Cumulative standard deviation
(Redirected from Standard Deviation)
You are encouraged to solve this task according to the task description, using any language you may know.
Task[|_}}%20*%2F%20 edit]
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.
- Test case
Use this to compute the standard deviation of this demonstration set, , which is .
- Related tasks
| ||||||||||||||||||||||||||||||||||||||||||||||
11l
T SD
sum = 0.0
sum2 = 0.0
n = 0.0
F ()(x)
.sum += x
.sum2 += x ^ 2
.n += 1.0
R sqrt(.sum2 / .n - (.sum / .n) ^ 2)
V sd_inst = SD()
L(value) [2, 4, 4, 4, 5, 5, 7, 9]
print(value‘ ’sd_inst(value))- Output:
2 0 4 1 4 0.942809042 4 0.866025404 5 0.979795897 5 1 7 1.399708424 9 2
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.
******** 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- 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
Action!
INCLUDE "H6:REALMATH.ACT"
REAL sum,sum2
INT count
PROC Calc(REAL POINTER x,sd)
REAL tmp1,tmp2,tmp3
RealAdd(sum,x,tmp1) ;tmp1=sum+x
RealAssign(tmp1,sum) ;sum=sum+x
RealMult(x,x,tmp1) ;tmp1=x*x
RealAdd(sum2,tmp1,tmp2) ;tmp2=sum2+x*x
RealAssign(tmp2,sum2) ;sum2=sum2+x*x
count==+1
IF count=0 THEN
IntToReal(0,sd) ;sd=0
ELSE
IntToReal(count,tmp1)
RealMult(sum,sum,tmp2) ;tmp2=sum*sum
RealDiv(tmp2,tmp1,tmp3) ;tmp3=sum*sum/count
RealDiv(tmp3,tmp1,tmp2) ;tmp2=sum*sum/count/count
RealDiv(sum2,tmp1,tmp3) ;tmp3=sum2/count
RealSub(tmp3,tmp2,tmp1) ;tmp1=sum2/count-sum*sum/count/count
Sqrt(tmp1,sd) ;sd=sqrt(sum2/count-sum*sum/count/count)
FI
RETURN
PROC Main()
INT ARRAY values=[2 4 4 4 5 5 7 9]
INT i
REAL x,sd
Put(125) PutE() ;clear screen
MathInit()
IntToReal(0,sum)
IntToReal(0,sum2)
count=0
FOR i=0 TO 7
DO
IntToReal(values(i),x)
Calc(x,sd)
Print("x=") PrintR(x)
Print(" sum=") PrintR(sum)
Print(" sd=") PrintRE(sd)
OD
RETURN- Output:
Screenshot from Atari 8-bit computer
x=2 sum=2 sd=0 x=4 sum=6 sd=1 x=4 sum=10 sd=.942809052 x=4 sum=14 sd=.86602541 x=5 sum=19 sd=.979795903 x=5 sum=24 sd=1 x=7 sum=31 sd=1.39970843 x=9 sum=40 sd=1.99999999
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;
- 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.
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
)- 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:
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)))
#
)- 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:
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- 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.
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.- 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 over a fold:
-------------- CUMULATIVE STANDARD DEVIATION -------------
-- 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 of a) + n, squaresSum:squaresSum, stages:stages}
end stdDevInc
--------------------------- TEST -------------------------
on run
set xs to [2, 4, 4, 4, 5, 5, 7, 9]
stages of foldl(stdDevInc, ¬
{sum:0, squaresSum:0, stages:[]}, xs)
--> {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 |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
-- 2nd class handler function lifted into 1st class script wrapper.
