns ), do: Enum.sum( ns ) / length( ns )
end
Standard_deviation.task</lang>
- Output:
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
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>
Emacs Lisp with built-in Emacs Calc
<lang emacs-lisp> (setq x '[2 4 4 4 5 5 7 9]) (string-to-number (calc-eval (format "sqrt(vpvar(%s))" x)))</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-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 ;
- st-add ( fnum stats -- )
dup 1e dup f+! float+ fdup dup f+! float+ fdup f* f+! std-stddev ;</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 -- )
dup 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 st-stddev ;</lang>
Usage example: <lang forth>create stats 0e f, 0e f, 0e f,
2e stats st-add f. \ 0. 4e stats st-add f. \ 1. 4e stats st-add f. \ 0.942809041582063 4e stats st-add f. \ 0.866025403784439 5e stats st-add f. \ 0.979795897113271 5e stats st-add f. \ 1. 7e stats st-add f. \ 1.39970842444753 9e stats st-add f. \ 2. </lang>
Fortran
<lang fortran> program standard_deviation
implicit none integer(kind=4), parameter :: dp = kind(0.0d0)
real(kind=dp), dimension(:), allocatable :: vals integer(kind=4) :: i
real(kind=dp), dimension(8) :: sample_data = (/ 2, 4, 4, 4, 5, 5, 7, 9 /)
do i = lbound(sample_data, 1), ubound(sample_data, 1)
call sample_add(vals, sample_data(i))
write(*, fmt='(#,I1,1X,value = ,F3.1,1X,stddev =,1X,F10.8)') &
i, sample_data(i), stddev(vals)
end do
if (allocated(vals)) deallocate(vals)
contains
! Adds value :val: to array :population: dynamically resizing array subroutine sample_add(population, val) real(kind=dp), dimension(:), allocatable, intent (inout) :: population real(kind=dp), intent (in) :: val
real(kind=dp), dimension(:), allocatable :: tmp integer(kind=4) :: n
if (.not. allocated(population)) then
allocate(population(1))
population(1) = val
else
n = size(population)
call move_alloc(population, tmp)
allocate(population(n + 1))
population(1:n) = tmp
population(n + 1) = val
endif
end subroutine sample_add
! Calculates standard deviation for given set of values real(kind=dp) function stddev(vals) real(kind=dp), dimension(:), intent(in) :: vals real(kind=dp) :: mean integer(kind=4) :: n
n = size(vals) mean = sum(vals)/n stddev = sqrt(sum((vals - mean)**2)/n) end function stddev
end program standard_deviation </lang>
- Output:
#1 value = 2.0 stddev = 0.00000000 #2 value = 4.0 stddev = 1.00000000 #3 value = 4.0 stddev = 0.94280904 #4 value = 4.0 stddev = 0.86602540 #5 value = 5.0 stddev = 0.97979590 #6 value = 5.0 stddev = 1.00000000 #7 value = 7.0 stddev = 1.39970842 #8 value = 9.0 stddev = 2.00000000
Old style, four ways
Early computers loaded the entire programme and its working storage into memory and left it there throughout the run. Uninitialised variables would start with whatever had been left in memory at their address by whatever last used those addresses, though some systems would clear all of memory to zero or possibly some other value before each load. Either way, if a routine was invoked a second time, its variables would have the values left in them by their previous invocation. The DATA statement allows initial values to be specified, and allows repeat counts when specifying such values as well. It is not an executable statement: it is not re-executed on second and subsequent invocations of the containing routine. Thus, it is easy to have a routine employ counters and the like, visible only within themselves and initialised to zero or whatever suited.
With more complex operating systems, routines that relied on retaining values across invocations might no longer work - perhaps a fresh version of the routine would be loaded to memory (perhaps at odd intervals), or, on exit, the working storage would be discarded. There was a half-way scheme, whereby variables that had appeared in DATA statements would be retained while the others would be discarded. This subtle indication has been discarded in favour of the explicit SAVE statement, naming those variables whose values are to be retained between invocations, though compilers might also offer an option such as "automatic" (for each invocation always allocate then discard working memory) and "static" (retain values), possibly introducing non-standard keywords as well. Otherwise, the routines would have to use storage global to them such as additional parameters, or, COMMON storage and in later Fortran, the MODULE arrangements for shared items. The persistence of such storage can still be limited, but by naming them in the main line can be ensured for the life of the run. The other routines with access to such storage could enable re-initialisation, additional reports, or multiple accumulations, etc.
