
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 (numberOfSigmas^{2}/2)) · 100%, 
where the numberOfSigmas is the radius of the "ellipse":
(x/σ_{x})^{2} + (y/σ_{y})^{2} = numberOfSigmas^{2}. 
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(1p/100)). 
For covering 95% of the data we calculate numberOfSigmas = 2.45.
© djmw, November 13, 2007