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 probability distribution function 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% of the data, the numberOfSigmas=2 ellipse 86.5%, and the numberOfSigmas=3 ellipse 98.9%.
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.
© djmw, November 13, 2007