Since the standard deviation can be calculated in a single pass through the data, producing values for the standard deviation of all values so far supplied is easily done without re-calculation. Accuracy is quite another matter. Calculations using deviances from a working mean are much better, and capturing the first X as the working mean would be easy, just test on N = 0. The sum and sum-of-squares method is quite capable of generating a negative variance, but the second method cannot, because the terms being added in to V are never negative. This is demonstrated by comparing the results computed from StdDev(A), StdDev(A + 10), StdDev(A + 100), StdDev(A + 1000), etc.
Incidentally, Fortran implementations rarely enable re-entrancy for the WRITE statement, so, since here the functions are invoked in a WRITE statement, the functions cannot themselves use WRITE statements, say for debugging. <lang Fortran>
REAL FUNCTION STDDEV(X) !Standard deviation for successive values.
REAL X !The latest value.
REAL V !Scratchpad.
INTEGER N !Ongoing: count of the values.
REAL EX,EX2 !Ongoing: sum of X and X**2.
SAVE N,EX,EX2 !Retain values from one invocation to the next.
DATA N,EX,EX2/0,0.0,0.0/ !Initial values.
N = N + 1 !Another value arrives.
EX = X + EX !Augment the total.
EX2 = X**2 + EX2 !Augment the sum of squares.
V = EX2/N - (EX/N)**2 !The variance, but, it might come out negative!
STDDEV = SIGN(SQRT(ABS(V)),V) !Protect the SQRT, but produce a negative result if so.
END FUNCTION STDDEV !For the sequence of received X values.
REAL FUNCTION STDDEVP(X) !Standard deviation for successive values.
REAL X !The latest value.
INTEGER N !Ongoing: count of the values.
REAL A,V !Ongoing: average, and sum of squared deviations.
SAVE N,A,V !Retain values from one invocation to the next.
DATA N,A,V/0,0.0,0.0/ !Initial values.
N = N + 1 !Another value arrives.
V = (N - 1)*(X - A)**2 /N + V !First, as it requires the existing average.
A = (X - A)/N + A != [x + (n - 1).A)]/n: recover the total from the average.
STDDEVP = SQRT(V/N) !V can never be negative, even with limited precision.
END FUNCTION STDDEVP !For the sequence of received X values.
REAL FUNCTION STDDEVW(X) !Standard deviation for successive values.
REAL X !The latest value.
REAL V,D !Scratchpads.
INTEGER N !Ongoing: count of the values.
REAL EX,EX2 !Ongoing: sum of X and X**2.
REAL W !Ongoing: working mean.
SAVE N,EX,EX2,W !Retain values from one invocation to the next.
DATA N,EX,EX2/0,0.0,0.0/ !Initial values.
IF (N.LE.0) W = X !Take the first value as the working mean.
N = N + 1 !Another value arrives.
D = X - W !Its deviation from the working mean.
EX = D + EX !Augment the total.
EX2 = D**2 + EX2 !Augment the sum of squares.
V = EX2/N - (EX/N)**2 !The variance, but, it might come out negative!
STDDEVW = SIGN(SQRT(ABS(V)),V) !Protect the SQRT, but produce a negative result if so.
END FUNCTION STDDEVW !For the sequence of received X values.
REAL FUNCTION STDDEVPW(X) !Standard deviation for successive values.
REAL X !The latest value.
INTEGER N !Ongoing: count of the values.
REAL A,V !Ongoing: average, and sum of squared deviations.
REAL W !Ongoing: working mean.
SAVE N,A,V,W !Retain values from one invocation to the next.
DATA N,A,V/0,0.0,0.0/ !Initial values.
IF (N.LE.0) W = X !Oh for self-modifying code!
N = N + 1 !Another value arrives.
D = X - W !Its deviation from the working mean.
V = (N - 1)*(D - A)**2 /N + V !First, as it requires the existing average.
A = (D - A)/N + A != [x + (n - 1).A)]/n: recover the total from the average.
STDDEVPW = SQRT(V/N) !V can never be negative, even with limited precision.
END FUNCTION STDDEVPW !For the sequence of received X values.
PROGRAM TEST
INTEGER I !A stepper.
REAL A(8) !The example data.
DATA A/2.0,3*4.0,2*5.0,7.0,9.0/ !Alas, another opportunity to use @ passed over.
REAL B !An offsetting base.
WRITE (6,1)
1 FORMAT ("Progressive calculation of the standard deviation."/
1 " I",7X,"A(I) EX EX2 Av V*N Ed Ed2 wAv V*N")
B = 1000000 !Provoke truncation error.
