Substandard deviation

Question

The mean of a population \$(x_1,\dots,x_n)\$ is defined as \$\bar x=\frac1n\sum_{i=1}^n x_i\$. The (uncorrected) standard deviation of the population is defined as \$\sqrt{\frac1n\sum (x_i-\bar x)^2}\$. It measures how dispersed the population is: a large standard deviation indicates that the values are far apart; a low standard deviation indicates that they are close. If all values are identical, the standard deviation is 0.

Write a program or function which takes as input a (non-empty) list of non-negative integers, and outputs its standard deviation. But check the scoring rule, as this is not code golf!

Input/Output

Input/Output is flexible. Your answer must be accurate to at least 2 decimal places (either rounding or truncating). The input is guaranteed to contain only integers between 0 and 255, and to not be empty.

Scoring

To compute your score, convert your code to integer code points (using ASCII or whatever code page is standard for your language) and compute the standard deviation. Your score is the number of bytes in your code multiplied by the standard deviation. Lower score is better. You should therefore aim for code which at the same time (a) is short and (b) uses characters with close codepoints.

Here is an online calculator to compute your score (assuming you use ASCII).

Test cases

Input              | Output
77 67 77 67        | 5
82                 | 0
73 73 73           | 0
83 116  97 116 115 | 13.336

A word of caution about built-ins: if your language has a built-in, that's fine (and good for you if it only uses one character!). But make sure that it uses \$n\$ and not \$n-1\$ as the denominator in the formula, or else your answer won't be valid.

Answer 1

MATL, score 65.30697

tYmhZs

Try it online! Or verify all test cases.

How it works

The built-in function Zs with its default arity (1 input, 1 output) computes the corrected standard deviation:

\$\sqrt{\frac 1 {n-1}\sum (x_i-\bar x)^2}\$

The uncorrected standard deviation can be obtained with the 2-input version of Zs: 1&Zs, where 1 as second input means uncorrected. l or T could be used instead of 1 to reduce the score, but & is very far from the other characters. 2$ or H$ could be used instead of &, but $ is even farther.

So it is better to use the default version of Zs (corrected standard deviation) on the input with its mean appended. This increases the input length by 1 and contributes 0 in the numerator, which causes the corrected standard deviation to become uncorrected.

t     % Implicit input: numeric vector. Duplicate
Ym    % Mean
h     % Concatenate the input vector with its mean
Zs    % Corrected standard deviation

Answer 2

J, 19 bytes, Score 119.8249

-~1 thanks to Bubbler

Tries to have most characters between 0x23 and 0x2F #$%&'()*+,-./, with : being a bit further away.

(+/%$)&.:*:&(-+/%#)

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How it works

(+/%$)&.:*:&(-+/%#)
            (-+/%#) x - sum divided by length
         *:&        and squared
(+/%$)&.:           mean of that
      &.:*:         reverse square -> square root

Answer 3

Google Sheets, Score 142.6885

=STDEVP(F:F

Google Sheets automatically closes parentheses, and using F as the input column minimizes the standard deviation. This saves one byte over Excel's uncorrected standard deviation, since Excel uses STDEV.P instead of STDEVP

Answer 4

R, 34 bytes 24 bytes, score 789.5923 723.4687 722.6112

sd(c(scan()->J,mean(J)))

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Edit: switched to a shorter formula to calculate the population sd (which I found here), which now only benefits from selecting the best variable name among the golfs outlined below for the previous version.

Edit2: score reduced by 0.8575 thanks to Robin Ryder

The (previous) ungolfed code is was: x=scan();sqrt(mean((x-mean(x))^2)) (which would have a score of 1104.484)

From this, sequential score-improving golfs are:

• x=scan();`?`=mean;sqrt(?(x-?x)^2) = re-define mean() as a single character unary operator (score 983.8933)
• x=scan();`?`=mean;(?(x-?x)^2)^.5 = exchange sqrt() for ()^.5 (score 918.6686)
• H=scan();`?`=mean;(?(H-?H)^2)^.5 = exchange x for H which is the closest codepoint value to the mean of the program, thereby reducing the standard deviation (score 801.4687)
• I=scan();`?`=mean;I=I-?I;(?I^2)^.5 = first calculate x-mean(x) separately, to reduce number of parentheses (which are at the far end of the ASCII range, and so increase the standard deviation), and re-adjust the variable name to I. Although this increases the code length by 2 characters, it reduces the score to 789.5923.