if script is class of f then
f
else
script
property |λ| : f
end script
end if
end mReturn
- Output:
{0.0, 1.0, 0.942809041582, 0.866025403784,
0.979795897113, 1.0, 1.399708424448, 2.0}
Or as a map-accumulation:
-------------- CUMULATIVE STANDARD DEVIATION -------------
-- cumulativeStdDevns :: [Float] -> [Float]
on cumulativeStdDevns(xs)
script go
on |λ|(sq, x, i)
set {s, q} to sq
set _s to x + s
set _q to q + (x ^ 2)
{{_s, _q}, ((_q / i) - ((_s / i) ^ 2)) ^ 0.5}
end |λ|
end script
item 2 of mapAccumL(go, {0, 0}, xs)
end cumulativeStdDevns
--------------------------- TEST -------------------------
on run
cumulativeStdDevns({2, 4, 4, 4, 5, 5, 7, 9})
end run
------------------------- GENERIC ------------------------
-- 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 |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- mapAccumL :: (acc -> x -> (acc, y)) -> acc -> [x] -> (acc, [y])
on mapAccumL(f, acc, xs)
-- 'The mapAccumL function behaves like a combination of map and foldl;
-- it applies a function to each element of a list, passing an
-- accumulating parameter from |Left| to |Right|, and returning a final
-- value of this accumulator together with the new list.' (see Hoogle)
script
on |λ|(a, x, i)
tell mReturn(f) to set pair to |λ|(item 1 of a, x, i)
{item 1 of pair, (item 2 of a) & {item 2 of pair}}
end |λ|
end script
foldl(result, {acc, []}, xs)
end mapAccumL
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
-- 2nd class handler function lifted into 1st class script wrapper.
if script is class of f then
f
else
script
property |λ| : f
end script
end if
end mReturn
- Output:
{0.0, 1.0, 0.942809041582, 0.866025403784, 0.979795897113, 1.0, 1.399708424448, 2.0}
Arturo
arr: new []
loop [2 4 4 4 5 5 7 9] 'value [
'arr ++ value
print [value "->" deviation arr]
]
- Output:
2 -> 0.0 4 -> 1.0 4 -> 0.9428090415820634 4 -> 0.8660254037844386 5 -> 0.9797958971132711 5 -> 0.9999999999999999 7 -> 1.39970842444753 9 -> 2.0
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))
}
- 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
# 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))
}
- 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:
)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)This can be called using:
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)BBC BASIC
Uses the MOD(array()) and SUM(array()) functions.
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)
- 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
#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);
}
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;
}
C#
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);
}
}
}
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.
#include <cassert>
#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";
}
}
Clojure
(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)))
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.
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
CoffeeScript
Uses a class instance to maintain state.
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 }
"""
- 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:
(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
In the REPL, one step at a time:
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
Component Pascal
BlackBox Component Builder
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.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
Crystal
Object
Use an object to keep state.
class StdDevAccumulator
def initialize
@n, @sum, @sum2 = 0, 0.0, 0.0
end
def <<(num)
@n += 1
@sum += num
@sum2 += num**2
Math.sqrt (@sum2 * @n - @sum**2) / @n**2
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}" }
- Output:
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 0.9428090415820634 adding 4: stddev of 4 samples is 0.8660254037844386 adding 5: stddev of 5 samples is 0.9797958971132712 adding 5: stddev of 6 samples is 1.0 adding 7: stddev of 7 samples is 1.3997084244475304 adding 9: stddev of 8 samples is 2.0
Closure
def sdaccum
n, sum, sum2 = 0, 0.0, 0.0
->(num : Int32) do
n += 1
sum += num
sum2 += num**2
Math.sqrt( (sum2 * n - sum**2) / n**2 )
end
end
sd = sdaccum
[2,4,4,4,5,5,7,9].each {|n| print sd.call(n), ", "}
- Output:
0.0, 1.0, 0.9428090415820634, 0.8660254037844386, 0.9797958971132712, 1.0, 1.3997084244475304, 2.0
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());
}
}
- 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.