DO I = 1,8 !Step along the data series,
WRITE (6,2) I,INT(A(I) + B), !No fractional part, so I don't want F11.0.
1 STDDEV(A(I) + B),STDDEVP(A(I) + B), !Showing progressive values.
2 STDDEVW(A(I) + B),STDDEVPW(A(I) + B) !These with a working mean.
2 FORMAT (I2,I11,1X,4F12.6) !Should do for the example.
END DO !On to the next value.
END
</lang>
Output: the second pair of columns have the calculations done with a working mean and thus accumulate deviations from that.
Progressive calculation of the standard deviation. I A(I) EX EX2 Av V*N Ed Ed2 wAv V*N 1 2 0.000000 0.000000 0.000000 0.000000 2 4 1.000000 1.000000 1.000000 1.000000 3 4 0.942809 0.942809 0.942809 0.942809 4 4 0.866025 0.866025 0.866025 0.866025 5 5 0.979796 0.979796 0.979796 0.979796 6 5 1.000000 1.000000 1.000000 1.000000 7 7 1.399708 1.399708 1.399708 1.399708 8 9 2.000000 2.000000 2.000000 2.000000
I A(I) EX EX2 Av V*N Ed Ed2 wAv V*N 1 12 0.000000 0.000000 0.000000 0.000000 2 14 1.000000 1.000000 1.000000 1.000000 3 14 0.942809 0.942809 0.942809 0.942809 4 14 0.866025 0.866025 0.866025 0.866025 5 15 0.979796 0.979796 0.979796 0.979796 6 15 1.000000 1.000000 1.000000 1.000000 7 17 1.399708 1.399708 1.399708 1.399708 8 19 2.000000 2.000000 2.000000 2.000000
I A(I) EX EX2 Av V*N Ed Ed2 wAv V*N 1 102 0.000000 0.000000 0.000000 0.000000 2 104 1.000000 1.000000 1.000000 1.000000 3 104 0.942809 0.942809 0.942809 0.942809 4 104 0.866025 0.866025 0.866025 0.866025 5 105 0.979796 0.979796 0.979796 0.979796 6 105 1.000000 0.999999 1.000000 1.000000 7 107 1.399708 1.399708 1.399708 1.399708 8 109 2.000000 1.999999 2.000000 2.000000
I A(I) EX EX2 Av V*N Ed Ed2 wAv V*N 1 1002 0.000000 0.000000 0.000000 0.000000 2 1004 1.000000 1.000000 1.000000 1.000000 3 1004 0.942809 0.942809 0.942809 0.942809 4 1004 0.866025 0.866028 0.866025 0.866025 5 1005 0.979796 0.979798 0.979796 0.979796 6 1005 1.000000 1.000004 1.000000 1.000000 7 1007 1.399708 1.399711 1.399708 1.399708 8 1009 2.000000 1.999997 2.000000 2.000000
I A(I) EX EX2 Av V*N Ed Ed2 wAv V*N 1 10002 -2.000000 0.000000 0.000000 0.000000 2 10004 -1.000000 1.000000 1.000000 1.000000 3 10004 -0.666667 0.942809 0.942809 0.942809 4 10004 1.936492 0.866072 0.866025 0.866025 5 10005 2.181742 0.979829 0.979796 0.979796 6 10005 2.309401 1.000060 1.000000 1.000000 7 10007 1.801360 1.399745 1.399708 1.399708 8 10009 2.645751 1.999987 2.000000 2.000000
I A(I) EX EX2 Av V*N Ed Ed2 wAv V*N 1 100002 19.493589 0.000000 0.000000 0.000000 2 100004 7.416198 1.000000 1.000000 1.000000 3 100004 -7.333333 0.942809 0.942809 0.942809 4 100004 20.093531 0.865650 0.866025 0.866025 5 100005 -1.280625 0.979531 0.979796 0.979796 6 100005 -16.492422 1.000305 1.000000 1.000000 7 100007 17.851427 1.399895 1.399708 1.399708 8 100009 20.566963 1.999835 2.000000 2.000000
I A(I) EX EX2 Av V*N Ed Ed2 wAv V*N 1 1000002 -80.024994 0.000000 0.000000 0.000000 2 1000004 158.767120 1.000000 1.000000 1.000000 3 1000004 -89.146576 0.942809 0.942809 0.942809 4 1000004 90.795097 0.869074 0.866025 0.866025 5 1000005 193.357590 0.981953 0.979796 0.979796 6 1000005 238.361069 0.999691 1.000000 1.000000 7 1000007 153.462296 1.399519 1.399708 1.399708 8 1000009 151.284500 1.997653 2.000000 2.000000
Speaking loosely, to square a number of d digits accurately requires the ability to represent 2d digits accurately, with more digits needed if many such squares are to be added together accurately. In this example, 1000 when squared, is pushing at the limits of single precision. The average&variance method is resistant to this problem (and does not generate negative variances either!) because it manipulates differences from the running average, but it is still better to use a working mean.
