Covariance and correlation

[jd]

I am having trouble with a simple statistics problem.  I have all of the numbers right but I am plugging them into or computing the formula wrong.
X          Y
-3        4
2        5
0        3
4       -2
-3       -3
I am coming up with a covariance of -.25 and I know that the correlation is -.022 but I am coming up with .017 Where am I going wrong?

[jd]

I figured it out! Covariance is correct at 0.25 and correlation is -0.022.  I was just reading the problem wrong. Thanks

[Yoni]

How did you do that? Describe in an overview and show the formulas you used please.

[Adron]

Hmm, I'm not getting the same results... Maybe I've forgotten statistics...


x        y
-3        4
2        5
0        3
4      -2
-3      -3

xmean = (-3 + 2 + 0 + 4 - 3) / 5 = 0
ymean = (4 + 5 + 3 - 2 - 3) / 5 = 7/5 = 1.4


xnorm  ynorm
-3        2.6
2         3.6
0         1.6
4         -3.4
-3        -4.4

cov = (-3*2.6 + 2*3.6 + 0*1.6 + 4*-3.4 + -3*-4.4)/5 = -0.2

[jd]

sum of X= 0
sum of Y = 7
sum of XY= -1
Sum of XY= -1
Sum of X squared= 38
sum of Y squared= 63

covariance

s=-1-  (0)(7)/5/4= -1- 0/5/4=-1/4=0.25

correlation coefficient

r= 5(-1)- (0)(7)/square root of [5(38)-0][5(63)-49=
r=-5-0/square root of (190)(266)= -5/sqaure root of 50540=-5/224.811  r= -0.022

sorry it took me so long to post this and sorry if it is a little confusing.  I couldn't paste the statistical notation into the message box.

[Yoni]

That looks interesting. What formulas did you use exactly? And what is the meaning of the covariance and correlation?

[Adron]

IIRC:

average(x) = E(x) = expected value for x

variance(x) = E((x-E(x))^2) = expected (difference of x from its average squared)

covariance(x, y) = E( (x-E(x)) * (y - E(y)) ) = expected (difference of x from its average times difference of y from its average)

y can be another signal/series or just a time-delayed version of x. When you do it with a time-delayed version, you're measuring frequency contents in the signal x.


Edit:

Correlation measures the same thing as covariance but is normalized.

standard deviation, stddev(x) = sqrt(variance(x))

correlation(x, y) = covariance(x, y) / (stddev(x) * stddev(y))

[Adron]

And on another note, Maple agrees with me about the covariance of those sequences:

with(stats):
describe[covariance]([-3,2,0,4,-3], [4,5,3,-2,-3]);

                                -1/5