I realize that the depth of field discussion was a couple of pages back, and also that (IMO) it is an old chestnut where people argue over different answers because they are holding different things constant. However, I made a figure to illustrate what I feel is a basic point, the size of the background blur relative to the foreground subject. (Because for pictorial purposes, frequently one has a foreground subject and either wishes background objects to be in focus, or to be blurred out.)
Roughly, the diameter of the background blur is proportional to the physical diameter of the entrance aperture. This illustration shows why: light from a background point source is a cone of light entering the lens aperture, and when it passes through the foreground plane of sharp focus, it is not a point but a circle of some diameter, so it shows up in the image as a spread-out circle. This blurred image is indicated by a red bar in the drawing. The blur circle is the same for a 50mm f/2 lens and a 100mm f/4 lens, for example.
View attachment 349722
Now, if you use a 50mm lens and a 100mm lens on the same camera, at the same position, you'll get different fields of view. If you back up with the 100mm lens to get a similar field of view, you'll change the perspective and relative sizes of subject to background.
If you use a 50mm lens on a small format and a 100mm lens on a 2x larger format, with the two cameras in the same place, you'll get the same field of view and the same perspective, keeping the same composition. But to get the same amount of background blur, you now need different f-stops, like 50/2 and 100/4. If you used the same f-number, like 50/4 and 100/4, the smaller format camera will have less blur. This is the experiment that Cholentpot and wiltw did (except that they used a multiplier of about 1.5x since APS-C is about 1.5x smaller than full frame).
Does this matter? Sometimes. For ex, if you have an APS-C DSLR and the kit lens, something like 18-55/3.5-5.6, your "normal" lens is about 35/4.5 or so (8mm aperture diameter), and you can never get the amount of blur that someone could with a full frame SLR and a humble 50/2 lens (25mm aperture diameter). If you went out and added the 35/1.8 APS-C lens (19mm aperture diameter) you are getting close. These are "normal" focal length lenses, so if you wanted to take portraits you might want a short telephoto, but the same rules apply.
In the film world, it is often said that medium (or large) format has less depth of field than 35mm, for the same reason. If you are trying to get subject and background in focus, holding the composition equal, you'll have to stop down more with the larger format.