In other words, a two-pass method will be more accurate (where the second pass calculates the variance by accumulating deviations from the actual average, itself calculated with a working mean) but at the cost of that second pass and the saving of all the values. Higher precision variables for the accumulations are the easiest way towards accurate results.
FreeBASIC
<lang freebasic>' FB 1.05.0 Win64
Function calcStandardDeviation(number As Double) As Double
Static a() As Double Redim Preserve a(0 To UBound(a) + 1) Dim ub As UInteger = UBound(a) a(ub) = number Dim sum As Double = 0.0 For i As UInteger = 0 To ub sum += a(i) Next Dim mean As Double = sum / (ub + 1) Dim diff As Double sum = 0.0 For i As UInteger = 0 To ub diff = a(i) - mean sum += diff * diff Next Return Sqr(sum/ (ub + 1))
End Function
Dim a(0 To 7) As Double = {2, 4, 4, 4, 5, 5, 7, 9}
For i As UInteger = 0 To 7
Print "Added"; a(i); " SD now : "; calcStandardDeviation(a(i))
Next
Print Print "Press any key to quit" Sleep</lang>
- Output:
Added 2 SD now : 0 Added 4 SD now : 1 Added 4 SD now : 0.9428090415820634 Added 4 SD now : 0.8660254037844386 Added 5 SD now : 0.9797958971132712 Added 5 SD now : 1 Added 7 SD now : 1.39970842444753 Added 9 SD now : 2
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>List samples = []
def stdDev = { def sample ->
samples << sample
def sum = samples.sum()
def sumSq = samples.sum { it * it }
def count = samples.size()
(sumSq/count - (sum/count)**2)**0.5
}
[2,4,4,4,5,5,7,9].each {
println "${stdDev(it)}"
}</lang>
- Output:
0 1 0.9428090416999145 0.8660254037844386 0.9797958971132712 1 1.3997084243469262 2
Haskell
We store the state in the ST monad using an STRef.
<lang haskell>{-# LANGUAGE BangPatterns #-}
import Data.List (foldl') -- ' import Data.STRef import Control.Monad.ST
data Pair a b = Pair !a !b
sumLen :: [Double] -> Pair Double Double sumLen = fiof2 . foldl' (\(Pair s l) x -> Pair (s+x) (l+1)) (Pair 0.0 0) --'
where fiof2 (Pair s l) = Pair s (fromIntegral l)
divl :: Pair Double Double -> Double divl (Pair _ 0.0) = 0.0 divl (Pair s l) = s / l
sd :: [Double] -> Double sd xs = sqrt $ foldl' (\a x -> a+(x-m)^2) 0 xs / l --'
where p@(Pair s l) = sumLen xs
m = divl p
mkSD :: ST s (Double -> ST s Double) mkSD = go <$> newSTRef []
where go acc x =
modifySTRef acc (x:) >> (sd <$> readSTRef acc)
main = mapM_ print $ runST $
mkSD >>= forM [2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0]</lang>
Or, simply accumulating across a fold:
<lang Haskell>type Index = Int type DataPoint = Float
type Sum = Float type SumOfSquares = Float
type Deviations = [Float] type Accumulator = (Sum, SumOfSquares, Deviations)
stdDevInc :: Accumulator -> (DataPoint, Index) -> Accumulator stdDevInc (s, q, ds) (x, i) = (_s, _q, _ds)
where _s = s + x _q = q + (x ^ 2) _i = fromIntegral i _ds = ds ++ [sqrt ((_q / _i) - ((_s / _i) ^ 2))]
sample :: [DataPoint] sample = [2, 4, 4, 4, 5, 5, 7, 9]
main :: IO () main = mapM_ print devns
where (_, _, devns) = foldl stdDevInc (0, 0, []) $ zip sample [1 .. length sample]</lang>
- Output:
0.0 1.0 0.9428093 0.8660254 0.97979593 1.0 1.3997087 2.0
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
Imperative
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
Functional (ES 5)
Accumulating across a fold
<lang JavaScript>(function (xs) {
return xs.reduce(function (a, x, i) {
var n = i + 1,
sum_ = a.sum + x,
squaresSum_ = a.squaresSum + (x * x);
return {
sum: sum_,
squaresSum: squaresSum_,
stages: a.stages.concat(
Math.sqrt((squaresSum_ / n) - Math.pow((sum_ / n), 2))
