concentration ellipse

The percentage of bivariate normally distributed data covered by an ellipse whose axes have a length of numberOfSigmas·σ can be obtained by integration of the p.d.f. over an elliptical area. This results in the following equation as can be verified from equation 26.3.21 in Abramowitz & Stegun (1970):

percentage = (1 - exp (-numberOfSigmas2/2))· 100%,

where the numberOfSigmas is the radius of the "ellipse":

(x/σx)2 + (y/σy)2 = numberOfSigmas2.

The numberOfSigmas = 1 ellipse covers 39.3%, the numberOfSigmas = 2 ellipse covers 86.5% and the numberOfSigmas = 3 ellipse covers 98.9% of the data.

From the formula above we can show that if we want to cover p percent of the data, we have to chose numberOfSigmas as:

numberOfSigmas = √(-2 ln(1-p/100)).

For covering 95%of the data we calculate numberOfSigmas = 2.45.

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© djmw, November 13, 2007