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]
}? 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
2.0EasyLang
global sum sum2 n .
proc sd x . r .
sum += x
sum2 += x * x
n += 1
r = sqrt (sum2 / n - sum * sum / n / n)
.
v[] = [ 2 4 4 4 5 5 7 9 ]
for v in v[]
sd v r
print v & " " & r
.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
- 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
(defun running-std (items)
(let ((running-sum 0)
(running-len 0)
(running-squared-sum 0)
(result 0))
(dolist (item items)
(setq running-sum (+ running-sum item))
(setq running-len (1+ running-len))
(setq running-squared-sum (+ running-squared-sum (* item item)))
(setq result (sqrt (- (/ running-squared-sum (float running-len))
(/ (* running-sum running-sum)
(float (* running-len running-len))))))
(message "%f" result))
result))
(running-std '(2 4 4 4 5 5 7 9))
- Output:
0.000000 1.000000 0.942809 0.866025 0.979796 1.000000 1.399708 2.000000 2.0
(let ((x '(2 4 4 4 5 5 7 9)))
(string-to-number (calc-eval "sqrt(vpvar($1))" nil (append '(vec) x))))
;; lexical-binding: t
(require 'generator)
(iter-defun std-dev-gen (lst)
(let ((sum 0)
(avg 0)
(tmp '())
(std 0))
(dolist (i lst)
(setq i (float i))
(push i tmp)
(setq sum (+ sum i))
(setq avg (/ sum (length tmp)))
(setq std 0)
(dolist (j tmp)
(setq std (+ std (expt (- j avg) 2))))
(setq std (/ std (length tmp)))
(setq std (sqrt std))
(iter-yield std))))
(let* ((test-data '(2 4 4 4 5 5 7 9))
(generator (std-dev-gen test-data)))
(dolist (i test-data)
(message "with %d: %f" i (iter-next generator))))
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 ).
- 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
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 ;
FOCAL
01.01 C-- TEST SET
01.10 S T(1)=2;S T(2)=4;S T(3)=4;S T(4)=4
01.20 S T(5)=5;S T(6)=5;S T(7)=7;S T(8)=9
01.30 D 2.1
01.35 T %6.40
01.40 F I=1,8;S A=T(I);D 2.2;T "VAL",A;D 2.3;T " SD",A,!
01.50 Q
02.01 C-- RUNNING STDDEV
02.02 C-- 2.1: INITIALIZE
02.03 C-- 2.2: INSERT VALUE A
02.04 C-- 2.3: A = CURRENT STDDEV
02.10 S XN=0;S XS=0;S XQ=0
02.20 S XN=XN+1;S XS=XS+A;S XQ=XQ+A*A
02.30 S A=FSQT(XQ/XN - (XS/XN)^2)- Output:
VAL= 2.00000 SD= 0.00000 VAL= 4.00000 SD= 1.00000 VAL= 4.00000 SD= 0.94281 VAL= 4.00000 SD= 0.86603 VAL= 5.00000 SD= 0.97980 VAL= 5.00000 SD= 1.00000 VAL= 7.00000 SD= 1.39971 VAL= 9.00000 SD= 2.00000
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 ;
This variation is more numerically stable when there are large numbers of samples or large sample ranges.
: 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 ;
Usage example:
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.
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
- 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.
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
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
' 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- 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.
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))
}
}
- Output:
0 1 0.9428090415820634 0.8660254037844386 0.9797958971132713 1 1.3997084244475304 2
Groovy
Solution:
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)}"
}
- 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.
{-# 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]
Or, perhaps more simply, as a map-accumulation over an indexed list:
import Data.List (mapAccumL)
-------------- CUMULATIVE STANDARD DEVIATION -------------
cumulativeStdDevns :: [Float] -> [Float]
cumulativeStdDevns = snd . mapAccumL go (0, 0) . zip [1.0..]
where
go (s, q) (i, x) =
let _s = s + x
_q = q + (x ^ 2)
in ((_s, _q), sqrt ((_q / i) - ((_s / i) ^ 2)))
--------------------------- TEST -------------------------
main :: IO ()
main = mapM_ print $ cumulativeStdDevns [2, 4, 4, 4, 5, 5, 7, 9]
- Output:
0.0 1.0 0.9428093 0.8660254 0.97979593 1.0 1.3997087 2.0
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));
}
}
0 1 0.942809041582063 0.866025403784439 0.979795897113271 1 1.39970842444753 2
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
ENDAdding 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
- 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
IS-BASIC
100 PROGRAM "StDev.bas"
110 LET N=8
120 NUMERIC ARR(1 TO N)
130 FOR I=1 TO N
140 READ ARR(I)
150 NEXT
160 DEF STDEV(N)
170 LET S1,S2=0
180 FOR I=1 TO N
190 LET S1=S1+ARR(I)^2:LET S2=S2+ARR(I)
200 NEXT
210 LET STDEV=SQR((N*S1-S2^2)/N^2)
220 END DEF
230 FOR J=1 TO N
240 PRINT J;"item =";ARR(J),"standard dev =";STDEV(J)
250 NEXT
260 DATA 2,4,4,4,5,5,7,9J
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.