)
};
}, {
sum: 0,
squaresSum: 0,
stages: []
}).stages
})([2, 4, 4, 4, 5, 5, 7, 9]);</lang>
- Output:
<lang JavaScript>[0, 1, 0.9428090415820626, 0.8660254037844386, 0.9797958971132716, 1, 1.3997084244475297, 2]</lang>
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>
Julia
Use a closure to create a running standard deviation function. <lang Julia> function makerunningstd()
a = zero(Float64)
b = zero(Float64)
n = zero(Int64)
function runningstd(x)
a += x
b += x^2
n += 1
std = sqrt(n*b - a^2)/n
return std
end
return runningstd
end
test = [2, 4, 4, 4, 5, 5, 7, 9]
rstd = makerunningstd()
println("Perform a running standard deviation of ", test) for i in test
println(i, " => ", rstd(i))
end </lang>
- Output:
Perform a running standard deviation of [2,4,4,4,5,5,7,9] 2 => 0.0 4 => 1.0 4 => 0.9428090415820635 4 => 0.8660254037844386 5 => 0.9797958971132712 5 => 1.0 7 => 1.3997084244475302 9 => 2.0
Kotlin
Using a class to keep the running sum, sum of squares and number of elements added so far: <lang scala>// version 1.0.5-2
class CumStdDev {
private var n = 0 private var sum = 0.0 private var sum2 = 0.0
fun sd(x: Double): Double {
n++
sum += x
sum2 += x * x
return Math.sqrt(sum2 / n - sum * sum / n / n)
}
}
fun main(args: Array<String>) {
val testData = doubleArrayOf(2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0)
val csd = CumStdDev()
for (d in testData) println("Add $d => ${csd.sd(d)}")
}</lang>
- Output:
Add 2.0 => 0.0 Add 4.0 => 1.0 Add 4.0 => 0.9428090415820626 Add 4.0 => 0.8660254037844386 Add 5.0 => 0.9797958971132708 Add 5.0 => 1.0 Add 7.0 => 1.399708424447531 Add 9.0 => 2.0
Liberty BASIC
Using a global array to maintain the state. Implements definition explicitly. <lang lb>
dim SD.storage$( 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>
Here is a vectorized one line solution as a function <lang Matlab> function stdDevEval(n) disp(sqrt(sum((n-sum(n)/length(n)).^2)/length(n))); end </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
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>
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>
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
Oforth
Oforth does not have global variables that can be used to create statefull functions.
Here, we create a channel to hold current list of numbers. Constraint is that this channel can't hold mutable objects. On the other hand, stddev function is thread safe and can be called by tasks running in parallel.
<lang Oforth>Channel new [ ] over send drop const: StdValues
- stddev(x)
| l |
StdValues receive x + dup ->l StdValues send drop #qs l map sum l size asFloat / l avg sq - sqrt ;</lang>
- Output:
>[ 2, 4, 4, 4, 5, 5, 7, 9 ] apply(#[ stddev println ]) 0 1 0.942809041582063 0.866025403784439 0.979795897113272 1 1.39970842444753 2 ok >
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]) Say '#'i 'value =' x[i] 'stdev =' sd
end
- 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>
- Output:
#1 value = 2 stdev = 0 #2 value = 4 stdev = 1 #3 value = 4 stdev = 0.94280905 #4 value = 4 stdev = 0.866025405 #5 value = 5 stdev = 0.979795895 #6 value = 5 stdev = 1 #7 value = 7 stdev = 1.39970844 #8 value = 9 stdev = 2
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
Std.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
Delphi
<lang Delphi>program prj_CalcStdDerv;
{$APPTYPE CONSOLE}
uses
Math;
var Series:Array of Extended;
UserString:String;
function AppendAndCalc(NewVal:Extended):Extended;
begin
setlength(Series,high(Series)+2); Series[high(Series)] := NewVal; result := PopnStdDev(Series);
