Correlation Coefficient of Words

Question

This is going to be relatively quick...

Challenge

Given a lower-case word as input, calculate the Pearson's Correlation Coefficient between the letter position of the word (nth letter within the word, x), and the letter position within the alphabet (nth letter of the alphabet, y).

Background

Pearson's Correlation Coefficient is worked out for data sets x and y, as:

where x and y are the respective data sets as explained above (and further below), and n is the length of the string.

Examples

  Input: forty
  Output: 0.964406804

Explanation:

  Letters: f o r t y
  Alphabet Number (y): 6 15 18 20 25
  Letter Position (x): 1 2 3 4 5
  Length of word (n): 5

Correlation coefficient is worked out from the above formula.

  Input: bells
  Output: 0.971793199

Explanation:

  Letters: b e l l s
  Alphabet Number (y): 2 5 12 12 19
  Letter Position (x): 1 2 3 4 5
  Length of word (n): 5

Correlation coefficient is once again worked out from the above formula.

Rules

1. Input must provide the word as a lower-case alphabetic string with no spaces (assumed input). The delivery of this string is entirely up to you (file, STDIN, function, array, carrier pigeon, etc.)
2. Output is in the form of a numeric that will, by the formula, provide r, such that -1 <= r <= 1. Delivery of the output, once again, is entirely up to you.
3. No silly loop holes
4. This is not code-golf, as I'm sure there will be many languages that may have native statistical functions. Instead, this will be a popularity contest, so most votes wins here, so make your entry count... I'll be throwing in a bounty for the most creative answer that attracts my attention a little later down the line...

Any questions/comments?

Answer 1

R, 63 bytes

s=strsplit(scan(,""),"")[[1]];cor(match(s,letters),1:length(s))

Ungolfed :

#Takes the input and splits it, letters by letters
s=strsplit(
           scan(,""), 
           "")[[1]];

#Finds the positions of each letters in the lowercase alphabet, and correlates them with
#their position in the word
cor(
    match(s,letters),
    1:length(s)
    )

Answer 2

Python

That's a lot of traversals, let's do all we need to do in one pass for efficiency.

Since y will be [1,2,...,len(x)] we can calculate it's sum and sum of squares using the formulas for triangle and square pyramid numbers respectively.

...and there's really no need for two square roots since the two are multiplied we can just square the product.

def p(l):
    y = sx = sxy = sx2 = 0
    for c in l:
        y += 1
        x = ord(c) - 96
        sx += x
        sxy += x * y
        sx2 += x * x
    sy = y * (y + 1) / 2
    sy2 = sy * (2 * y + 1) / 3
    return (n * sxy - sx * sy ) / ((n * sx2 - sx * sx) * (n * sy2 - sy * sy)) ** .5

Answer 3

Python 3

Simply uses the formula in the challenge. Doesn't use any libraries like numpy and friends.

from math import sqrt

def correlation_coefficient(word):
  alphabet = 'abcdefghijklmnopqrstuvwxyz'

  x = list(range(len(word)))
  y = list(map(alphabet.index, word))
  n = len(word)

  top = n * sum(a*b for a, b in zip(x, y)) - sum(x) * sum(y)
  bottom = (sqrt(n * sum(a ** 2 for a in x) - sum(x) ** 2)
          * sqrt(n * sum(a ** 2 for a in y) - sum(y) ** 2))

  r = top / bottom
  return r

Ideone it!