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
Alternatives:
Using verbose names for J primitives.
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
Or we could take a cue from the R implementation and reverse the Bessel correction to stddev:
require'stats'
(%:@:(%~<:)@:# * stddev)\ 2 4 4 4 5 5 7 9
0 1 0.942809 0.866025 0.979796 1 1.39971 2
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));
}
}
}
JavaScript
Imperative
Uses a closure.
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(', ');
- Output:
0, 1, 0.942809041582063, 0.866025403784439, 0.979795897113273, 1, 1.39970842444753, 2
Functional
ES5
Accumulating across a fold
(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]);
- Output:
[0, 1, 0.9428090415820626, 0.8660254037844386,
0.9797958971132716, 1, 1.3997084244475297, 2]
ES6
As a map-accumulation:
(() => {
'use strict';
// ---------- CUMULATIVE STANDARD DEVIATION ----------
// cumulativeStdDevns :: [Float] -> [Float]
const cumulativeStdDevns = ns => {
const go = ([s, q]) =>
([i, x]) => {
const
_s = s + x,
_q = q + (x * x),
j = 1 + i;
return [
[_s, _q],
Math.sqrt(
(_q / j) - Math.pow(_s / j, 2)
)
];
};
return mapAccumL(go)([0, 0])(ns)[1];
};
// ---------------------- TEST -----------------------
const main = () =>
showLog(
cumulativeStdDevns([
2, 4, 4, 4, 5, 5, 7, 9
])
);
// --------------------- GENERIC ---------------------
// mapAccumL :: (acc -> x -> (acc, y)) -> acc -> [x] -> (acc, [y])
const mapAccumL = f =>
// A tuple of an accumulation and a list
// obtained by a combined map and fold,
// with accumulation from left to right.
acc => xs => [...xs].reduce((a, x, i) => {
const pair = f(a[0])([i, x]);
return [pair[0], a[1].concat(pair[1])];
}, [acc, []]);
// showLog :: a -> IO ()
const showLog = (...args) =>
console.log(
args
.map(x => JSON.stringify(x, null, 2))
.join(' -> ')
);
// MAIN ---
return main();
})();
- Output:
[ 0, 1, 0.9428090415820626, 0.8660254037844386, 0.9797958971132716, 1, 1.3997084244475297, 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.
# 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:
simulateExample 1
# observations.txt 2 4 4 4 5 5 7 9
- Output:
$ jq -s -f Dynamic_standard_deviation.jq observations.txt
0
1
0.9428090415820634
0.8660254037844386
0.9797958971132711
0.9999999999999999
1.3997084244475302
1.9999999999999998
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.
# requires jq version > 1.4
def simulate(stream):
foreach stream as $observation
(initial_state;
update_state($observation);
standard_deviation);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.
#!/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
Example 3
$ ./standard_deviation_server.sh
Next observation: 10
0
Next observation: 20
5
Next observation: 0
8.16496580927726
Julia
Use a closure to create a running standard deviation function.