end;
const data:array[0..7] of Extended =
(2,4,4,4,5,5,7,9);
var rr: Extended; begin
setlength(Series,0);
for rr in data do
begin
writeln(rr,' -> ',AppendAndCalc(rr));
end;
Readln;
end. </lang>
- Output:
2.0000000000000000E+0000 -> 0.0000000000000000E+0000 4.0000000000000000E+0000 -> 1.0000000000000000E+0000 4.0000000000000000E+0000 -> 9.4280904158206337E-0001 4.0000000000000000E+0000 -> 8.6602540378443865E-0001 5.0000000000000000E+0000 -> 9.7979589711327124E-0001 5.0000000000000000E+0000 -> 1.0000000000000000E+0000 7.0000000000000000E+0000 -> 1.3997084244475303E+0000 9.0000000000000000E+0000 -> 2.0000000000000000E+0000
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
Phix
demo\rosetta\Standard_deviation.exw contains a copy of this code and a version that could be the basis for a library version that can handle multiple active data sets concurrently. <lang Phix>atom sdn = 0, sdsum = 0, sdsumsq = 0
procedure sdadd(atom n)
sdn += 1 sdsum += n sdsumsq += n*n
end procedure
function sdavg()
return sdsum/sdn
end function
function sddev()
return sqrt(sdsumsq/sdn - power(sdsum/sdn,2))
end function
--test code: constant testset = {2, 4, 4, 4, 5, 5, 7, 9} integer ti for i=1 to length(testset) do
ti = testset[i]
sdadd(ti)
printf(1,"N=%d Item=%d Avg=%5.3f StdDev=%5.3f\n",{i,ti,sdavg(),sddev()})
end for</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=0.943 N=4 Item=4 Avg=3.500 StdDev=0.866 N=5 Item=5 Avg=3.800 StdDev=0.980 N=6 Item=5 Avg=4.000 StdDev=1.000 N=7 Item=7 Avg=4.429 StdDev=1.400 N=8 Item=9 Avg=5.000 StdDev=2.000
PHP
This is just straight PHP class usage, respecting the specifications "stateful" and "one at a time": <lang PHP><?php class sdcalc {
private $cnt, $sumup, $square;
function __construct() {
$this->reset();
}
# callable on an instance
function reset() {
$this->cnt=0; $this->sumup=0; $this->square=0;
}
function add($f) {
$this->cnt++;
$this->sumup += $f;
$this->square += pow($f, 2);
return $this->calc();
}
function calc() {
if ($this->cnt==0 || $this->sumup==0) {
return 0;
} else {
return sqrt($this->square / $this->cnt - pow(($this->sumup / $this->cnt),2));
}
}
}
- start test, adding test data one by one
$c = new sdcalc(); foreach ([2,4,4,4,5,5,7,9] as $v) {
printf('Adding %g: result %g%s', $v, $c->add($v), PHP_EOL);
}</lang>
This will produce the output:
Adding 2: result 0 Adding 4: result 1 Adding 4: result 0.942809 Adding 4: result 0.866025 Adding 5: result 0.979796 Adding 5: result 1 Adding 7: result 1.39971 Adding 9: result 2
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
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
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
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>
Python: 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>
Python: Callable class
You could rename the method sd to __call__ this would make the class instance callable like a function so instead of using sd_inst.sd(value) it would change to sd_inst(value) for the same results.
Python: 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>
Python: 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>
Python: In a couple of 'functional' lines
<lang python>>>> myMean = lambda MyList : reduce(lambda x, y: x + y, MyList) / float(len(MyList)) >>> myStd = lambda MyList : (reduce(lambda x,y : x + y , map(lambda x: (x-myMean(MyList))**2 , MyList)) / float(len(MyList)))**.5
>>> print myStd([2,4,4,4,5,5,7,9]) 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
These REXX versions use running sums.
