Calculate the correlation coefficient

Mathematica 34 bytes

Here are a few ways to obtain the Pearson product moment correlation. They all produce the same result. From Dr. belisarius: 34 bytes

Dot@@Normalize/@(#-Mean@#&)/@{x,y}

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Built-in Correlation function I: 15 chars

This assumes that x and y are lists corresponding to each variable.

x~Correlation~y

0.76909

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Built-in Correlation function II: 31 chars

This assumes d is a list of ordered pairs.

d[[;;,1]]~Correlation~d[[;;,2]]

0.76909

The use of ;; for All thanks to A Simmons.

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Relying on the Standard Deviation function: 118 115 chars

The correlation can be determined by:

s=StandardDeviation;
m=Mean;
n=Length@d;
x=d[[;;,1]];
y=d[[;;,2]];
Sum[((x[[i]]-m@x)/s@x)((y[[i]]-m@y)/s@y),{i,n}]/(n-1)

0.76909

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Hand-rolled Correlation: 119 chars

Assuming x and y are lists...

s=Sum;n=Length@d;m@p_:=Tr@p/n;
(s[(x[[i]]-m@x)(y[[i]]-m@y),{i,n}]/Sqrt@(s[(x[[i]]-m@x)^2,{i,n}] s[(y[[i]] - m@y)^2,{i,n}]))

0.76909

PHP 144 bytes

<?
for(;fscanf(STDIN,'%f%f',$$n,${-$n});$f+=${-$n++})$e+=$$n;
for(;$$i;$z+=$$i*$a=${-$i++}-=$f/$n,$y+=$a*$a)$x+=$$i*$$i-=$e/$n;
echo$z/sqrt($x*$y);

Takes the input from STDIN, in the format provided in the original post. Result:

0.76909044055492

Using the vector dot product:

where are the input vectors adjusted downwards by and respectively.

Perl 112 bytes

/ /,$e+=$`,$f+=$',@v=($',@v)for@u=<>;
$x+=($_-=$e/$.)*$_,$y+=($;=$f/$.-pop@v)*$;,$z-=$_*$;for@u;
print$z/sqrt$x*$y

0.76909044055492

Same alg, different language. In both cases, new lines have been added for 'readability', and are not required. The only notable difference in length is the first line: the parsing of input.

Q

Assuming builtins are allowed and x,y data are seperate vectors (7 chars):

x cor y

If data are stored as orderded pairs, as indicated by David Carraher, we get (for 12 characters):

{(cor).(+)x}