function makerunningstd(::Type{T} = Float64) where T
∑x = ∑x² = zero(T)
n = 0
function runningstd(x)
∑x += x
∑x² += x ^ 2
n += 1
s = ∑x² / n - (∑x / n) ^ 2
return s
end
return runningstd
end
test = Float64[2, 4, 4, 4, 5, 5, 7, 9]
rstd = makerunningstd()
println("Perform a running standard deviation of ", test)
for i in test
println(" - add $i → ", rstd(i))
end
- Output:
Perform a running standard deviation of [2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0] - add 2.0 → 0.0 - add 4.0 → 1.0 - add 4.0 → 0.8888888888888875 - add 4.0 → 0.75 - add 5.0 → 0.9600000000000009 - add 5.0 → 1.0 - add 7.0 → 1.9591836734693864 - add 9.0 → 4.0
Kotlin
Using a class to keep the running sum, sum of squares and number of elements added so far:
// 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 => ${csd.sd(d)}")
}
- Output:
Add 2.0 => 0.0 Add 4.0 => 1.0 Add 4.0 => 0.9428090415820626 Add 4.0 => 0.8660254037844386 Add 5.0 => 0.9797958971132708 Add 5.0 => 1.0 Add 7.0 => 1.399708424447531 Add 9.0 => 2.0
Liberty BASIC
Using a global array to maintain the state. Implements definition explicitly.
dim SD.storage$( 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, 9New 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
Lobster
// Stats computes a running mean and variance
// See Knuth TAOCP vol 2, 3rd edition, page 232
class Stats:
M = 0.0
S = 0.0
n = 0
def incl(x):
n += 1
if n == 1:
M = x
else:
let mm = (x - M)
M += mm / n
S += mm * (x - M)
def mean(): return M
//def variance(): return (if n > 1.0: S / (n - 1.0) else: 0.0) // Bessel's correction
def variance(): return (if n > 0.0: S / n else: 0.0)
def stddev(): return sqrt(variance())
def count(): return n
def test_stdv() -> float:
let v = [2,4,4,4,5,5,7,9]
let s = Stats {}
for(v) x: s.incl(x+0.0)
print concat_string(["Mean: ", string(s.mean()), ", Std.Deviation: ", string(s.stddev())], "")
test_stdv()- Output:
Mean: 5.0, Std.Deviation: 2.0
Lua
Uses a closure. Translation of JavaScript.
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
Mathematica/Wolfram Language
runningSTDDev[n_] := (If[Not[ValueQ[$Data]], $Data = {}];StandardDeviation[AppendTo[$Data, n]])
MATLAB / Octave
The simple form is, computing only the standand deviation of the whole data set:
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
When the intermediate results are also needed, one can use this vectorized form:
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
Here is a vectorized one line solution as a function
function stdDevEval(n)
disp(sqrt(sum((n-sum(n)/length(n)).^2)/length(n)));
end
MiniScript
StdDeviator = {}
StdDeviator.count = 0
StdDeviator.sum = 0
StdDeviator.sumOfSquares = 0
StdDeviator.add = function(x)
self.count = self.count + 1
self.sum = self.sum + x
self.sumOfSquares = self.sumOfSquares + x*x
end function
StdDeviator.stddev = function()
m = self.sum / self.count
return sqrt(self.sumOfSquares / self.count - m*m)
end function
sd = new StdDeviator
for x in [2, 4, 4, 4, 5, 5, 7, 9]
sd.add x
end for
print sd.stddev
- Output:
2
МК-61/52
0 П4 П5 П6 С/П П0 ИП5 + П5 ИП0
x^2 ИП6 + П6 КИП4 ИП6 ИП4 / ИП5 ИП4
/ x^2 - КвКор БП 04
Instruction: В/О С/П number С/П number С/П ...
Nanoquery
class StdDev
declare n
declare sum
declare sum2
def StdDev()
n = 0
sum = 0
sum2 = 0
end
def sd(x)
this.n += 1
this.sum += x
this.sum2 += x*x
return sqrt(sum2/n - sum*sum/n/n)
end
end
testData = {2,4,4,4,5,5,7,9}
sd = new(StdDev)
for x in testData
println sd.sd(x)
end- Output:
0.0 1.0 0.9428090415820634 0.8660254037844386 0.9797958971132712 1.0 1.3997084244475304 2.0
Nim
Using global variables
import math, strutils
var sdSum, sdSum2, sdN = 0.0
proc sd(x: float): float =
sdN += 1
sdSum += x
sdSum2 += x * x
sqrt(sdSum2 / sdN - sdSum * sdSum / (sdN * sdN))
for value in [float 2,4,4,4,5,5,7,9]:
echo value, " ", formatFloat(sd(value), precision = -1)
- Output:
2 0 4 1 4 0.942809 4 0.866025 5 0.979796 5 1 7 1.39971 9 2
Using an accumulator object
import math, strutils
type SDAccum = object
sdN, sdSum, sdSum2: float
var accum: SDAccum
proc add(accum: var SDAccum; value: float): float =
# Add a value to the accumulator. Return the standard deviation.
accum.sdN += 1
accum.sdSum += value
accum.sdSum2 += value * value
result = sqrt(accum.sdSum2 / accum.sdN - accum.sdSum * accum.sdSum / (accum.sdN * accum.sdN))
for value in [float 2, 4, 4, 4, 5, 5, 7, 9]:
echo value, " ", formatFloat(accum.add(value), precision = -1)
- Output:
Same output.
Using a closure
import math, strutils
func accumBuilder(): auto =
var sdSum, sdSum2, sdN = 0.0
result = func(value: float): float =
sdN += 1
sdSum += value
sdSum2 += value * value
result = sqrt(sdSum2 / sdN - sdSum * sdSum / (sdN * sdN))
let std = accumBuilder()
for value in [float 2, 4, 4, 4, 5, 5, 7, 9]:
echo value, " ", formatFloat(std(value), precision = -1)
- Output:
Same output.
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();
}
}
}Objective-C
#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;
}
Blocks
#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;
}
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)
- 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.
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 ;- 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
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
- 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.
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])Pascal
Std.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.
- 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
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.
- 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
{
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;
}
}
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";
A much shorter version using a closure and a property of the variance:
# <(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);
- Output:
0 1 0.942809041582063 0.866025403784439 0.979795897113272 1 1.39970842444753 2
one-liner:
perl -MMath::StdDev -e '$d=new Math::StdDev;foreach my $v ( 2,4,4,4,5,5,7,9 ) {$d->Update($v); print $d->variance(),"\n"}'
small script:
use Math::StdDev;
$d=new Math::StdDev;
foreach my $v ( 2,4,4,4,5,5,7,9 ) {
$d->Update($v);
print $d->variance(),"\n"
}
- 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.
with javascript_semantics 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
- 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":
<?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);
}
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
(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)) ) )- 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
*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;- 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.
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 / $nums.Length)
}
}
Usage as follows:
PS> 2,4,4,4,5,5,7,9 | Get-StandardDeviation 0 1 0.942809041582063 0.866025403784439 0.979795897113271 1 1.39970842444753 2
PureBasic
;Define our Standard deviation function
Declare.d Standard_deviation(x)
; Main program
If OpenConsole()
Define i, x
Restore MyList
For i=1 To 8
Read.i x
PrintN(StrD(Standard_deviation(x)))
Next i
Print(#CRLF$+"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- 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
>>> 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)
>>>
Python: Using a class instance
>>> 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))
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
>>> 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
Python: Using an extended generator
>>> 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
Python: In a couple of 'functional' lines
>>> 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
R
To compute the running sum, one must keep track of the number of items, the sum of values, and the sum of squares.