show running sums
<lang rexx>/*REXX program calculates and displays the standard deviation of a given set of numbers.*/ parse arg # /*obtain optional arguments from the CL*/ if #= then #=2 4 4 4 5 5 7 9 /*None specified? Then use the default*/ n=words(#); $=0; $$=0 /*N: # items; $,$$: sums to be zeroed*/ n=words(#); L=length(n); $=0; $$=0 /*# items; item width; couple of sums*/
/* [↓] process each number in the list*/
do j=1 for n; _=word(#,j); $=$+_; $$=$$+_**2
say ' item' right(j,L)":" right(_,4) ' average=' left($/j,12),
' standard deviation=' sqrt($$/j - ($/j)**2)
end /*j*/ /* [↑] prettify output with whitespace*/
say 'standard deviation: ' sqrt($$/n - ($/n)**2) /*calculate & display the std deviation*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); h=d+6; m.=9; numeric form
numeric digits; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g * .5'e'_ % 2
do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/
numeric digits d; return g/1</lang>
output when using the default input of: 2 4 4 4 5 5 7 9
item 1: 2 average= 2 standard deviation= 0 item 2: 4 average= 3 standard deviation= 1 item 3: 4 average= 3.33333333 standard deviation= 0.942809047 item 4: 4 average= 3.5 standard deviation= 0.866025404 item 5: 5 average= 3.8 standard deviation= 0.979795897 item 6: 5 average= 4 standard deviation= 1 item 7: 7 average= 4.42857143 standard deviation= 1.39970843 item 8: 9 average= 5 standard deviation= 2 standard deviation: 2
only show standard deviation
<lang rexx>/*REXX program calculates and displays the standard deviation of a given set of numbers.*/ parse arg # /*obtain optional arguments from the CL*/ if #= then #=2 4 4 4 5 5 7 9 /*None specified? Then use the default*/ n=words(#); $=0; $$=0 /*N: # items; $,$$: sums to be zeroed*/
/* [↓] process each number in the list*/ do j=1 for n; _=word(#,j); $=$+_; $$=$$+_**2 /*perform summation on two sets of #'s.*/ end /*j*/
say 'standard deviation: ' sqrt($$/n - ($/n)**2) /*calculate & display the std deviation*/ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); h=d+6; m.=9; numeric form
numeric digits; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g * .5'e'_ % 2
do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/
numeric digits d; return g/1</lang>
output when using the default input of: 2 4 4 4 5 5 7 9
standard deviation: 2
Ruby
Object
Uses an object to keep state.
"Simplification of the formula [...] for standard deviation [...] can be memorized as taking the square root of (the average of the squares less the square of the average)." c.f. wikipedia.
<lang ruby>class StdDevAccumulator
def initialize @n, @sum, @sumofsquares = 0, 0.0, 0.0 end def <<(num) # return self to make this possible: sd << 1 << 2 << 3 # => 0.816496580927726 @n += 1 @sum += num @sumofsquares += num**2 self end def stddev Math.sqrt( (@sumofsquares / @n) - (@sum / @n)**2 ) end def to_s stddev.to_s end
end
sd = StdDevAccumulator.new i = 0 [2,4,4,4,5,5,7,9].each {|n| puts "adding #{n}: stddev of #{i+=1} samples is #{sd << n}" }</lang>
adding 2: stddev of 1 samples is 0.0 adding 4: stddev of 2 samples is 1.0 adding 4: stddev of 3 samples is 0.942809041582063 adding 4: stddev of 4 samples is 0.866025403784439 adding 5: stddev of 5 samples is 0.979795897113272 adding 5: stddev of 6 samples is 1.0 adding 7: stddev of 7 samples is 1.39970842444753 adding 9: stddev of 8 samples is 2.0
Closure
<lang ruby>def sdaccum
n, sum, sum2 = 0, 0.0, 0.0 lambda do |num| n += 1 sum += num sum2 += num**2 Math.sqrt( (sum2 / n) - (sum / n)**2 ) end
end
sd = sdaccum [2,4,4,4,5,5,7,9].each {|n| print sd.call(n), ", "}</lang>
0.0, 1.0, 0.942809041582063, 0.866025403784439, 0.979795897113272, 1.0, 1.39970842444753, 2.0,
Run BASIC
<lang runbasic>dim sdSave$(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
Generic for any numeric type
<lang Scala>import scala.math.sqrt
object StddevCalc extends App {
def calcAvgAndStddev[T](ts: Iterable[T])(implicit num: Fractional[T]): (T, Double) = {
def avg(ts: Iterable[T])(implicit num: Fractional[T]): T = {
num.div(ts.sum, num.fromInt(ts.size)) // Leaving with type of function T
}
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 (standart-deviation-generator)
(let ((nums '()))
(lambda (x)
(set! nums (cons x nums))
(let* ((mean (/ (apply + nums) (length nums)))
(mean-sqr (lambda (y) (expt (- y mean) 2)))
(variance (/ (apply + (map mean-sqr nums)) (length nums))))
(sqrt variance)))))
(let loop ((f (standart-deviation-generator))
(input '(2 4 4 4 5 5 7 9)))
(if (not (null? input))
(begin
(display (f (car input)))
(newline)
(loop f (cdr input)))))
</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.