---

R + multicon, 15 bytes, score 273.5032

multicon::popsd

Trivial solution using built-in popsd function from multicon library.
Not installed at TIO, but you can try it at rdrr.io by copy-pasting this code:

x=c(67,77,67,77)  # data
multicon::popsd(x)

Answer 5

Wolfram Language (Mathematica), Score 537.0884

A@((#-A@#)^2)^.5&;A=Mean

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@att saved 17.6142 points

Answer 6

Python 3, Score 680.5175

Where the golfiest solution is not the best. I doubt any non-builtin could be better but I might be wrong.

import statistics;statistics.pstdev

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Python 3, Score 733.6818

from statistics import*;pstdev

Python 3, Score 798.5587

__import__('statistics').pstdev

Answer 7

05AB1E, Score: 531.168 431.516 360.278 (10 15 14 bytes)

Osg/nsn-Osg/(t

Uses the 05AB1E coding page. The characters used have the codepoints [79,73,103,47,110,73,110,45,68,79,73,103,47,40,116].

Try it online or verify all test cases.

Explanation:

                # Get the arithmetic mean of the (implicit) input-list by:
O               #  Summing the (implicit) input-list
 I              #  Push input-list again
  g             #  Pop and push its length
   /            #  Divide the sum by this length
                #  (which gives a better score than the builtin `ÅA`)
    n           # Square it
     I          # Push the input again (better score than `s` or `¹`)
      n         # Square each value in the input as well
       -        # Subtract each from the squared mean
                # Take the arithmetic mean of that list again by:
        O       #  Summing it
         Ig     #  Push the input-list again, and pop and push its length
           /    #  Divide the sum by this length
            (   # Negate it
             t  # And take its square-root
                # (after which the result is output implicitly)

Answer 8

JavaScript (ES7), Score  1359 1228  1156.077

Saved 72 points thanks to @edc65

D=>D[F='map'](C=>B-=(C+E/A)**2/A,D[F](C=>E+=--A?C:9,A=B=E=0))&&B**.5

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Character breakdown

 char. | code | count
-------+------+-------
   0   |  48  |   1
   2   |  50  |   1
   5   |  53  |   1
   9   |  57  |   1
   &   |  38  |   2
   '   |  39  |   2
   (   |  40  |   3
   )   |  41  |   3
   *   |  42  |   4
   +   |  43  |   2
   ,   |  44  |   2
   -   |  45  |   3
   .   |  46  |   1
   /   |  47  |   2
   :   |  58  |   1   <-- mean ≈ 59.43
   =   |  61  |   9
   >   |  62  |   3
   ?   |  63  |   1
   A   |  65  |   4
   B   |  66  |   3
   C   |  67  |   4
   D   |  68  |   3
   E   |  69  |   3
   F   |  70  |   2
   [   |  91  |   2
   ]   |  93  |   2
   a   |  97  |   1
   m   | 109  |   1
   p   | 112  |   1

Answer 9

Arn, score = 925.3172 655.6836 602.7985 123.2274

sdev:s

Uses the builtin standard deviation function. Go to the old answer for a more interesting one

Old Answer

I don't compress it because Standard Deviation would be way higher. I have updated this answer, since I found a much shorter method (sitting at 14 bytes). Link here (this is the program the score refers to). I will leave the original program for posterity's sake

:/(+v{:*v-(:s.mean}\)/((:s)#

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Explained

$$\large\sqrt {\frac1n \sum(x_i-\bar x)^2}$$ Just made use of the formula. :/ is the sqrt prefix, :* is the square prefix, +v{:*v-(:s.mean}\ Folds with + (addition) after mapping with the block v{:*v-(:s.mean}. v is current entry, :s splits on space (no variable is provided, so it assumes the variable _, which is STDIN). Then it just divides by the length (# suffix).