If the goal is to get a vector of running standard deviations, the simplest is to do it with cumsum:
cumsd <- function(x) {
n <- seq_along(x)
sqrt(cumsum(x^2) / n - (cumsum(x) / n)^2)
}
set.seed(12345L)
x <- rnorm(10)
cumsd(x)
# [1] 0.0000000 0.3380816 0.8752973 1.1783628 1.2345538 1.3757142 1.2867220 1.2229056 1.1665168 1.1096814
# Compare to the naive implementation, i.e. compute sd on each sublist:
Vectorize(function(k) sd(x[1:k]) * sqrt((k - 1) / k))(seq_along(x))
# [1] NA 0.3380816 0.8752973 1.1783628 1.2345538 1.3757142 1.2867220 1.2229056 1.1665168 1.1096814
# Note that the first is NA because sd is unbiased formula, hence there is a division by n-1, which is 0 for n=1.The task requires an accumulator solution:
accumsd <- function() {
n <- 0
m <- 0
s <- 0
function(x) {
n <<- n + 1
m <<- m + x
s <<- s + x * x
sqrt(s / n - (m / n)^2)
}
}
f <- accumsd()
sapply(x, f)
# [1] 0.0000000 0.3380816 0.8752973 1.1783628 1.2345538 1.3757142 1.2867220 1.2229056 1.1665168 1.1096814Racket
#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)))
Raku
(formerly Perl 6)
Using a closure:
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 $_ for 2, 4, 4, 4, 5, 5, 7, 9;
Using a state variable (remember that <(x-<x>)²> = <x²> - <x>²):
sub stddev($x) {
sqrt
( .[2] += $x²) / ++.[0]
- ((.[1] += $x ) / .[0])²
given state @;
}
say .&stddev for <2 4 4 4 5 5 7 9>;
- Output:
0 1 0.942809041582063 0.866025403784439 0.979795897113271 1 1.39970842444753 2
REXX
These REXX versions use running sums.
show running sums
/*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; $$= 0; L= length(n) /*N: # items; $,$$: sums to be zeroed*/
/* [↓] 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 0 /*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
- 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
/*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 /*perform summation on two sets of #'s.*/
_= word(#, j); $= $ + _
$$= $$ + _**2
end /*j*/
say 'standard deviation: ' sqrt($$/n - ($/n)**2) /*calculate&display the std, deviation.*/
exit 0 /*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
- output when using the default input of: 2 4 4 4 5 5 7 9
standard deviation: 2
Ring
# Project : Cumulative standard deviation
decimals(6)
sdsave = list(100)
sd = "2,4,4,4,5,5,7,9"
sumval = 0
sumsqs = 0
for num = 1 to 8
sd = substr(sd, ",", "")
stddata = number(sd[num])
sumval = sumval + stddata
sumsqs = sumsqs + pow(stddata,2)
standdev = pow(((sumsqs / num) - pow((sumval /num),2)),0.5)
sdsave[num] = string(num) + " " + string(sumval) +" " + string(sumsqs)
see "" + num + " value in = " + stddata + " Stand Dev = " + standdev + nl
nextOutput:
1 value in = 2 Stand Dev = 0 2 value in = 4 Stand Dev = 1 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 7 value in = 7 Stand Dev = 1.399708 8 value in = 9 Stand Dev = 2
RPL
Basic RPL
≪ CL∑ { } SWAP
1 OVER SIZE FOR j
DUP j GET ∑+
IF j 1 > THEN
SDEV ∑DAT SIZE 1 GET DUP 1 - SWAP / √ *
ROT SWAP + SWAP END
NEXT
DROP CL∑
≫ 'CSDEV' STO
RPL 1993
≪ CL∑
1 ≪ ∑+ PSDEV ≫ DOSUBS CL∑
≫ 'CSDEV' STO
- Output:
1: { 0 1 0.942809041582 0.866025403784 0.979795897113 1 1.39970842445 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.
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}" }
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
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), ", "}
0.0, 1.0, 0.942809041582063, 0.866025403784439, 0.979795897113272, 1.0, 1.39970842444753, 2.0,
Run BASIC
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 num1 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
Rust
Using a struct:
pub struct CumulativeStandardDeviation {
n: f64,
sum: f64,
sum_sq: f64
}
impl CumulativeStandardDeviation {
pub fn new() -> Self {
CumulativeStandardDeviation {
n: 0.,
sum: 0.,
sum_sq: 0.
