Calculate Standard Deviation

Question

Challenge

Given a list of numbers, calculate the population standard deviation of the list.

Use the following equation to calculate population standard deviation:

Input

The input will a list of integers in any format (list, string, etc.). Some examples:

56,54,89,87
67,54,86,67

The numbers will always be integers.

Input will be to STDIN or function arguments.

Output

The output must be a floating point number.

Rules

You may use built in functions to find the standard deviation.

Your answer can be either a full program or a function.

Examples

10035, 436844, 42463, 44774 => 175656.78441352615

45,67,32,98,11,3 => 32.530327730015607

1,1,1,1,1,1 => 0.0

Winning

The shortest program or function wins.

Leaderboard

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Answer 1

Clip, 3

.sk

.s is the standard deviation, k parses the input in the form {1,2,3}.

Answer 2

Mathematica, 24 22 bytes

Nice, Mathematica has a built-in StandardDevi... oh... that computes the sample standard deviation, not the population standard deviation.

But what if we use Variance... oh... same deal.

But there is yet another related built-in:

CentralMoment[#,2]^.5&

Yay. :)

This also works for 22 bytes:

Mean[(#-Mean@#)^2]^.5&

And this for 27:

N@RootMeanSquare[#-Mean@#]&

Answer 3

Octave, 14 bytes

g=@(a)std(a,1)

Try it on ideone.

Answer 4

kdb+, 3 bytes

dev

One of the APL derviates had to have this as a built-in.

Test run

q)dev 56, 54, 89, 87
16.53028
q)f:dev
q)f 10035, 436844, 42463, 44774
175656.8
q)f 45,67,32,98,11,3
32.53033

Answer 5

Dyalog APL, 24 23 21 20 19 17 bytes

*∘.5∘M×⍨-M×M←+/÷≢

This defines an unnamed, monadic function train, which is equivalent to the following function.

{.5*⍨M(×⍨⍵)-M⍵×(M←{(+/⍵)÷≢⍵})⍵}

Try them online on TryAPL.

How it works

The code consists of several trains.

M←+/÷≢

This defines a monadic 3-train (fork) M that executes +/ (sum of all elements) and ≢ (length) for the right argument, then applies ÷ (division) to the results, returning the arithmetic mean of the input.

M×M

This is another fork that applies M to the right argument, repeats this a second time, and applies × (product) to the results, returning μ².

×⍨-(M×M)

This is yet another fork that calculates the square of the arithmetic mean as explained before, applies ×⍨ (product with itself) to the right argument, and finally applies - (difference) to the results.

For input (x₁, …, x_N), this function returns (x₁ - μ², …, x_N - μ²).

*∘.5∘M

This composed function is applies M to its right argument, then *∘.5. The latter uses right argument currying to apply map input a to a*0.5 (square root of a).

(*∘.5∘M)(×⍨-(M×M))

Finally, we have this monadic 2-train (atop), which applies the right function first, then the left to its result, calculating the standard deviation as follows.

Answer 6

R, 41 40 39 36 30 28 bytes

code

Thanks to beaker, Alex A. and MickyT for much bytes.

cat(sd(c(v=scan(),mean(v))))   

old codes

v=scan();n=length(v);sd(v)/(n/(n-1))**0.5
m=scan();cat(sqrt(sum(mean((m-mean(m))^2))))
m=scan();cat(mean((m-mean(m))^2)^.5) 

This should yield the population standard deviation.

Answer 7

Julia, 26 19 bytes

x->std([x;mean(x)])

This creates an unnamed function that accepts an array and returns a float.

Ungolfed, I guess:

function f(x::Array{Int,1})
    # Return the sample standard deviation (denominator N-1) of
    # the input with the mean of the input appended to the end.
    # This corrects the denominator to N without affecting the
    # mean.
    std([x; mean(x)])
end

Answer 8

Pyth, 20 19 17 13 bytes

@.O^R2-R.OQQ2

Thanks to @FryAmTheEggman for golfing off 4 bytes!

Try it online.

How it works

        .OQ    Compute the arithmetic mean of the input (Q).
      -R   Q   Subtract the arithmetic mean of all elements of Q.
   ^R2         Square each resulting difference.
 .O            Compute the arithmetic mean of the squared differences.
@           2  Apply square root.

Answer 9

CJam, 24 22 21 bytes

q~_,_@_:+d@/f-:mh\mq/

Thanks to @aditsu for golfing off 1 byte!

Try it online in the CJam interpreter.

