       Re: Difference Fit vs. Correlation

• To: mathgroup at smc.vnet.net
• Subject: [mg96815] Re: Difference Fit vs. Correlation
• From: Bill Rowe <readnews at sbcglobal.net>
• Date: Wed, 25 Feb 2009 04:06:41 -0500 (EST)

```On 2/24/09 at 5:48 AM, clausenator at gmail.com (Claus) wrote:

>Hi, in the code below, using Correlation I get 0.501338, using Fit I
>get 0.514093. How come?

<snip>

>In:= lm = LinearModelFit[GaltonDat, {1, x}, x]

>In:= lm["BestFit"]

>Out= 33.8866 + 0.514093 x

>In:= r2 = Correlation[GaltonX, GaltonY]

>Out= 0.501338

You seem to expect the slope of the best fit line to have the
same value as the correlation between x and y. These are two
separate things and will never have the same value with real data.

The slope of the best fit line is given by

Covariance[x,y]/Variance[y]

The correlation between x and y (Pearson's correlation) is given by

Covariance[x,y]/Sqrt[ Variance[x] Variance[y] ]

The only case where the slope and correlation have the same
value is the trivial case where x = y and there is no error
term, i.e., perfect correlation with a slope of 1.

```

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