Let's go for a "purely" mathematical reasoning.
Let's assume that the eye has a FOV = afov.
With ho an object height and hs the sensor height, d the distance to the object and f the focal length, you have the following schema :
Finding a relation between all those values is straightforward : hs / f = tan(0.5* afov)
Unless I'm mistaken, you "just" have to settle for an estimation of the eye FOV to get your ratio.
Now, to get an estimation of an human's eye FOV, you can visit Biology SE, it says Monocular field of view (measured from central fixation) is 160 deg (width w) x 175 deg (height h), so afov = 175 deg.
This answer from mattdm might help too : https://photo.stackexchange.com/a/5924/26456
=> For a human eye, the angle of view happens to be about 95°, but since your eyes move around unconsciously and your brain fills in the details, it feels much wider than that.
You can probably find other sources (and other values).
I let you compute the tangent value :)