How it works

q~                    e# Read all input and evaluate it.
  _,                  e# Copy the array and push its length.
    _@                e# Copy the length and rotate the array on top.
      _:+d            e# Copy the array and compute its sum. Cast to Double.
          @/          e# Rotate the length on top and divide the sum by it.
            f-        e# Subtract the result (μ) from the array's elements.
              :mh     e# Reduce by hypotenuse.
                      e# a b mh -> sqrt(a^2 + b^2)
                      e# sqrt(a^2 + b^2) c mh -> sqrt(sqrt(a^2 + b^2)^2 + c^2)
                      e#                           = sqrt(a^2 + b^2 + c^2)
                      e# ⋮
                 \mq/ e# Divide the result by the square root of the length.

Answer 10

APL, 24 bytes

{.5*⍨+/(2*⍨⍵-+/⍵÷≢⍵)÷≢⍵}

A little different approach than Dennis' Dyalog APL solution. This should work with any APL implementation.

This creates an unnamed monadic function that computes the vector (x - µ)² as 2*⍨⍵-+/⍵÷≢⍵, divides this by N (÷≢⍵), takes the sum of this vector using +/, and then takes the square root (.5*⍨).

Try it online

Answer 11

TI-BASIC, 7 bytes

stdDev(augment(Ans,{mean(Ans

I borrowed the algorithm to get population standard deviation from sample standard deviation from here.

The shortest solution I could find without augment( is 9 bytes:

stdDev(Ans√(1-1/dim(Ans

Answer 12

Haskell, 61 bytes

d n=1/sum(n>>[1])
f a=sqrt$d a*sum(map((^2).(-)(d a*sum a))a)

Straightforward, except maybe my custom length function sum(n>>[1]) to trick Haskell's strict type system.

Answer 13

Python 3.4+, 30 bytes

from statistics import*;pstdev

Imports the builtin function pstdev, e.g.

>>> pstdev([56,54,89,87])
16.53027525481654

Answer 14

Jelly

11 bytes

S÷L
Dz_²ÇN½

This is a direct translation of my APL answer to Jelly. Try it online!

How it works

S÷L        Helper link. Argument: z (vector)

S          Compute the sum of z.
  L        Compute the length of z.
 ÷         Divide the former by the latter.
           This computes the mean of z.

Dz_²ÇN½    Main link. Argument: z (vector)

Ç          Apply the previous link, i.e., compute the mean of z.
 ²         Square the mean.
   ²       Square all number in z.
  _        Subtract each squared number from the squared mean.
    Ç      Take the mean of the resulting vector.
     N     Multiply it by -1.
      ½    Take the square root of the result.

Answer 15

Prolog (SWI), 119 bytes

Code:

q(U,X,A):-A is(X-U)^2.
p(L):-sumlist(L,S),length(L,I),U is S/I,maplist(q(U),L,A),sumlist(A,B),C is sqrt(B/I),write(C).

Explanation:

q(U,X,A):-A is(X-U)^2.   % calc squared difference of X and U
p(L):-sumlist(L,S),      % sum input list
      length(L,I),       % length of input list
      U is S/I,          % set U to the mean value of input list
      maplist(q(U),L,A), % set A to the list of squared differences of input and mean
      sumlist(A,B),      % sum squared differences list
      C is sqrt(B/I),    % divide sum of squares by length of list
      write(C).          % print answer

Example:

p([10035, 436844, 42463, 44774]).
175656.78441352615

Try it out online here

Answer 16

J, 18 bytes

[:%:@M*:-M*M=:+/%#

This is a direct translation of my APL answer to J.

Try it online!

Answer 17

Haskell, 45 bytes

(?)i=sum.map(^i)
f l=sqrt$2?l/0?l-(1?l/0?l)^2

Try it online!

The value i?l is the sum of the i'th powers of elements in l, so that 0?l is the length and 1?l is the sum.

Answer 18

05AB1E, 10 7 bytes

ÅA-nÅAt

-3 bytes thanks to @ovs.

Try it online or verify all test cases.

Explanation:

ÅA       # Get the arithmetic mean of the (implicit) input-list
  -      # Subtract it from each value in the (implicit) input-list
   n     # Square each of those
    ÅA   # Take the arithmetic mean of that
      t  # And take the square-root of that
         # (after which it is output implicitly as result)

Answer 19

Brachylog, 16 bytes

⟨∋-⟨+/l⟩⟩ᶠ^₂ᵐ↰₂√

Try it online! (all cases at once)

I feel like maybe there's some shorter way to compute the squared deviations than 13 bytes.