}
}
fn push(&mut self, x: f64) -> f64 {
self.n += 1.;
self.sum += x;
self.sum_sq += x * x;
(self.sum_sq / self.n - self.sum * self.sum / self.n / self.n).sqrt()
}
}
fn main() {
let nums = [2, 4, 4, 4, 5, 5, 7, 9];
let mut cum_stdev = CumulativeStandardDeviation::new();
for num in nums.iter() {
println!("{}", cum_stdev.push(*num as f64));
}
}
- Output:
0 1 0.9428090415820626 0.8660254037844386 0.9797958971132708 1 1.399708424447531 2
Using a closure:
fn sd_creator() -> impl FnMut(f64) -> f64 {
let mut n = 0.0;
let mut sum = 0.0;
let mut sum_sq = 0.0;
move |x| {
sum += x;
sum_sq += x*x;
n += 1.0;
(sum_sq / n - sum * sum / n / n).sqrt()
}
}
fn main() {
let nums = [2, 4, 4, 4, 5, 5, 7, 9];
let mut sd_acc = sd_creator();
for num in nums.iter() {
println!("{}", sd_acc(*num as f64));
}
}
- Output:
0 1 0.9428090415820626 0.8660254037844386 0.9797958971132708 1 1.399708424447531 2
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;
- 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
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: T = avg(ts) // Leave val type of T
// Root of mean diffs
val stdDev = sqrt(ts.map { x =>
val diff = num.toDouble(num.minus(x, mean))
diff * diff
}.sum / ts.size)
(mean, stdDev)
}
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))))
println(s"Successfully completed without errors. [total ${scala.compat.Platform.currentTime - executionStart}ms]")
}
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)))
(unless (null? input)
(display (f (car input)))
(newline)
(loop f (cdr input))))
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.
T=[2,4,4,4,5,5,7,9];
stdev(T)*sqrt((length(T)-1)/length(T))
- 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:
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}"
}
- 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:
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(_) }
- Output:
0 1 0.942809041582063365867792482806465385713114583585 0.866025403784438646763723170752936183471402626905 0.979795897113271239278913629882356556786378992263 1 1.39970842444753034182701947126050936683768427466 2
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 ]
].
|sdacc sd|
sdacc := SDAccum new.
#( 2 4 4 4 5 5 7 9 ) do: [ :v | sd := sdacc value: v ].
('std dev = %1' % { sd }) displayNl.
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;
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
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()
- 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:
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 + $1})
return sqrt(sumOfSquaredAvgDiff / length)
}
let responseTimes = [ 18.0, 21.0, 41.0, 42.0, 48.0, 50.0, 55.0, 90.0 ]
standardDeviation(responseTimes) // 20.8742514835862
standardDeviation([2,4,4,4,5,5,7,9]) // 2.0
Tcl
With a Class
or
oo::class create SDAccum {
variable sum sum2 num
constructor {} {
set sum 0.0
set sum2 0.0
set num 0
}
method value {x} {
set sum2 [expr {$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"
- Output:
the standard deviation is: 2.0
With a Coroutine
# 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"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
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- 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.
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- Output:
0 1 0.942809041582063 0.866025403784439 0.979795897113271 1 1.39970842444753 2
Wren
import "./fmt" for Fmt
import "./math" for Nums
var cumStdDev = Fiber.new { |a|
for (i in 0...a.count) {
var b = a[0..i]
System.print("Values : %(b)")
Fiber.yield(Nums.popStdDev(b))
}
}
var a = [2, 4, 4, 4, 5, 5, 7, 9]
while (true) {
var sd = cumStdDev.call(a)
if (cumStdDev.isDone) return
Fmt.print("Std Dev : $10.8f\n", sd)
}- Output:
Values : [2] Std Dev : 0.00000000 Values : [2, 4] Std Dev : 1.00000000 Values : [2, 4, 4] Std Dev : 0.94280904 Values : [2, 4, 4, 4] Std Dev : 0.86602540 Values : [2, 4, 4, 4, 5] Std Dev : 0.97979590 Values : [2, 4, 4, 4, 5, 5] Std Dev : 1.00000000 Values : [2, 4, 4, 4, 5, 5, 7] Std Dev : 1.39970842 Values : [2, 4, 4, 4, 5, 5, 7, 9] Std Dev : 2.00000000
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);
]- Output:
0.00000 1.00000 0.94281 0.86603 0.97980 1.00000 1.39971 2.00000
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())
}- 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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