Sidef
Using an object to keep state: <lang ruby>class StdDevAccumulator(n=0, sum=0, sumofsquares=0) {
method <<(num) {
n += 1
sum += num
sumofsquares += num**2
self
}
method stddev {
sqrt(sumofsquares/n - pow(sum/n, 2))
}
method to_s {
self.stddev.to_s
}
} var i = 0 var sd = StdDevAccumulator() [2,4,4,4,5,5,7,9].each {|n|
say "adding #{n}: stddev of #{i+=1} samples is #{sd << n}"
}</lang>
- Output:
adding 2: stddev of 1 samples is 0 adding 4: stddev of 2 samples is 1 adding 4: stddev of 3 samples is 0.942809041582063365867792482806465385713114583585 adding 4: stddev of 4 samples is 0.866025403784438646763723170752936183471402626905 adding 5: stddev of 5 samples is 0.979795897113271239278913629882356556786378992263 adding 5: stddev of 6 samples is 1 adding 7: stddev of 7 samples is 1.39970842444753034182701947126050936683768427466 adding 9: stddev of 8 samples is 2
Using static variables: <lang ruby>func stddev(x) {
static(num=0, sum=0, sum2=0)
num++
sqrt(
(sum2 += x**2) / num -
(((sum += x) / num)**2)
)
} %n(2 4 4 4 5 5 7 9).each { say stddev(_) }</lang>
- Output:
0 1 0.942809041582063365867792482806465385713114583585 0.866025403784438646763723170752936183471402626905 0.979795897113271239278913629882356556786378992263 1 1.39970842444753034182701947126050936683768427466 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>
SQL
<lang SQL>-- the minimal table create table if not exists teststd (n double precision not null);
-- code modularity with view, we could have used a common table expression instead create view vteststd as
select count(n) as cnt, sum(n) as tsum, sum(power(n,2)) as tsqr
from teststd;
-- you can of course put this code into every query create or replace function std_dev() returns double precision as $$
select sqrt(tsqr/cnt - (tsum/cnt)^2) from vteststd;
$$ language sql;
-- test data is: 2,4,4,4,5,5,7,9 insert into teststd values (2); select std_dev() as std_deviation; insert into teststd values (4); select std_dev() as std_deviation; insert into teststd values (4); select std_dev() as std_deviation; insert into teststd values (4); select std_dev() as std_deviation; insert into teststd values (5); select std_dev() as std_deviation; insert into teststd values (5); select std_dev() as std_deviation; insert into teststd values (7); select std_dev() as std_deviation; insert into teststd values (9); select std_dev() as std_deviation; -- cleanup test data delete from teststd; </lang> With a command like psql <rosetta-std-dev.sql you will get an output like this: (duplicate lines generously deleted, locale is DE)
CREATE TABLE
FEHLER: Relation »vteststd« existiert bereits
CREATE FUNCTION
INSERT 0 1
std_deviation
---------------
0
(1 Zeile)
INSERT 0 1
std_deviation
---------------
1
0.942809041582063
0.866025403784439
0.979795897113272
1
1.39970842444753
2
DELETE 8
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
Functional:
<lang Swift> func standardDeviation(arr : [Double]) -> Double {
let length = Double(arr.count)
let avg = arr.reduce(0, { $0 + $1 }) / length
let sumOfSquaredAvgDiff = arr.map { pow($0 - avg, 2.0)}.reduce(0, {$0 + $1})
return sqrt(sumOfSquaredAvgDiff / length)
}
let responseTimes = [ 18.0, 21.0, 41.0, 42.0, 48.0, 50.0, 55.0, 90.0 ]
standardDeviation(responseTimes) // 20.8742514835862 standardDeviation([2,4,4,4,5,5,7,9]) // 2.0 </lang>
Tcl
With a Class
or
<lang tcl>oo::class create SDAccum {
variable sum sum2 num
constructor {} {
set sum 0.0
set sum2 0.0
set num 0
}
method value {x} {
set sum2 [expr {$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
VBScript
<lang vb>data = Array(2,4,4,4,5,5,7,9)
For i = 0 To UBound(data) WScript.StdOut.Write "value = " & data(i) &_ " running sd = " & sd(data,i) WScript.StdOut.WriteLine Next
Function sd(arr,n) mean = 0 variance = 0 For j = 0 To n mean = mean + arr(j) Next mean = mean/(n+1) For k = 0 To n variance = variance + ((arr(k)-mean)^2) Next variance = variance/(n+1) sd = FormatNumber(Sqr(variance),6) End Function</lang>
- Output:
value = 2 running sd = 0.000000 value = 4 running sd = 1.000000 value = 4 running sd = 0.942809 value = 4 running sd = 0.866025 value = 5 running sd = 0.979796 value = 5 running sd = 1.000000 value = 7 running sd = 1.399708 value = 9 running sd = 2.000000
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
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