Answer 20

Arn, 19 18 14 bytes

¯‡iʠØbmÅQƥªÈªÆ

Try it!

Explained

Unpacked: :/mean(n{:*n-mean:s}\

:/                      Square root
  mean                  Mean function
    (                   Begin expression
        n{              Block with key of n
          :*            Square
              n
            -           Subtraction
              mean
                    _   Variable initialized to STDIN; implied
                  :s    Split on spaces
         }              End of block
       \                Mapped over
         _              Implied
    )                   End of expression; Implied

Answer 21

Simplex v.0.5, 43 bytes

Just 'cuz. I really need to golf this one more byte.

t[@u@RvR]lR1RD@wA@T@{j@@SR2ERpR}u@vR@TR1UEo   
t[      ]                                     ~~ Applies inner function to entire strip (left-to-right)
  @                                           ~~ Copies current value to register
   u                                          ~~ Goes up a strip level
    @                                         ~~ Dumps the register on the current byte
     R                                        ~~ Proceeds right (s1)
      v                                       ~~ Goes back down
       R                                      ~~ Proceeds right (s0)
                                              ~~ Go right until an empty byte is found
         lR1RD                                ~~ Push length, 1, and divide.
              @                               ~~ Store result in register (1/N)
               wA                             ~~ Applies A (add) to each byte, (right-to-left)
                 @T@                          ~~ Puts 1/N down, multiplies it, and copies it to the register
                    {          }              ~~ Repeats until a zero-byte is met
                     j@@                      ~~ inserts a new byte and places register on it
                        SR                    ~~ Subtract it from the current byte and moves right
                          2E                  ~~ Squares result
                            RpR               ~~ Moves to the recently-created cell, deletes it, and continues
                                u@v           ~~ takes 1/N again into register
                                   R@T        ~~ multiplies it by the new sum
                                      R1UE    ~~ takes the square root of previous
                                          o   ~~ output as number

Answer 22

JavaScript (ES6), 73 bytes

a=>Math.sqrt(a.reduce((b,c)=>b+(d=c-eval(a.join`+`)/(l=a.length))*d,0)/l)

Answer 23

Perl5, 39 38

---

 16 for the script
+22 for the M switch
+ 1 for the E switch
=39

perl -MStatistics::Lite=:all -E"say stddevp@ARGV" .1 .2 300

Tested in Strawberry 5.20.2.

---

Oh, but then I realized that you said our answers can be functions instead of programs. In that case,

{use Statistics::Lite":all";stddevp@_}

has just 38. Tested in Strawberry 5.20.2 as

print sub{use Statistics::Lite":all";stddevp@_}->( .1, .2, 300)

Answer 24

MathGolf, 10 7 bytes

ë_▓-²▓√

Port of my 05AB1E answer.

Try it online.

Explanation:

ë        # Read all inputs as float-list
 _       # Duplicate that list
  ▓      # Get the average of that list
   -     # Subtract that average from each value in the list
    ²    # Square each value
     ▓   # Take the average of that again
      √  # And take the square root of that
         # (after which the entire stack joined together is output implicitly as result)

Answer 25

Whispers v2, 105 bytes

> Input
>> #1
>> ∑1
>> 3÷2
>> L-4
>> L²
>> Each 5 1
>> Each 6 7
>> ∑8
>> 9÷2
>> √10
>> Output 11

Try it online!

In the latest version of Whispers, the builtin σ can be used to shave off around 70 bytes.

How it works

For those unfamiliar with Whispers, the language works by using numbers as line references in order to pass values between lines. For example, the line >> 3÷2 doesn't calculate \$3 \div 2\$, rather it takes the values of lines 3 and 2 and calculates their division. Execution always begins on the last line.

This program simply implements the standard formula for standard deviation:

$$\sigma = \sqrt{\frac{1}{N}\sum^N_{i=1}{(x_i-\bar{x})^2}}$$ $$\bar{x} = \frac{1}{N}\sum^N_{i=1}{x_i}$$

Lines 2, 3 and 4 define \$\bar{x}\$, with it's specific value accessible on line 4. Line 2 stores \$N\$. We then calculate \$(x_i-\bar{x})^2\$ for each \$x_i \in x\$ with the lines 5, 6, 7 and 8:

>> L-4
>> L²
>> Each 5 1
>> Each 6 7

Line 7 runs line 5 over each element in the input, which takes the difference between each element in the input and the mean, We then square these differences using lines 8 and 6. Finally, we take the sum of these squares (line 9), divide by \$N\$ (line 10) and take the square root (line 11). Finally, we output this result.

