How do I find the value for the correlation coefficient r?

Question

X 5 1 4 2 5

Y 5 10 12 4 8

Answer 1

Unfortunately, the correlation coefficient is quite complicated to calculate by hand.  Believe it or not, the simplest form of the formula (to substitute into) is the following:

r = [n∑xy - ∑x∑y] ÷ √[n∑x^2 - (∑x)^2][n∑y^2 - (∑y)^2]  <-- everything after the √ is under the square root

where n is the number of data points in the set and ∑ means "the sum of" (the expression immiedately following the symbol).  For example, ∑xy means add up the products of each x-value with its corresponding y-value.  But be careful, note that ∑x^2 is different from (∑x)^2.  I hope that's not too confusing.  If it is, read on.  It might make sense with some actual numbers.

Ok, let's get started by first calculating each ∑ in the formula from left to right:

n = 5 (# of data points in the set)

∑xy = (5 x 5) + (1 x 10) + (4 x 12) + (2 x 4) + (5 x 8) = 131

∑x = 5 + 1 + 4 + 2 + 5 = 17

∑y = 5 + 10 + 12 + 4 + 8 = 39

∑x^2 = (5^2) + (1^2) + (4^2) + (2^2) + (5^2) = 71

∑y^2 = (5^2) + (10^2) + (12^2) + (4^2) + (8^2) = 349

Whew, now we can substitute these values into the formula for r:

r = (n∑xy - ∑x∑y) ÷ √[n∑x^2 - (∑x)^2][n∑y^2 - (∑y)^2]

= [(5)(131) - (17)(39)] ÷ √[(5)(71) - (17)^2][(5)(349) - (39)^2]

= -8 ÷ √[66][224]  <-- remember that everything after √ is under the square root

= -0.065795169

I hope that helps.