# What is a common way to measure the “goodness of fit” of an individual data point to a correlation?

Let's say I have a collection of data points (X & Y values) that show some correlation when, eg, Pearson's correlation formula is applied. What is a good measure for determining which data points are "wild ducks" in the collection?

I would guess that the orthogonal distance from the data point to the line of the computed correlation formula would be a good measure, but I've not found any formula for computing this (and I'm kind of hoping it's a common concept that, eg, Excel knows about, if I can just pin a name on it).

(I realize that this is a bit of a novice question, but I've searched Google and here and not found anything, probably because I'm not up on the lingo. And because "correlation" gets 85 million hits.)

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You could look at my paper in the American Journal of Mathematical and Management Science 1982 "The Influence Function and its Application to Data Validation" or Gnanadesikan's Multivariate Analysis book. Influence functions determine the effect of a point on a parameter estimate. It essentially tells you how much the estimate will change if the particular observation is left out. For Pearson's product moment correction you will find in Gnanadeskians book that for points in the (x,y) plane the contours of constant value for the influence function are hyperbolae. So these curves show you the direction in the 2 dimensional space where the moving in that direction creates a maximum change in the estimate of correlation. For the pair of correlated random variables (X1, Y1) Assume E(X1)=μ1 and E(Y1)=μ2 and Var(X1) = σ1$^2$ and Var(Y1)= σ2$^2$ Let Z1=(X1-μ1)/σ1 and W1 =(Y1-μ2)/σ2. The pair (Z1, W1) has the same correlation ρ as (X1,Y1) now for any point z1=(x1-μ1)/σ1 and w1= =(y1-μ2)/σ2, the influence function for the correlation ρ at (z1,w1) is z1 w1 - ρ(z1$^2$ +w1$^2$)/2. So Let C be a constant then z1 w1 - ρ(z1$^2$ +w1$^2$)/2 = C is a contour of constant influence. The proof of this result was given by Mallows in an unpublished paper in 1976 and is noted by Gnanadesikan in his book. So I think the influence function for the correlation gives you the measure you are looking for.