Answer 26

Shasta ^v0.0.10, 65 bytes

{(**(/(sum(map${|x|(**(- x(/(sum$)(length$)))2)}))(length$)).5)}

Online interpreter not set up yet, tested on my machine

Explanation:

{...} Function taking argument $

(** ... .5) Square root of

(/ ... (length $)) Dividing ... by the length of $

(sum ...) The sum of

(map $ {|x| ...}) Mapping each x in $ to

(** ... 2) The square of

(- x ...) The difference between x and

(/ (sum $) (length $)) The mean of $

Answer 27

Python, 57 bytes

lambda l:(sum((x-sum(l)/len(l))**2for x in l)/len(l))**.5

Takes input as a list

Thanks @xnor

Answer 28

PowerShell, 122

:\>type stddev.ps1
$y=0;$z=$args -split",";$a=($z|?{$_});$c=$a.Count;$a|%{$y+=$_};$b=$y/$c;$a|%{$x+
=(($_-$b)*($_-$b))/$c};[math]::pow($x,0.5)

explanation

<#
$y=0                            init
$z=$args -split","              split delim ,
$a=($z|? {$_})                  remove empty items
$c=$a.Count                     count items
$a|%{$y+=$_}                    sum
$b=$y/$c                        average
$a|%{$x+=(($_-$b)*($_-$b))/$c}  sum of squares/count
[math]::pow($x,0.5)             result
#>

result

:\>powershell -nologo -f stddev.ps1 45,67,32,98,11,3
32.5303277300156

:\>powershell -nologo -f stddev.ps1 45,  67,32,98,11,3
32.5303277300156

:\>powershell -nologo -f stddev.ps1 45,  67,32, 98 ,11,3
32.5303277300156

:\>powershell -nologo -f stddev.ps1 10035, 436844, 42463, 44774
175656.784413526

:\>powershell -nologo -f stddev.ps1 1,1,1,1,1,1
0

Answer 29

Fortran, 138 bytes

Just a straightforward implementation of the equation in Fortran:

double precision function std(x)
integer,dimension(:),intent(in) :: x
std = norm2(dble(x-sum(x)/size(x)))/sqrt(dble(size(x)))
end function

Answer 30

SmileBASIC, 105 bytes (as a function)

I just noticed it's allowed to be a function. Whoops, that reduces my answer dramatically. This defines a function S which takes an array and returns the population standard deviation. Go read the other one for an explanation, but skip the parsing part. I don't want to do it again.

DEF S(L)N=LEN(L)FOR I=0TO N-1U=U+L[I]NEXT
U=1/N*U FOR I=0TO N-1T=T+POW(L[I]-U,2)NEXT RETURN SQR(1/N*T)END

As a program, 212 bytes

Unfortunately, I have to take the input list as a string and parse it myself. This adds over 100 bytes to the answer, so if some input format other than a comma-separated list is allowed I'd be glad to hear it. Also note that because VAL is buggy, having a space before the comma or trailing the string breaks the program. After the comma or at the start of the string is fine.

DIM L[0]LINPUT L$@L I=INSTR(O,L$,",")IF I>-1THEN PUSH L,VAL(MID$(L$,O,I-O))O=I+1GOTO@L ELSE PUSH L,VAL(MID$(L$,O,LEN(L$)-O))
N=LEN(L)FOR I=0TO N-1U=U+L[I]NEXT
U=1/N*U FOR I=0TO N-1T=T+POW(L[I]-U,2)NEXT?SQR(1/N*T)

Ungolfed and explained:

DIM L[0]  'define our array
LINPUT L$ 'grab string from input

'parse list
'could've used something cleaner, like a REPEAT, but this was shorter
@L
I=INSTR(O,L$,",")                 'find next comma
IF I>-1 THEN                      'we have a comma
 PUSH L,VAL(MID$(L$,O,I-O))       'get substring of number, parse & store
 O=I+1                            'set next search location
 GOTO @L                          'go again
ELSE                              'we don't have a comma
 PUSH L,VAL(MID$(L$,O,LEN(L$)-O)) 'eat rest of string, parse & store
ENDIF                             'end

N=LEN(L) 'how many numbers we have

'find U
'sum all of the numbers, mult by 1/N
FOR I=0 TO N-1
 U=U+L[I]
NEXT
U=1/N*U

'calculate our popstdev
'sum(pow(x-u,2))
FOR I=0 TO N-1
 T=T+POW(L[I]-U,2)
NEXT
PRINT SQR(1/N*T) 'sqrt(1/n*sum)