DOF comparison between two lenses

What your friend told you is a way to estimate background blur, not depth of field. If you look at the DOF formula, you'll see it's inversely proportional to f²/N, not f/N. This corresponds with your statement, "focal length has an higher impact on DOF than the F-stop".

DOF = 2 u² N C / f²

N = aperture F-number
C = circle of confusion
u = distance to subject
f = focal length

Depth of Field (DOF), background blur, and bokeh are related, but different concepts. There is also subject-background isolation/separation.

• Depth of field is based on focal length, aperture, distance, and a predefined acceptable sharpness level. It is concerned with what parts of the image are expected to be sharp, not what parts of the image will be blurry, or how blurry unsharp portions of the image will be.

• Background blur – How blurry is the background? I think of it as something that can be quantified. How big are bokeh balls a given distance from the lens? Different lenses with the same focal lengths, apertures, and distances can create different, though similar, amounts of blur because of different amounts of distortion, aberration, and field curvature. (There's also foreground blur, but people tend to be less interested.)

• Bokeh is a qualitative description of the blur that is produced. Are the bokeh balls round? Do they vary in shape throughout the frame? Are they smooth? Do they have edge highlights? Are they smeared? Some people refer to how lenses "render" images.

• Subject-background isolation refers to (subjectively) how well the subject stands out from the background. This can be achieved with depth of field and background blur, as well as appropriate lighting (such as rim lighting and creative use of "glow"). The common formula is to try to use narrow depth of field with high background blur. However, some types of bokeh can achieve good subject isolation with high depth of field and low background blur. For instance, Sonnar lenses create bokeh balls with an edge highlight toward the center of the frame, but a smear toward the edge of the frame. This tends to emphasize the sharpness of the subject toward the center, while also emphasizing the blurriness of the background toward the periphery.

See also:

• What affects DOF more...distance to subject or focal length?
• How can I maximize the "blurry background, sharp subject" (bokeh) effect?

If you change the aperture/F# by a factor of 2 (2x, 1/2) it changes the DOF by a factor of 2 as well. But if you change the FL or the subject/focus distance by a factor of 2 it changes the DOF by a factor of 4... they have 2x the effect that aperture does, only in opposite directions from each other.

"So what (correct) statement can be made to compare the DOF of two lenses (with adjusted focus distances) based on the size of the entrance pupil?"

In your example, if you double the FL (1/4 DOF), and simultaneously increase the subject distance by 2x (4x DOF), they cancel each other; that leaves you with only the 2x F# remaining (longer lens, same dia entrance pupil), which results in 2x the DOF. So the answer is the difference will be the difference in the F#'s; and 2 stops is 2x (or 1/2) the DOF... (it's actually 1.4x/stop; 1.96x for 2 stops).

However, DOF isn't the only factor in rendering BG blur ("nicer backgrounds"); it's actually less effective than controlling BG selection/distance... When you double the distance to the subject, in order to negate the 2x increased magnification of the subject, you do not necessarily double the distance to the BG; and therefore you do not fully negate the increased magnification at that greater distance.

E.g. 100mm f/2 with subject distance 10ft and background is 30ft behind (40ft from camera). Switch to 200mm f/4 and 20ft subject distance which results in same subject size/magnification. But the BG is now only at 50ft; not the 80ft required to negate the increased magnification at that distance... you've only negated 25% of the increased magnification (10ft of the 40ft required). So less of the BG is included and what is included is larger; which generally results in less busy/smoother backgrounds which are more pleasing and provide more separation; even if there is more DOF at the focal plane and less blur radius in the BG.