Answer 10

Io, score = 1454.7164672196433912

-19.58295474318379 thanks to @ManishKundu

method(:,:map(Z,(Z- :average)squared)average sqrt)

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Answer 11

Jelly, (14 bytes), score 218.314

^{(218.31399405443526)}

+/÷LN+*2+/÷L*.

Try it online! Or see a self-evaluation.

Bytecode: 2b 2f 1c 4c 4e 2b 2a 32 2b 2f 1c 4c 2a 2e

How?

A naive program would be _Æm²Æm½for 348.47 (subtract the mean from each, square each, take the mean of that and then square root it).

We know that to get rid of the two byte monad Æm whose code-points are quite far apart (0x0d and 0x6d) we need to either:

• divide using ÷ (0x1c), or
• multiply, × (0x11), and invert, İ (0xc6)

But the latter bytes are also fairly far apart, so this answer attempts to use bytes close to ÷ (0x1c).

+/÷LN+*2+/÷L*. - Link: list of numbers, A
 /             - reduce (A) by:
+              -   addition             -> sum(A)
   L           - length (A)
  ÷            - divide                 -> mean(A)
    N          - negate
     +         - add (to A, vectorised) -> [mean(A)-v for v in A]
       2       - two
      *        - exponentiate           -> [(mean(A)-v)² for v in A]
         /     - reduce by:
        +      -   addition             -> sum((mean(A)-v)² for v in A)
           L   - length (A)
          ÷    - divide                 -> sum((mean(A)-v)² for v in A)/n
             . - a half
            *  - exponentiate           -> √(sum((mean(A)-v)² for v in A)/n)

Answer 12

Wolfram Language (Mathematica), 31 bytes, score 478.3451

a[a_]=RootMeanSquare[a-Mean[a]]

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Answer 13

Charcoal, 15 bytes, stddev 46.741654, score 701.12481

Ｉ₂∕ΣＸ⁻θ∕ΣθＬθ²Ｌθ

Try it online! Link is to verbose version of code. Link test case is the byte values in the Charcoal code page of the code. Explanation:

         θ      Input `x`
        Σ       Summed
       ∕  Ｌθ    Divided by `n`
     ⁻θ         Vectorised subtracted from `x`
    Ｘ       ²   Squared
   Σ            Summed
  ∕          Ｌθ Divided by `n`
 ₂              Square rooted
Ｉ               Cast to string
                Implicitly printed

Note that the alternative formula for the standard deviation, \$ \sqrt{\bar{x^2}-\bar x^2} \$, while having a slightly smaller standard deviation, takes 17 bytes, and therefore results in a higher score of 755.6.

Answer 14

Setanta, score: 2728.508

gniomh(g){f:=0h:=0e:=fad@g le i idir(0,e){d:=g[i]f+=d h+=d*d}toradh freamh@mata((h-f*f/e)/e)}

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Answer 15

C (gcc), ^{107 104} 99 bytes, stddev ^{25.25 \$\cdots\$25.32} 25.00, score ^{2702.01 \$\cdots\$ 2634.27} 2475.426270

Saved 3 bytes and 46.95288 points thanks to ceilingcat!!!
Saved 5 bytes and 158.848632 points thanks to att!!!

E;float D,G,H;float F(F,C)int*C;{E=F;for(H=G=0;E>-F;0>E?G+=D*D:(H+=*C++))D=H/F-C[--E];G=sqrt(G/F);}

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Answer 16

MATLAB/Octave, 12 bytes, score 336.32

(changed according to guidance by Giuseppe to conform with rules)

@(A)std(A,1)

Argument with name A provides the lowest deviation for score, output to standard output variable Ans and actually written to command window.
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std is a built-in function. By default it uses \$N-1\$ as demoninator but by passing 1 as second argument it's changed to \$N\$.