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Latest revision as of 15:32, 21 November 2023

Task

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, $\{2,4,4,4,5,5,7,9\}$ , which is $2$ .

Related tasks

Random numbers

11l

Translation of: Python:_Callable_class

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!

Library: Action! Tool Kit

Library: Action! Real Math

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

Translation of: C

Works with: ALGOL 68 version Standard - no extensions to language used

Works with: ALGOL 68G version Any - tested with release 2.8-win32

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

Translation of: python

Works with: ALGOL 68 version Standard - no extensions to language used

Works with: ALGOL 68G version Any - tested with release 2.8-win32

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

Translation of: python

Works with: ALGOL 68 version Standard - no extensions to language used

Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny

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

Translation of: ALGOL 68

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:

Axiom

This example is incorrect. Please fix the code and remove this message.

Details: It does not return the running standard deviation of the series.

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

Works with: BBC BASIC for Windows

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

Works with: OpenCOBOL version 1.1

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

This example is incorrect. Please fix the code and remove this message.

Details: Function does not take numbers individually.

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.

Translation of: Ruby

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

Translation of: Ruby

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

Translation of: Perl

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
2.0

EasyLang

Translation of: Pascal

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

Translation of: Erlang

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:

Library: Calc

(let ((x '(2 4 4 4 5 5 7 9)))
  (string-to-number (calc-eval "sqrt(vpvar($1))" nil (append '(vec) x))))

Library: generator.el

;; 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

Works with: Fortran version 2003 and later

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:

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
 END

Adding 2 stdev = 0
Adding 4 stdev = 1
Adding 4 stdev = 0.9428090416
Adding 4 stdev = 0.8660254038
Adding 5 stdev = 0.9797958971
Adding 5 stdev = 1
Adding 7 stdev = 1.399708424
Adding 9 stdev = 2

Icon and Unicon

procedure main()

stddev() # reset state / empty
every  s := stddev(![2,4,4,4,5,5,7,9]) do
   write("stddev (so far) := ",s)

end

procedure stddev(x)  # running standard deviation
static X,sumX,sum2X

   if /x then {   # reset state
      X := []
      sumX := sum2X := 0.
      }
   else {         # accumulate
      put(X,x)
      sumX +:= x
      sum2X +:= x^2
      return sqrt( (sum2X / *X) - (sumX / *X)^2 )
      }
end

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 2,4,4,4,5,5,7,9

J

J is block-oriented; it expresses algorithms with the semantics of all the data being available at once. It does not have native lexical closure or coroutine semantics. It is possible to implement these semantics in J; see Moving Average for an example. We will not reprise that here.

   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

Translation of: R

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:
simulate

Example 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

Translation of: Java

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, 9

New data 2 so S.D. now =   0.000000
New data 4 so S.D. now =   1.000000
New data 4 so S.D. now =   0.942809
New data 4 so S.D. now =   0.866025
New data 5 so S.D. now =   0.979796
New data 5 so S.D. now =   1.000000
New data 7 so S.D. now =   1.399708
New data 9 so S.D. now =   2.000000

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:

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

Translation of: Java

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:

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

Translation of: Java

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

Works with: Mac OS X version 10.6+

Works with: iOS version 4+

#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

Works with: 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

Translation of: AWK

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

Works with: Python version 3.x

>>> 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

Works with: Python version 2.5+

>>> 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 1.1096814

Racket

#lang racket
(require math)
(define running-stddev
  (let ([ns '()])
    (λ(n) (set! ns (cons n ns)) (stddev ns))))
;; run it on each number, return the last result
(last (map running-stddev '(2 4 4 4 5 5 7 9)))

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
next

Output:

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 num

1 value in = 2 Stand Dev =   0.000000
2 value in = 4 Stand Dev =   1.000000
3 value in = 4 Stand Dev =   0.942809
4 value in = 4 Stand Dev =   0.866025
5 value in = 5 Stand Dev =   0.979796
6 value in = 5 Stand Dev =   1.000000
7 value in = 7 Stand Dev =   1.399708
8 value in = 9 Stand Dev =   2.000000

Rust

Using a struct:

Translation of: Java

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

Library: Scala

import scala.math.sqrt

object StddevCalc extends App {

  def calcAvgAndStddev[T](ts: Iterable[T])(implicit num: Fractional[T]): (T, Double) = {
    def avg(ts: Iterable[T])(implicit num: Fractional[T]): T =
      num.div(ts.sum, num.fromInt(ts.size)) // Leaving with type of function T

    val mean: 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

Works with: GNU 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

Works with: Postgresql

-- 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

Works with: Tcl version 8.6

or

Library: TclOO

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

Works with: Tcl version 8.6

# 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

Library: Wren-fmt

Library: Wren-math

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