Regression slope =  Sxy/ Sx (sum of deviations cross product)/ (sum of squares of x deviations)

Sy = sum of squares of y deviations

Ellipse semi axes are:

Major ,minor  = sqrt k  sqrt { Sx + Sy +  sqrt {(Sx+Sy)2 – 4(SxSy-[Sxy]2)}

And k = F /n(n-2) for confidence ellipse at designated  alpha or k = F(n+1)/n(n-2) for tolerance ellipse

n=  no. of paired observations; F from F distribution

Dear Professor James Leigh, Thank you for you answer. Could you please see the figure I have recently added. I need that kind of envelope.

Regards

sorry, I don't follow. please give more explanation. what ellipses are you trying to define?

Hello Professor james Leigh

I have to make elliptical envelops. I have scattered data and I have to make elliptical envelops that enclose 70%, 20%, 9% points. please find example data with this message and probable outcome too.

Regards

 I'm not in this field. where does fig come from?

however my simple suggestion would be  to calculate the statistical tolerance ellipses for each probability (x) by method already given.  These would include x% of data.. then a 1/4 ellipse of this size would include x/4 % of data. Your envelopes are 1/4 ellipses so if you multiplied each semi axis by 2 the area would be 4x  (A = pi x ab, so new area = pi x 2a x 2b = 4 pixab =4A). So the semi axes of your envelope would be twice those calculated by the tolerance ellipse.

Dear Professor James Leigh,

I was able to make the best fitting ellipse, but could not find the different probability ellipses such as 70%, 20% and 9%.

You need the F distribution tables.

the degrees of freedom are 2 and Number of paired points - 2.

for 90% tolerance ellipse you want F 0.10, for 70% F0. 30 etc.

eg for 42 data points for 90% tolerance ellipse, F 2, 40 = 2.44,  for 75% tolerance ellipse F 2,40= 1.44